· iethods of Mathemat11cal ·•·hysics
· olu
1
METHODS OF
MATHEMATICAL PHYSICS
By R. COURANT and D. HILBERT
First English edition
Translated and Revised Jrom the German Original
VOLUME I
INTERSCIENCE PUBLISHERS , INC., NEW YORK
a division of J ohn Wiley & Sons, New York · London • Sydney
Copyright 1937 by Julius Springer, Berlin
vested in the Attorney General
of the United States, pursuant to law.
Manufactured and sold under License A-1556
Copyright 1953 by Interscience Publishers, Inc.
Library of Congress Catalog Card Number 53-7164
ISBN 0 470 17952 X
20 19 18 17 16 15 14
b 5~l 4-~~<+
P ESTON
POLYTECH
~ IC
(... 58 fJ5J
5?
9>'
6
PRI N'J 'EO IN THE UN ITED STATES OF AMERICA
PREFACE
The first German edition of this volume was published by Julius
Springer, Berlin, in 1924. A second edition, revised and improved
with the help of K. 0. Friedrichs, R. Luneburg, F. Rellich, and other
unselfish friends, followed in 1930. The second volume appeare d in
1938. In the meantim e I had been forced to leave German y and was
fortuna te and grateful to be given the opportunities open in the
United States. During the Second World War the German book became unavailable and later was even suppressed by the Nationa l
Socialist rulers of Germany. Thus the survival of the book was
secu;ed when the United States Govern ment seized the copyright
and licensed a reprint issued by Interscience Publishers, New York.
Such a license also had to be obtained from the Alien Propert y Custodian for the present English edition.
This edition follows the German original fairly closely but contains
a large number of addition s and modifications. I have had to postpone a plan to completely rewrite and modernize the book in collaboration with K. 0. Friedrichs, because the pressure for publication of
an English "Couran t-Hilbe rt" has become irresistible. Even so, it
is hoped that the work in its present form will be useful to mathematicians and physicists alike, as the numerous demands from all
sides seem to indicate.
The objective of the book can still today be e>.'Pressed almost as
in the preface to the first German edition. "Since the seventeenth
century , physical intuition has served as a vital source for mathematical problems and methods. Recent trends and fashions have,
however, weakened the connection between mathem atics and physics;
mathem aticians, turning away from the roots of mathem atics in
intuitio n, have concentrated on refinement and emphasized the postula.tional side of mathem atics, and at times have overlooked the unity
of their science with physics a.nd other fields. In many cases, physicists have ceased to appreciate the attitude s of mathematicians.
Trus rift is unquestionably a serious threat to science as a whole; the
broad stream of scientific development may split into smaller and
v
Vl
PREFACJ,;
smaller rivulets and dry out. It seems therefore important to direct
our efforts toward reuniting divergent trends by clarifying the common features and interconnections of many distinct and diverse
scientific facts. Only thus can the student attain some mastery of
the material and the basis be prepared for further organic development of research.
"The present work is designed to serve this purpose for the field of
mathematical physics. Mathematical methods originating in problems of physics are developed and the attempt is made to shape results into unified mathematical theories. Completeness is not attempted, but it is hoped that access to a rich and important field
will be facilitat.ed by the book.
"The responsibility for the present book rests with me. Yet the
name of my teacher, colleague, and friend, D. Hilbert, on the title
page seems justified by the fact that much material from Hilbert's
papers and lectures has been used, as well as by the hope that the
book expresses some of HilberL's spirit, which bas had such a decisive
influence on mathematical research and education."
I am greatly indebted to many helpers in all phases of the task of
preparing this edition : to Pet.er Ceike, Ernest Courant, and Anneli
Lax, who provided most of the first draft of the translation; to Hanan
Rubin and Herbert Kranzer, who have given constructive criticism;
to Wilhelm Magnus, who is responsible for the appendix to Chapter
VII; and to Natascha Artin and Lucile Gardner, who carried the
burden of the editorial work. Most cordial thanks also are due to
Interscience Publishers for their patient and helpful attitude and to
my old friend and publisher, Dr. Ferdinand Springer in Heidelberg,
the great pioneer of modern scientific publishing, for his sympathetic
understanding of the situation, which has so greatly changed since the
old days of our close cooperation.
R. CouRA!\T
New Rochelle, New York
June 196S
CONTENTS
I.
The Algebra or Linear Transformations and Quadratic Forms . . . .
1
§1. Linear equations and linear transformations ........... . ... .. ..
1
1. Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
2. Orthogonal systems of vectors. Completeness . . . . . . . . . . . . . . 3
3. Linear transformations. Matrices.... ... .. . . ...... . ....... . 5
4. Bilinear, quadratic, and Hermitian forms.. ...... . .... .. . . . 11
5. Orthogonal and unitary transformations.... . .............. 14
§2. Linear transformations with a linear parameter . . . . . . . . . . . . . . . 17
§3 . Transformation to principal axes of quadratic and Hermitian
forms .................... . ... . ................ . ............ 23
1. Transformation to principal axes on the basis of a maximum
principle . . . . . . . . . . ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2. Eigenvalues.... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3. Generalization to Hermitian forms . . . . . . . . . . . . . . . . . . . . . . . . 28
4. Inertial theorem for quadratic forms . . . . . . . . . . . . . . . . . . . . . . 28
5. Representation of the resolvent of a form . . . . . . . . . . . . . . . . . 29
6. Solution of systems of linear equations associated with
forms .. . . . .. . .... . . .. . . . .. ...................... . ....... 30
§4. Minimum-maximum property of eigenvalues .... . . . . . . . . . . . . . . . 31
1. Characterization of eigenvalues by a minimum-maximum
problem ... . . ... ..... .... . .. . .. . .. . .... ................. 31
2. Applications. Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
§5. Supplement and problems... . ... . .. . ....... . . ... ............. . 34
1. Linear independence and the Gram determinant ........ . .. 34
2. Hadamard's inequality for determinants............. . ..... 36
3. Generalized treatment llf canonical transformations ........ 37
4. Bilinear and quadratic forms of infinitely many variables .. 41
5. Infinitesimal linear transformations . .'.................... . 41
6. P erturbations... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
7. Constraints.. . ... . ............. . .................. . ....... 44
8. Elementary divisors of a matrix or a bilinear form ......... 45
9. Spectrum of a unitary matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
J
II.
Series Expansions or Arbitrary Functions . . . . . . . . . . . . . . . . . . . . . . . . . 48
§1. Orthogonal systems of fun ctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1. Definitions. ............. .. . . ... . ...........................
2. Orthogonalization of functions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3. Bessel's inequality. Completeness relation . Approximation
in the mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
VII
49
49
50
51
viii
CONTENTS
4. Orthogonal and unitary t ransformations with infinitely
many variables ................................ ... . .... .
5. Validity of the results for several independent variables.
More general assumptions........ . . . . . .. . . . .............
6. Construction of complete systems of functions of several
variables ...... ............................ ....... . ... .
§2. The accumulation principle for functions ......... ..... . ...... .
1. Convergence in function space ....... ... ..... ..... . ...... .
§3. Measure of independence and dimension number ... . . .. . . ... . .
1. Measure of independence . .... .... . ..... ... ..... ..... . ... .
2. Asymptotic dimension of a sequence of functions .. ....... .
§4. Weierstrass's approximation theorem. Completeness of powers
and of trigonometric functions .. .......... . . ...... ......... .
1. Weierstrass's approximation theorem ......... .. ...... .... .
2. 1i:xtension to functions of several variables .. . . . . .. . .... .. .
3. Simultaneous approximation of derivatives ........ .. .. .. . .
4. Completeness of the trigonometric func tions ......... . . . . .
§5. Fourier series ............. . ... ..... . ....... ..... ... ...... .... .
1. Proof of the fundamental theorem ...... . ... . . . .......... .
2. Multiple Fourier series .. ... ... . .............. .. . ..... . .. . .
3. Order of magnitude of Fourier coefficients ..... . ........ .. .
4. Change in length of basic interval. ... . . . . . . . . . . . . . . . . . . . . .
5. Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
§6. The Fourier integral ... . .......................... . ..... ..... .
1. The fundamental theorem .. ... .................. . ....... .
2. Extension of the result to several variables . . . . . . . . . . . . . . .
3. Reciprocity formulas ........... . . . . .. . . . . . . . . . . . . . . . . . . . . .
§7. Examples of Fourier integrals .................. . ..............
§8. Legendre polynomials. ........................... . . . . . . . . . . .
1. Construction of the Legendre polynomials by orthogonalizaof the powers 1, x, x 2 , • • • • • • . • • • • • • • • • . • • • . . . . . . . . . . . . . .
2. The generating function ............ . . . . . ..... . . . . . . . . . . . . .
3. Other properties of the Legendre polynomin.IR ............ .
(a) Recursion formula
(b) Differential equation
(c) Minimum property
.
~9. Examples of other orthogonal systems ........ .............. . .
1. Generalization of the problem leading to Legendre polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2. Tchebycheff polynomials. ......... .... .. . ........ .. ...... .
3. Jacobi polynomials... .... .. . ... ..... . .. ... .. . .. . ... ... ....
4. Hermite polynomials ... . ................. ... .. ......... . ..
5. Laguerre polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6. Completeness of the Laguerre and Hermite func tions.
§10. Supplement and problems ............................... . ... . .
1. Hurwitz's solution of the isoperimetric problem .... . .
2. Reciprocity formulas .. . ...... ........................ . .. . .
3. The Fourier integral and convergence in the mean ..... ... .
55
56
56
57
57
61
61
63
65
65
68
68
68
69
69
73
74
74
74
77
77
79
80
81
82
82
R'i
~7
87
88
90
91
93
95
97
97
98
98
CONTE NTS
4.
5.
6.
7.
8.
9.
10.
11.
Spectral decomposition by Fourier series and integrals.. . ..
Dense systems of functions ............... . . . .. : ...........
A Theorem of H. Miintz on the completeness of powers. .. .
Fejer's summation theorem .... . ................. .........
The Mellin inversion formulas ................. ............
The Gibbs phenomen0n ................. .... . . ....... .....
A theorem on Gram's determinant ............. .... ..... ..
Application of the Lebesgue integral ...... .. ......... . ....
lX
99
100
102
102
103
105
107
108
References ................ . ................. ................. ...... 111
Ill.
'·
Linear Integral Equations ... ... ... ........ .... . . . . ... ....... .. ... 112
§1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
1. Notation and basic concepts ................. .............. 112
2. Functions in integral representation ..... .. . ... ........... . 113
3. Degenerate kernels ..... .... ... .............. ..... ... ... ... 114
§2. Fredholm's theorems for degenerate kernels . ...... .. . ......... 115
§3. Fredholm's theorems for arbitrary kernels ................. .... 118
§4. Symmetric kernels and their eigenvalues . . . . . . . . . . . . . . . . . . . . . . 122
1. Existence of an eigenvalue of a symmetric kernel. . . . . . . . . . 122
2. The totality of eigenfunctions and eigenvalues ............. 126
3. Maximum-minim um property of eigenvalues ............... 132
§5. The expansion theorem and its applications ............... . . .. 134
1. Expansion theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
2. Solution of the inhomogeneous linear integral equation . . . . 136
3. Bilinear formula for iterated kernels. . . . . . . . . . . . . . . . . . . . . . 137
4. Mercer 's theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
§6. Neumann series and the reciprocal kernel .... ........ ......... 140
§7. The Fredholm formulas ................. . . .... . ............... 142
§8. Another derivation of the theory .... . . . . . . . . . . . . . . . . . . . . . . . . . . 147
1. A lemma..... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
2. Eigenfunctions of a symmetric kernel . . . . . . . . . . . . . . . . . . . . . 148
3. Unsymmetric kernels .... ........ .. ................. ..... .. 150
4. Continuous dependence of eigenvalues and eigenfunctions on
the kernel .... .. . . ................. ... ....... ·. . . . . . . . . . . 151
§9. Exten!?ions of the theory . .. .' ................. ................. 152
§10. Supplement and problems for Chapter III ................. .... 153
1. Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
2. Singular integral equations ... . . . ........ ....... ...... ..... 154
3. E. Schmidt's derivation of the Fredholm theorems ......... 155
4. E nskog's method for solving symmetric integral equations. 156
5. Kellogg's method for the determination of eigenfunctions.. 156
6. Symbolic functions of a kernel and their eigenvalues ....... 157
7. Example of an unsymmetric kernel without null solutions .. 157
8. Volterra integral equation ................. ................ 158
9. Abel's integral equation ..... ..... .......... . ... ......... .. 158
10. Adjoint orthogonal systems belonging to an unsymmetric
kernel ....... . ................. ................. . .... ... 159
11. Integral equations of the first kind ........ ...... .......... 159
CONTENTS
x
12.
13.
14.
15.
16.
Method of infiniLely many variables ................... . ...
Minimum properties of eigenfunctions .....................
Polar integral equations ..... . . ... ..... .... ... .......... . . .
Symmetrizable kernels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Determination of the resolvent kernel by functional equations ....... ................. . ...... ..... ...............
17. Continuity of definite kernels . . . . ....... ................. .
18. Hammerstein's theorem ......... . . .... ... .. ........ . .....
160
161
161
161
162
162
162
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
IV.
The Calculus of Variations .... ...... . ........ . ........... . ...... . 164
§1. Problems of the calculus of variations . . .. . .............. ..... .
l. Maxima and minima of functions .... .. . . .... . . ....... . .. . .
2. Functionals .................. .... ................ . ...... . .
3. Typical problems of the calculus of variations ..... ....... .
4. Characteristic difficulties of the calculus of variations .... .
§2. Direct solutions . .... . . . ...... . .......... .. ..... . .... ... ..... .
1. The isoperimetric problem .. . . .. .. ..... ... ... . . ........ . . .
2. The Rayleigh-Ritz method. Minimizing sequences ... ... .. .
3. Other direct methods. Method of finite differences. Infinitely many variables . ........... . ............ .... .... .
4. General remarks on direct methods of the cal1Julus of variations .............................................. .... .
§3. The Euler equations . . ..... . . . ....... ....... ... . ...... ....... .
1. "Simplest problem" of the variational calculus ...... . .... .
2. Several unknown functions ......... ..... ... . . ............ .
3. Higher derivatives ................... . ........... . ....... . .
4. Several independent variables ........... ........ . ........ .
5. Identical vanishing of the Euler differential expression .. . .
6. Euler equations in homogeneous form . ......... .......... .
7. Relaxing of conditions. Theorems of du Bois-Reymond and
Haar .. .. ..... ....... . ... ........................ .. .... .
8. Variational problems and functional equations .... .. ...... .
§4. Integration of the Euler differential equation . ... .... . . ...... .
§5. Boundary conditions .. .. ... ......... ..... . .... .... .. . ........ .
·
1. Natural boundary conditions for free boundaries . .. ...... .
2. Geometrical problems. Transversality ............... . . . . . ,
§6. The second variation and the Legendre condition ... ... ...... .
§7. Variational problems with subsidiary conditions .. .. ... ....... .
1. Isoperimetric problems ............................ . ...... .
2. Finite subsidiary conditions .. . ....... .. .......... .. . .. .. . .
3. Differential equations as subsidiary conditions ...... ... . .. .
§8. Invariant character of the Euler equations ............. ...... .
1. The Euler expression as a gradient in function space. Invariance of the Euler expression. . . . . . . . . . . . . . . . . . . . . . . . .
2. Transformation of 6.u. Spherical coordinates . .... . . .. . . . ...
3. Ellipsoidal coordinates .. . . . . ... . . .... .. . ... ... . ... ..... .. .
164
164
167
169
173
174
174
175
176
182
183
184
187
190
191
193
196
199
205
206
208
208
211
214
216
216
219
221
222
222
224
226
CONTEN TS
xi
§9. Transfor mation of variation al problems lo canonica l :ind involutory form . . .... . . . ......... ......... .... . ......... ..... 231
l. Transfor mation of an ordinary minimum problem with subsidiary condition s ....... . ......... ......... ...... . ...... 231
2. lnvoluto ry transform ation of the simplest variati onal problems . . ... .... . . . ... . .... . .. .... . . . ...... .. . .. . . ... ... ... 233
3. Transfor mation of variation al problems to canonica l form .. 238
4. Generali zations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
§10. Variation al calculus and the different ial equation s of mathematical physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
1. General remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
2. The vibrating string and the vibrating rod ......... ....... 244
3. Membra ne and plate ... .. ..... .. ........ ... ......... . . ... . 246
§11. Reciproc al quadrati c variation al problems ......... ....... . ... 252
§12. Supplem entary remarks and exercises . ... ......... ......... ... 257
1. Variatio nal problem for a given different ial equation ...... 257
2. Reciproc ity for isoperim etric problems ..... . ......... . .... 258
3. Circular light rays . .. . . . . ......... ..... . . . . . ... . ..... . .... 258
4. The problem of Dido .... . ......... . . ... . . . ... . . . ......... . 258
5. Example s of problems in space ......... .. .. .... . ......... . 258
6. The indicatri x and applications. . . . . . . . ......... ......... 258
7. Variable domains . . . . ... . ......... . ....... . ......... ... ... 260
8. E. Noether' s theorem on invarian t variation a l problems .
Integrals in particle mechani cs ......... ......... ....... 262
. ... '. ....... 266
9. Transver sality for multiple integrals . . . . .
10. Euler's different ial expressio ns on s urfaces.. ... . . . . . . . . . . 267
11. Thomson 's principle in electrost atics ......... ......... .... 267
12. Equilibr ium problems for elastic bodies . Cast igliano's principle ...... .. ......... ...... . ......... ......... ......... . 268
13. The variation al problem of buckling ......... ......... ..... 272
References . . .. ... .. . . . . ... . ......... ... . ......... ......... ... . ..... 274
V.
Vibration and Eigenval ue Problem s . . . . . . . . . . . . . . .... . ......... . 275
275
§1. Prelimin ary remarks about linear difTerentia.1 equation s .......
l. Principle of superpos ition ......... ......... ......... ...... 275
2. Homogen eous and nonhomo geneous problems . Boundary
condition s ......... .... . .... .. ......... ......... ........ 277
3. Formal relations . Adjoint different ial expressio ns . Green's
formulas .. . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . .. . . . . 277
4. Linear function al equation s as limiting cases and analogues
of systems of linear equation s ......... ... . ..... . ... . .... 280
§2. Systems or a finite number of degrees of freedom ......... ..... 281
1. Kormal modes of vibration . Normal coordina tes. General
theory of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
2. General propertie s of vibrating systems . .. ......... ....... 285
§3. The vibrating string ......... ..... . ... . . . ......... ......... ... 286
l. Free motion of the homogen eous string . .. .. . ........ .. .... 287
Xll
CONTENTS
2. Forced motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
3. The general nonhomogeneous string and the Sturm-Liouville eigenvalue problem ..... . ....... ... . ..... ........ .. 291
§4. The vibrating rod .. .... .. . . . ...... ......... .. ....... . ...... .. 295
§5. The vibrating membrane ... ..... .. .. . . . ... . ... . ... . ........... 297
1. General eigenvalue problem for the homogeneous membrane 297
2. Forced motion ...................... ................... ... 300
3. Nodal lines ...................... ...................... ... 300
4. Rectangular membrane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300
5. Circular membrane. Bessel functions .. .. . . . ........... . . . . 302
6. Nonhomogeneous membrane .............. .. ...... .... .. ... 306
§6. The vibro.ting plo.te ................ ... ..... ....... . .. .... .... . 307
1. Genero.l remarks ...................... ... . ................ 307
2. Circular boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307
§7. General remarks on the eigenfunction method . . . . . . . . . . . . . . . . . 308
1. Vibration and equilibrium problems .... ....... ........ .. .. 308
2. Heat conduction and eigenvalue problems . . ......... .. .... 311
§8. Vibration of three-dimensional continua. Separation of variables .... ...................... ...................... ....... 313
§9. Eigenfunctions and the boundary value problem of potential
theory .. ..... ·....... .. ............... ..... ...... ....... ..... 315
1. Circle, sphere, and spherical shell ...................... ... 315
2. Cylindrical domain .................... . .. ........... .. .. . 319
3. The Lame problem . ...................... ................ 319
§10. Problems of the Sturm-Liouville type. Singular boundary
points .. ...................... ............... . .. . . ........ .. 324
1. Bessel functions .. .................. . ............. ..... ... 324
2. Legendre functions of arbitrary order . .... .... . .......... . 325
3. Jacobi and Tchebycheff polynomials ............. . . . . . .... 327
4. Hermite and Laguerre polynomials . ........... . . . . ... . . . . . 328
§11. The asymptotic behavior of the solutions of Sturm-Liouville
equations .... ... ........... ..... ........... . . ... ....... . . ... 331
1. Boundedness of the solution as the independent variable
t ends to infinity ...................... .. .. ................. . 331
2. A sharper result. (Bessel functions) . . ... . ................. 332
3. Boundedness as the parameter increases . .... ... . ...... .... 334
4. Asymptotic representation of the solutions ... ............. 335
5. Asymptotic representation of Sturm-Liouville eigenfunctions .. . .. . .............. ....... ... .... . ..... . ....... ... 336
§12. E igenvalue problems with a continuous spectrum .. ........... . 339
1. Trigonometric functions ... . .. .. . .... ... . . . . . . . .. ........ .. 340
2. Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340
3. E igenvalue problem of the membrane equation for t he infinite plane ...................... ...................... . 341
4. The Schrodinger eigenvalue problem ....... .. . ... . ... . ..... 341
§13. Perturbation theory . . ...... ... .. ......... ... . . ....... ..... . . . 343
CONTENTS
xiii
1. Simple eigenvalues .............. .............. ............
2. Multiple eigenvalues ........ .. ...... ......... .............
3. An example .......... . ......... .. ............. ... ....... . .
§14. Green's function (influence function ) and reduction of differential equations to integral equations .............. ...........
1. Green 's function and boundary value problem for ordinary
differential equations ... . .............. ............ ... ..
2. Construction of Green's function; Green 's function in the
generalized sense ... ,_ .............. .............. .......
3. Equivalence of integral and differential equations ... . . .. ..
4. Ordinary differential equations of higher order .. ... . ......
5. Partial differential equations . .............. .............. .
§15. Examples of Green's function ... ...... ... ... ....... .. .. . ... .. .
1. Ordinary differential equations . ..... . ... ..... .... ..... ....
2. Green's function for Au : circle and sphere .............. ...
3. Green's fun ction and conformal mapping .............. ....
4. Green's function for the potential equation on the surface
of a sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5. Green's function for Au= 0 in a rectangular parallelepiped .
6. Green's function for Au in the interior of a rectangle ... .. .
7. Green's function for a circular ring .. .... . . ..... .. ........
§16. Supplemen t to Chapter V . . ...... ........ . ........ ...... ... ..
1. Examples for the vibrating string .. . .............. ...... ..
2. Vibrations of a freely suspended rope; Bessel functions ....
3. Examples for the explicit solution of the vibration equation. Mathieu functions .............. ........ .. .........
4. Boundary conditions with parameters .............. .......
5. Green's tensors for systems of differential equations ......
6. Analytic continuation of the solutions of the equation
Au+ >.u = 0 ............. ............. ........ : ........
7. A theorem on the nodal curves of the solutions of Au +
>.u = 0 ............. ............. ............. ..........
8. An example of eigenvalues of infinite multiplicity ..........
9. Limits for the validity of the expansion theorems ..........
References . ... .. ... . ........ ..... .... . ... ........ .............. ...
344
346
348
351
351
354
358
362
36.~
371
371
377
377
378
378
384
386
388
388
390
391
392
393
395
395
395
395
396
VI. Application of the Calculus of Variations to Eigenvalue Problems .. 397
§1. Extremurn properties of eigenvalues ............. ........... . 398
398
1. Classical extremurn properties ... .. . ..... ..... . ....... .
2. Generalizatio ns .. ... .... . ........... ..... ....... .. ... . .. .
3. Eigenvalue problems for regions with separate components.
4.. The maximum-min imum property of eigenvalues .... .......
§2. General consequences of the extremum properties of the eigenvalues .......... ........ ...... .. . . . . ..... .. ........ . ..... . . .
1. General theorems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
402
405
405
407
407
xiv
CON'rENTS
2. Infinite growth of the eigenvalues .............. .......... . 412
3. Asymptotic behavior of the eigenvalues in the Sturm-Liouville problem ... ........ ........ .............. .......... 414
4. Singular differential equations .......... . .......... ... . .... 415
5. Further remarks concerning the growth of eigenvalues.
Occurrence of negative eigenvalues .............. ...... .. 416
6. Continuity of eigenvalues ...... ... .............. .... . ... .. 418
§3. Completeness and expansion theorems . .. . . . .............. .... 424
1. Completeness of the eigenfunction s .............. ..... . .... 424
2. The expansion theorem .... ...... ....... .. ....... .. ... .... 426
3. Generalizatio n of the expansion theorem .............. .... 427
~4. Asymptotic distribution of eigenvalues .............. .......... 429
1. The equation t.u + Alt = 0 for a rectangle .............. ... 429
2. The equation t.u + )\u = 0 for domains consisting of a finite
number of squares or cubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431
3. Extension to the general differential equation L[1t] + )..pu =
0 ........... ..... .............. ....... ....... ........... 434
4. Asymptotic distribution of eigenvalues for an arbitrary
domain .............. .............. .............. ....... 436
5. Sharper form of the laws of asymptotic distribution of
eigenvalues for the differential equation t.u + )\u = 0 ... 443
§5. Eigenvalue problems of the Schrodinger type .............. ... . 445
§6. Nodes of eigenfunction s .............. . ............ .. ......... 451
§7. Supplementar y remarks and problems .............. .. . ....... 455
1. Minimizing properties of eigenvalues. Derivation from
completeness ...... ... .............. .............. ....... 455
2. Characterizat ion of the first eigenfunction by absence of
nodes ............. ........ ..... .............. .......... 458
3. Further minimizing properties of eigenvalues .............. 459
4. Asymptotic distribution of eigenvalues .............. ..... . 460
5. Parameter eigenvalue problems..... . . . . ........... .. .... 460
6. Boundary conditions containing parameters .... . ....... .. 461
7. Eigenvalue problems for closed surfaces .. .. .............. . 461
8. Estimates of eigenvalues when singular points occur ....... 461
9. Minimum theorems for the membrane and plate ........... 463
10. Minimum problems for variable mass distribution . ........ 463
11 . Nodal points for the Sturm-Liouvi lle problem. Maximumminimum principle .............. .............. .......... 463
References... ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....... . ............ 464
VII. Special Functions Defined by Eigenvalue Problems .. ........ ......
§1. Preliminary discussion of linear second order differential equations ............. ............. ............. ............
§2. Bessel functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . .............. .
1. Application of the integral transformatio n ...... ........
2. Hankel functions . . . . . . . . . . . . . . . . . . . . .
..... .... .. . ..
3. Bessel and Neumann functions ............. .............
4. Integral representations of Bessel func t ions. . ......... ...
466
466
467
468
469
471
474
CONTENTS
xv
5. Another integral representation of the Hankel and Bessel
functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476
6. Power series expansion of Bessel functions . .............. . 482
7. Relations between Bessel functions .. . . . . . .... . ....... . . . . . 485
8. Zeros of Bessel functions .............. . . ..... . ......... ... 492
9. Neumann functions ... . . . ........ . . . . . .......... . ..... ... . 496
§3. Legendre functions . . ... ............. . .............. ... .. .... . 501
l. Schliifii 's integral. .............. .. .. .............. ........ 501
2. Integral representatio ns of Laplace .............. .......... 503
3. Legendre functions of the second kind ...... . ... . ........ . . 504
4. Associated Legendre functions. (Legendre functions of
higher order.) ... .... .............. .............. . ... . ... 505
§4. Application of the method of integral transformatio n to Legendre, Tchebycheff, Hermite, and Laguerre equations .. . ... 506
l. Legendre functions ...... . .. . ...... . . . . . .. . .... . ....... . ... 506
2. Tchebycheff functions .. . .. . .............. ......... .. ...... 507
3. Hermite functions ........ . .. . .............. .............. 508
4. Laguerre functions ............. . ..... . ...... . ............. 509
§5. Laplace spherical harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510
l. Determinatio n of 2n + 1 spherical harmonics of n-th order ... 511
2. Completeness of the system of functions .............. ..... 512
3. Expansion theorem .. ...... . ........ . ......... .. .. . ..... . . 513
4 . The Poisson integral . . . . . . . . . . . ......... , .... . ... . .... . . 513
5. The Maxwell-Sylv ester representation of spherical harmonics ............ . .... .. . . .. . ....... . .............. . . .. 514
§6. Asymptotic expansions ............. ............. ........... .. 522
l. Stirling's formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522
2. Asymptotic calculation of Hankel and Bessel functions for
large values of the arguments .............. .......... . . . 524
3. The saddle point method ............ . .............. ... .... 526
4. Application of the saddle point method to the calculation of
Hankel and Bessel functions for large parameter and
large argument . .. .... . ........ . .... . .. . . . . . ............ 527
5. General remarks on the saddle point method ............. . 532
6. The Darboux method .............. .............. ......... 532
7. Application of the Darboux method to the asymptotic expansion of Legendre polynomials .. ... ....... .. . . ....... . 533
§7. Appendix to Chapter VII. Transformati on of Spherical Harmonics . . . . . . .... . ... . ... . ... . . . ....... . .............. .. 535
l. Introduction and notation ... . .. . .......... . ......... . .... 535
2. Orthogonal transformatio ns ... .. . .. ..... . .............. ... 536
3. A generating function for spherical harmonics .... . . ... .... 539
4. Transformati on formula ..... . .... . . . .............. ........ 542
5. Expressions in terms of angular coordinates .......... .. . .. 543
Additional Bibliography ........ . .............. .............. ........... 547
Index . . . . ..... . ... . . ... . ....... . . . . ............. ..... . . . . . .. . . . . . . . . . .. 551
CHAPTER I
The Algebra of Linear Transformatio ns
and Quadratic Forms
In the present volume we shall be concerned with many topics in
mathematical analysis which are intimately related to the theory
of linear transformations and quadratic forms. . A brief resume of
pertinent aspects of this field will, therefore, be given in Chapter I.
The reader is assumed to be familiar with the subject in general.
§1. Linear Equations and Linear Trans/ormations
The results of the theory of linear equations can be
expressed concisely by the notation of vector analysis. A system
of n real numbers X1 , X2 , • • • , Xn is called an n-dimensional vector or a
vector in n-dimensional space and denoted by the bold face letter x;
the numbers Xi (i = 1, · · · , n) are called the components of the vector
x. If all components vanish, the vector is said to be zero or the null
vector; for n = 2 or n = 3 a vector can be interpreted geometrically
as a "position vector" leading from the origin to the point with the
rectangular coordinates Xi. For n > 3 geometrical visualization is
no longer possible but geometrical terminology remains suitable.
µy = z
Given two arbitrary real numbers A and µ, the vector AX
is defined as the vector whose components Zi are given by Zi
= AXi µ.1J; • Thus. in particular, the sum and cliffPrence of two
vectors arP dPfined.
I. Vectors.
+
+
The number
(I)
x·y =
X1Y1
+ ... + XnYn =
Y1X1
+ ... + YnXn =
y·x
is called the "inner product" of the vectors x and y.
Occasionally we shall call the inner product x · y the component of
the vector y with respect to x or vice versa.
If the inner product x·y vanishes we say that the vectors x and y
3 this terminology has an immeare orthogonal; for n = 2 and n
1
2
I. LINEAR TRANSFORMATIONS AND QUADRATIC FORMS
diate geometrical meaning. The inner product x · x = x2 of a vector
with itself plays a special role; it is called the norm of the vector.
The positive square root of x2 is called the length of the vector and
denoted by I x I = Vi2. A vector whose length is unity is called a
normalized vector or unit vector.
The following inequality is satisfied by the inner product of two
vectors a == (a1 , • · • , an) and b = (bi , · ; · , bn) :
(a· b) 2 ~ a 2b 2
or, without using vector notation,
where the equality holds if and only if the ai and the bi are proportional, i.e. if a re]ation of the form Xa + µb = 0 with X2 + µ 2 ~ 0 is
satisfied.
The proof of this "Schwarz inequality" 1 follows from the fact that
the roots of the quadratic equation
n
L (aix
i-1
+ bi)
n
2
=
x La~
2
i-1
n
+ 2x L
i-1
n
aibi
+ Lb~
i-1
=
0
for the unknown x can never be real and distinct, but must be imaginary, unless the ai and bi are proportional. The Schwarz inequality
is. merely an expression of this fact in terms of the discriminant of
the equation. Another proof of the Schwarz inequality follows immediate]y from the identity
n
n
! L L (a;bk - akb;) 2 •
i=-1 k-l
Vectors X1 , x2 , • • • , Xm are said to be linearly dependent if a set
of numbers X1 , X2, · · · , Am (not all equal to zero) exists such that
the vector equation
A1X1
+ ••• + AmXm = 0
is satisfied, i.e. such that all the components of the vector on the left
vanish. Otherwise the vectors are said to be linearly independent.
The n vectors e1, e2, · · · , en inn-dimensional space whose com1
This relation was, as a matter of fact, used by Cauchy before Schwarz.
LINEAR EQUATIONS AND LINEAR TRANSFORMATIO NS
3
ponents are given, respectively, by the first, second, · · · , and n-th
rows of the array
1 0
0
1
0
0 0
1,
0
form a system of n linearly independent vectors. For, if a relation
1
A1e1 + · · · + Anen = 0 were satisfied, we could multiply this relation
by eh and obtain Xh = 0 for every h, since e~ = 1 and eh· ek = 0 if
h F- k. Thus, systems of n linearly independent vectors certainly
exist. However, for any n + 1 vectors U1 , U2 , • · · , Un+1 (in ndimensional space) there is at least one linear equation of the form
µ1U1 +
• • • +µn+Illn+i =
0,
with coefficients that do not all vanish, since n homogeneous linear
equations
(k
=
1, · · · , n)
for the n + 1 unknowns µ1 , µ2 , · · · , µn+1 always have at least one
nontrivial solution (cf. subsection 3).
2. Orthogonal Systems of Vectors. Completeness. The above "coordinate vectors" ei form a particular system of orthogonal unit
vectors. In general a system of n orthogonal unit vectors e1 , e2, · · · , en
is defined as a system of vectors of unit length satisfying the relations
e: = 1,
(h F- k)
1, 2, · · · , n. As above, we see that the n vectors
en are linearly independent.
If xis an arbitrary vector, a relation of the form
for h, k,
e1 ,
=
e2 ' . • • ,
eox--
c1e1 -
••• -
enen
=
0
with constants Ci that do not all vanish must hold; for, as we have
seen, any n + 1 vectors are linearly dependent. Since the ei are
linearly independent, Co cannot be zero; we may therefore, without
1
To multiply two vectors is to take their inner product.
4
I. LINEAR TRANSFORMATIONS AND QUADRATIC FORMS
loss of generality, take it to be equal to unity. Every vector x can
thus be expressed in terms of a system of orthogonal unit vectors in
the form
(2)
The coefficients
e1,
Ci , the components of x with respect to the system
ez, · · · , en, may be found by multiplying (2) by each of the vec-
tors ei ; they are
From any arbitrary system of m linearly independent vectors
we may, by the following orthogonalization process
due to E. Schmidt, obtain a system of m orthogonal unit vectors
e1 , ez , · · · , em : First set e1 = vi/ I V1 I . Then choose a number
I.
h
I
•
• set c1I = V2 · ei •
c1 m sue a way t h at V2 - c1e1
is
orth ogona1 to e1 , i.e.
Since V1 and V2 , and therefore ei and V2 , are linearly independent,
the vector v 2 - c~e1 is different from zero. We may then divide this
vector by its length obtaining a unit vector ez which is orthogonal
to e1 . We next find two numbers c~ , c; such that Va - c~ e1 - c; ez
is orthogonal to both e1 and e2, i.e. we set c~ = Va· e1 and
= v3 • e2.
This vector is again different from zero and can, therefore, be normalized; we divide it by its length and obtain the unit vector e3 •
By continuing this procedure we obtain the desired orthogonal system.
For m < n the resulting orthogonal system is called incomplete,
and if m = n we speak of a complete orthogonal system. Let us
denote the components of a vector x with respect to e1, ez, · · · , em by
C1, c2, · · · , Cm as before. The self-evident inequality
V1, V2, · • • , Vm,
c;
(x -
c1e1 -
••• -
Cmem)
2
2 0
is satisfied. Evaluating the left side term by term according to the
usual rules of algebra (which hold for vectors if the inner product of
two vectors is used whenever two vectors are multiplied), we find
x
or
(3)
2
-
2x ·
f
i-1
ciei
+ fi-1 c~
= x
2
-
2
f
i=l
c~ + i=l
f c~ 2
0
LINEAR EQUATIONS AND LINEAR TRANSFORMATIONS
where m
~ n
5
and Ci = x · ei ; the following equality holds for m = n:
m
x2
(4)
L-'
= "
i-1
c,.
2
Relations (3) and (4)-(4) expresses the theorem of Pythagoras in
vector notation-have an intuitive significance for n ~ 3; they
are called, respectively, Bessel's inequality and the completeness relation. Relation (4), if it is satisfied for every vector x, does in fact indicate that the given orthogonal system is complete since (4) could not
be satisfied for a unit vector orthogonal to all vectors e1, e2, · · · , em,
and such a vector necessarily exists if m < n.
The completeness relation may also be expressed in the more general
form
m
(5)
X•XI
= £-Cs-Cs,
i-1
"'
I
which follows from the orthogonality of the e, .
So far these algebraic relations are all purely formal. Their significance lies in the fact that they occur again in a similar manner in
transcendental problems of analysis.
3. Linear Transformations. Matrices. A system .of n linear equations
(6)
with coefficients aik , assigns a unique set of quantities Y1 , Y2 , · · · , y.,,,
to every set of quantities x1, x2, · · · , x.,,,. Such an assignment is
called a linear transformation of the set x1 , X2, • • • , Xn into the set
Yi, Y2, · · · , y.,,,, or, briefly, of the vector x into the vector y. This
transformation is clearly linear since the vector AiY1 + AiY2 correA2X2.
sponds to the vector A1X1
The most important problem in connection with linear transformations is the p· oblem of inversion, the que3tion, in other words, of the
+
6
I. LINEAR TRANSFORM ATIONS AND QUADRATIC FORMS
existence of a solution of a system of linear equations. The answer
is given by the following fundamen tal theorem of the theory of linear
equations, whose proof we assume to be known:
For the system of equations
aux1
+ llnXt + · · · + atnX"
= Y2 ,
or, briefly,
(7)
(i =I, · · · , n),
with given coefficients aik, the following alternative holds: Either it
has one and only one solution x for each arbitrarily given vector y,
in particular the solution x = 0 for y = 0; or, alternatively, the homogeneous equations arising from (7) for y = 0 have a positive number
p of nontrivial (not identically zero) linearly independent solutions
X1 , X2 , • • • , Xp , which may be assumed to be normalized. In the latter
case the "transposed" homogeneous system of equations
(8)
(i =I, · · · , n),
where a~k
aki , also has exactly p linearly independent nontrivial solutions x~ , x~ , · · · , x; . The inhomogeneous system (7) then possesses
solutions for just those vectors y which are orthogonal to x~ , x~ , · · · , x; .
These solutions are determined only to within an additive term which is
an arbitrary solution of the homogeneous system of equations, i.e. if x
is ·a solution of the inhomogeneous system and Xu is any solution of the
homogeneous system, then x + Xu is also a solution of the inhomogeneous
system.
In this formulation of the fundamen tal theorem reference to the
theory of determina nts has been avoided. Later, to obtain explicit
expressions for the solutions of the system of equations, determina nts
will be required.
LINEAR EQUATIONS AND LINEAR TRANSFORMATIO NS
7
The essential features of such a linear transformation are contained
in the array of coefficients or matrix of the equations (7) :
(9)
A
(ai11) =
with the determinant
It is sometimes useful to denote the transformation itself (also called
1
tensor or operator) by a special letter A. The elements aik of the
matrix A are called the components of the tensor. The linear transformation (7) may be regarded as a "muUiplication " of the tensor A
by the vector x, written symbolically in the form
Ax= y.
Many results in the algebra of linear transformations may be expressed concisely in terms of matrices or tensors, once certain simple
rules and definitions known as matrix .algebra have been introduced.
First we define matrix multiplication; this concept arises if we suppose that the vector x, which is transformed in equations (7), is itself
the product of a tensor B with components bik and another vector w:
L" b,,;w;. =
Xk
(k = 1, · · · , n).
;-1
Multiplying w by a tensor C we obtain the vector y. The matrix C
which corresponds to the tensor C is obtained from A and B by the
rule of matrix multiplication, C = AB, which states that the element
Ci; is the inner product of the i-th row of A and the j-th column of B:
(10)
Ci;=
L"
lc-1
~1cb1;
(i, j
=
1, · · · , n).
1 In modern usage the term "operator" is customary to denote linear transformations.
8
I. LINEAR TRANSFORMATIONS AND QUADRATIC FORMS
The tensor or transformation C is therefore called the inner product or simply the product of the tensors or transformations A and B.
Henceforth tensors and the equivalent matrices will not be distinguished from each other. Note that matrix products obey the
associative law
(AB)C
=
A (BC),
so that the product AiA2 · · · Ah of any number of matrices written in
a fixed order has a unique meaning. For A 1 = A 2 = · · · =Ah= A
we write this product as the h-th power Ah of the matrix A. It is,
on the other hand, essential to note that the commutative law of multiplication is in general not valid; AB, in other words, differs in
general from BA. Finally the matrix AA + µB is defined as the
matrix whose elements are "Aaik + µbik ; thus the null matrix is the
matrix in which all components van~ab. 1 The validity of the disributive law
(A
+ B)C =
AC
+ BC
is immediately evident.
The unit matrix is defined by
1 0
0
0
1
0
0 0
1
It is characterized by the fact that the equation
AE =EA= A
holds for an arbitrary matrix A.
the identity transformation
The unit matrix corresponds to
(i = 1, · · · , n).
The zero-th power of every matrix A is defined as the unit matrix:
A
0
=
E.
Note that in matrix algebra it does not necessarily follow from the matrix
equation AB = (0) that one of the two factors vanishes, as can be seen from
i
the example A
= (~ g), B = (g
n·
LINEAR EQUATIONS AND LINEAR TRANSFORMATIO NS
9
Since the powers A" of a matrix are defined, we can also define
polynomials whose argument is a matrix. Thus, if
f(x) =
ao +
a1x
+ . . . + amxm
is a polynomial of the m-th degree in the variable x, then .f(A) is
defined by
f(A) = aoE + a1A + · · · + amAm
as a (symbolic) polynomial in the matrix A. This definition of a
matrix as a function f (A) of A can even, on occasion, be extended to
functions which are not polynomials but which can be expressed as
power series. The matrix eA, for example, may be defined by
~
A3
A2
B
A•
- + ··· =I:-.
= e.t = E +A+ -2! + 3!
,_o 11!
Note that in such a series one first considers the sum of the first N
2
terms and then investigates whether each of the n eJements of the
resulting matrix converges to a limit with increasing N; if this is the
2
case, the matrix formed from the n limiting values is considered to
be the sum of the series. In the particular case of the matrix eA it
turns out, as will be shown below, that the series always converges.
A particularly important relation is obtained for a matrix S defined
by a geometric series with partial sums Sm given by
Sm
::s:
E
+ A + A + ... + Am .
2
Multiplying the equation which defines Sm by A, we obtain the
equation
from which it follows that
Sm(E - A)
=
E - A m+l
•
Now if the matrix Sm approaches a limit S with increasing m, so
that Am+i tends to zero, we obtain the relation
S(E - A)= E
for the matrix S defined by the infinite geometric series
s
= E +A
+ A + ...
2
~
=
A·.
L
,,_o
10
I. LINEAR TRANSFORMA TIONS AND QUADRATIC FORMS
Under what circumstanc es an infinite geometric series of matrices or
a Neumann series, as it is occasionally called, converges will be investigated in the next section.
Matrix polynomials may be handled very much like ordinary polynomials in x. For example, an identity between two polynomials in
x implies the correspondin g identity for an arbitrary matrix A. Thus
the identity
x
3
+ 2x + 3x + 4 = (x + 1) (x + 2) + (2x + 2)
2
2
corresponds to the relation
A
3
+ 2A + 3A + 4E = (A + E) (A+ 2E) + (2A + 2E)
2
2
valid for every matrix A.
f(x) = <lo
The factorization
+ a1X + ••• + am.Xm =
am(X -
X1)
(x -
X2) • • •
(x - Xm)
where X1, X2, • • · , Xm are the zeros of the polynomialf (x), leads to the'
matrix equation
f(A) = aoE
=
+ a1A + ·· · + amAm
am(A - x1E)(A - X2E) ... (A - XmE)
for every matrix A.
Every matrix A with components aik , which may in general be
complex, is associated with certain other matrices. If llik is the complex number conjugate to aik, then the matrix A = (aik) is called
the conjugate matrix; the matrix A' = (aki) obtained by interchangin g
correspondin g rows and columns of A is called the transposed matrix
or the transpose of A and A* = A' = (aki) the conjugate transpose
of A. The conjugate transpose is thus obtained by replacing the
elements by their complex conjugates and interchangin g rows and
columns.
The equation
(AB)'
= B'A'
is immediately verifiable. A matrix for which A
symmetric; a real matrix which satisfies
AA'= E
A' is caUed
LINEAR l1QUATIO NS AND LINEAR TRANSFO RMATION S
is called orthogonal.
it satisfies
11
Finally, a complex matrix is called unitary if
AA*= E.
The inversion of the linear transfor mation (7) is possible for arbitrary Yi, as is known from the theory of determinants, if and only if
the determi nant A = I ai1: I does not vanish. In this case the solution
is uniquely determined and is given by a corresponding transformation
n
xi
(11)
The coefficients
aik
=
L: aikYk
k-1
(i = 1, · · · , n).
are given by
(12)
where Aki is the cofactor to the element aki in the matrix A. The
matrix A = (aik) is called the reciprocal or inverse of A and is distinguished by the fact that it satisfies
AA= AA= E.
by A - i instead of A ;
We denote this uniquely determined matrix
1
the solution of a
Thus
•
Avalue
the
has
l
A
of
nant
the determi
determi nant
shing
nonvani
a
has
A
matrix
whose
s
system of equation
matrix B =
a
by
algebra,
matrix
of
e
languag
the
in
is characterized,
s
relation
the
satisfies
which
A -i
AB= BA= E.
4. Bilinear, Quadrat ic, and Hermiti an Forms. To write the linear
equations (7) concisely we may employ the bilinear form which corresponds to the matrix A. This bilinear form
n
(13)
A(u, x) =
L
~1:1'iX1:
i,k-1
is obtaine d by multiplying the linear forms in x1 , X2 , • • • , Xn on
the left-hand side in equation (7) by undetermined quantities
u 1 , u2 , · · · , Un and adding. In this way we obtain from the system
of equations (7) the single equation
(14)
A(u, x) = E(u, y)
12
I. LINEAR TRANSFORMATIO NS AND QUADRATIC FORMS
:..1
valid for all u; here E(u, y) = I: u.yds the bilinear form corresponding to the unit matrix, the unit bilinear form. The symbolic product
of two bilinear forms A(u, x) and B(u, x) with matrices A and Bis
defined as the bilinear form C(u, x) with the matrix C = AB; the
h-th power A"(u, x) is often called the h1old iterated form. The
"reciprocal bilinear form" A-1 (u, x) with the matrix A-1 may, according to the theory of determinants, be written in the form
( 15)
-1(
A
)
u,x
= -
A(u, x)
-A--,
where
A(u, x)
I:"
=
i,k-1
ant
Xn
•••
AilcXiUk.
ann
The symmetric linear transformations, characterized by the condition aik = a1ci , are of special interest. To investigate them it is
sufficient to consider the quadratic form
A (x, x)
I:"
=
i ,k-1
aikXiX1c
which is obtained from the bilinear form by putting ui = xi. For,
from a quadratic form A (x, x) one can obtain a symmetric bilinear
form
~
L..J
i ,k-l
ai1cUiX1c --
!.
2
~ U· aA(x, x)
' -
L..J
--
d<Xi
i-1
A(x
+ u,x + u) -
A(x, x) - A(u,u)
2
which is called the polar form corresponding to the quadratic form
A(x, x).
If A(u, x) = _L;,k-1 aikUiXk is an arbitrary nonsymmetric bilinear
form (with real coefficients), then AA'(u, x) and A'A(u, x) are always
symmetric bilinear forms; specifically we have
AA' (u, x)
=
f (£ aikXi "£ a;kui)
k-1
A'A(u, x)
=
i-1
j-1
f (f aikXk f
i-1
k-1
i-l
ai;u;).
TIONS
LINEA R EQUAT IONS AND LINEA R TRANS FORMA
13
The corresponding quadra tic forms
2
f (f OtikXi)
)2
A' A(x, x) = L L
AA' (x, x) =
,
i-1
k-1
n
(
n
aikXk
,
k-1
i-1
. Forms
which are sums of squares, assume only non-negative values
of this kind are called positive definite quadra tic forms.
Hermi tian
An impor tant generalization of the quadra tic form is the
form. A Hermi tian form is a bilinear form
n
A(u; x)
whose coefficients
aik
=
L
auu.-x1c
i,k-1
have complex values subjec t to the condition
Ui are
Thus a Hermi tian form assumes real values if the variables
n in
writte
taken to be the complex conjugates of Xi; it is usually
the form
• n
H (x,
x) =
L:
n
L:
~kXiXk =
a1c1XiXk •
i,k-l
i,k-1
To an arbitra ry bilinear form
n
A(u, x)
=
L:
OuUiXk
i,k-1
forms
with complex coefficients there correspond the Hermi tian
2
AA *(x, x) = AA'(x , x) =
't I E~kXi l
A*A(x , x) = A'A(x , x) =
L: L: ~kx1c 12•
k-1
and
n
i-1
i-1
I"
k-1
If the variables of a bilinear form
n
A (x, y)
L aax,y1c
= i,k-1
14
I. LINEAR TRANSFORMATIONS AND QUADRATIC FORMS
are subjected to the two transformations
n
Xi= LCiiSi
;-1
with matrices C and B, respectively, we obtain
n
A(x, y) =
L
i,k-1
n
L
aikXiY1c
ai1cCi;b1clSi''I i
i,f,k,l-1
n
n
L
i,l-1
Pil
PilSiT/l;
=
L
i ,k-1
aikCi;b1c1 .
Thus A is transformed into a bilinear form with the matrix
(p;i) = C'AB,
whose determinant is, according to the theorem on the multiplication of determinants, equal to ABr. In particular, if A is a quadrfttic
form
n
K(x, x)
=
L: kpq Xp Xq
p,q-1
with the symmetric matrix K = (kpq) and the determinant K = I kpq I ,
and if we set C = B, and transform the variables x we obtain a
quadratic form with the symmetric matrix C' KC whose determinant
is Kr2•
5. Orthogonal and Unitary Transformations. We now consider
the problem of finding "orthogonal" linear transformations L
n
(16)
Xp =
L
lpqyq = L,,(y)
q-1
(p
= 1, · · ·, n),
with the real matrix L = (lpq) and the determinant A = I l11q I , i.e.
transformations which transform the unit quadratic form
n
E(x, x) =
L
x;
P-1
into itself, thus satisfying the relation
(17)
E(x, x)
=
E(y, y)
for arbitrary y.
Applying our rules of transformation to the quadratic form
15
LINEAR EQUATIONS AND LINEAR TRANSFORMATIONS
A(x, x) = E(x, x), we find that requirement (17) yields the equations
L'EL
(18)
= L'L =LL'= E;
as a necessary and sufficient condition for the orthogonality of L.
Thus the transposed matrix of an orthogonal transformation is identical with its reciprocal matrix; therefore the solution of equations (16)
is given by the transformation
n
YP ~
(19)
L
lqpXq = L~(x),
q-1
which is likewise orthogonal. We see that an orthogonal transformation is one whose matrix is orthogonal as defined in subsection 3.
Written out in detail, the orthogonality conditions become
(p ¢ q)
(20)
or, equivalently,
(p ¢ q).
(21)
To express an orthogonal transformation in vector notation we prescribe a system of n orthogonal unit vectors 11 , 12 , · · · , ln into which
the coordinate vectors e1, e2, ···,en are to be transformed. Then
the vector x is represented by
X
= X1e1
+ X2e2 + ··· + Xnen
Multiplying by ep we obtain
Xp
lpq
=
= Y1li
+
Y2l2
+ · ·· + Ynln .
:E:...1 y 9(eplq); hence
= e,,·lq.
From (18) it follows that A = 1, i.e. that the determinant of an
orthogonal transformation is either +1 or -1. Therefore the determinant of an arbitrary quadratic form is invariant with respect to
orthogonal transformations.
Furthermore, the relation L' (AB)L = (L' AL) (L'BL) follows from
(18) for the matrices A, B, and L of any two bilinear forms and any
orthogonal transformation. This means that the symbolic product
of a number of bilinear forms may be transformed orthogonally by
2
16
I. LINEAR TRANSFO RMATION S AND QUADRAT IC FORMS
subjecting each factor to the same orthogonal transformation. In
particular, it follows that the orthogonal transforms of two reciprocal
forms are also reciprocal.
The generalization of these considerations to Hermitia n forms
n
H(x, x) =
leads to unitary transformations.
L hpqXpXq
p,q-1
A unitary transformation
(p
=
1, .. · n)
is defined as a transformation (with compJex coefficients lpq) which
transforms the unit Hermitian form
into itself, i.e. for which
In exactly the same way as above one obtains the matrix equation
LL*= L*L
=E
as a necessary and sufficient condition for the unitary character of
the transformation whose matrix is L. Here L* = L' is the conjugate
transpose of L. L must therefore be a unitary matrix as defined in
subsection 3. Specifically, a transformation is unitary if the following conditions hold:
n
(22)
LI l"P 1
2
P-1
n
= 1,
L
.,_1
l.,Pll'q = 0
(p ~ q),
or, equivalently,
n
(23)
:L
1zp.,1 2
P-1
n
=
1,
L: zp11lq"
11-1
=
o
(p ¢. q).
The determin ant of a unitary transformation has the absolute
value 1, as follows immediately from the equation LL* = E.
LINEAR TRANSFORMATIONS WITH A LINEAR PARAMETER
17
§2. Linear Transformations with a Linear Parameter
In many problems the system of equations of a linear transformation takes the form
n
(24)
Xi -
A
.L:
k-1
tikXk
=
Yi
(i = 1, · · · , n)
where A is a parameter (in general complex). The corresponding
bilinear form is E (u, x) - XT (u, x), where T (u, x) is the form whose
matrix is (tik). As we have seen in the preceding section, the problem
of solving the system of equations (24) is equivalent to the problem
of finding the reciprocal bilinear form R(u, y; A) with the matrix R
which satisfies the equation (E- AT)R = E. We know that this
reciprocal matrix R exists if and only if the determinant IE - XT I is
different from zero.
Let us consider the zeros of the determinant IE- XT I or, equivalently, for K = 1/X ~ 0, the zeros of the determinant I T - KE I .
Clearly, I T - KE l is a polynomial in K of then-th degree. Therefore there exist n values of K. (namely the zeros of the polynomial) for
which the form R(u, y; A.) fails to exist. These values Ki are known
as the "characteristic values," "proper values," or "eigenvalues" of
T with respect to the unit matrix E; they form the so-called
"spectrum" of the matrix T. 1
The particular form of equations (~4) suggests a solution by iteration: In the equation
n
Xi
=
Yi
+ A k-1
L ta.:x11:
we substitute for the quantities xk on the right the expressions
n
Yk
+AL
tk;X;,
i-1
and then again repeat this substitution. The procedure is conveniently described if we write R = E
ATR and continue:
+
R = E
= E
+ ATR = E + AT + A2T2R ,
+ AT + AT + AT R = · ...
2
2
3
3
1 Sometimes the set of values>.. = 1/K,, for which no reciprocal
of E - >..T
exists, is called the spectrum. We shall call this the "reciprocal spectrum"
and the Xi the "reciprocal eigenvalues."
18
I. LINEAR TRANSFORMATIONS AND QUADRATIC FORMS
We thus obtain an expression for R as an infinite series
which-assuming that it converges-represents the reciprocal matrix
of E - "AT. To see this we simply multiply the series by E - >-..T
and remember that symbolic multiplication may be carried
out term by term provided the result converges. It is then immediately clear that the representation
R
=
(E - >-..T)-1
E
=
+ >-..T + >-. T + >-.. T + ...
2
3
2
3
is, formally, completely equivalent to the ordinary geometric series.
(Compare the discussion of geometric series on page 9, where we
need only put A = "AT to obtain equivalence.)
Let us now represent our original system of equations using bilinear
forms instead of the corresponding matrices:
= E(u, y).
E(u, x) - XT(u, x)
We may write the solution of this equation in the form
+ XT(u, y; >-..)
E(u, y)
=
E(u, x),
which is completely symmetric to it; here
T(u, y; X)
+ >-..T2 -t >-.. T3 + · · ·
2
=
T
=
R(u, y; X) - E(u, y)
>-..
The form T is called the resolvent of T.
The convergence of the above Neumann series for R or T for sufficiently small I >-.. I is easily proved: If M is an upper bound of the
absolute values of the numbers tik , it follows immediately that upper
bounds for the absolute values of the coefficients of the forms T 2 ,
T3, · · · , T"' are given by nM2, n 2M 3, • • • , nh---1M"' . Thus
(M
+ XnM + Xn M + ·· · )
· (I
+ Ut + · •' + I Un l )(
2
2 2
U1 J
3
J
J
J
Yi I
+ '· ' + I Yn I )
is a majorant of the Neumann series for T(u, y; X); it is certainly
convergent for I>-.. I < 1/nM. Therefore our Neumann series also
LINEAR TRANSFORMATIONS WITH A LINEAR PARAMETER
19
converges for sufficiently small I "'A I and actually represents the
1
resolvent of T(u, x).
The above estimate proves, incidentally, that in any everywhere
convergent power series f(x) = L:..o c,,x" we may replace x by an
arbitrary matrix A and obtain a new matrix f(A) = L:...o c.,A".
Thus, in particular, the matrix eA. always exists.
While the above expression for R or T converges only for sufficiently
small IX I, we may obtain from equation (15) of the previous section
1
an expression for the reciprocal form or matrix R = (E - >..Tr
which retains its meaning even outside. the region of convergence.
For, if we identify the form E - >.T with the form A(u, x), we immediately obtain, for the reciprocal form,
R(
. >..)
u, y,
= _ A(u, y; X)
A(>..)
1 The convergence of the majorant obtained above evidently becomes worse
· with increasing n. It may, however, be pointed out that, by slightly refining
the argument, an upper bound for the coefficients of the form T can be obtained
which is independent of n and which, therefore, can be used for the generalization to infinitely many variables. We denote the elements of the matrix r• by
and set
t <.•>
.~
~
~
a.-1
cu I = Zp.
I tpa
.
Then, if Mis an upper bound for all then quantities
shown below by induction, that
Zp ,
it follows, as will be
therefore,
for p, q = 1, 2, · · · , n a_!!d every v. From this we see that our Neumann series
converges for I>. I < 1/M. We thus have a bound which does not depend on n
explicitly.
To prove the above inequality for arbitrary v we assume it to be proved for
the index v - 1; we then have
Since the inequality is valid for v = 1, it is proved for arbitrary P.
20
I. LINEAR TRANSFORMATIONS AND QUADRATIC FORMS
and, for the resolvent T,
_ A(u, y; A) _ ~ E(
)
AA(A)
A u, y'
T(u, y; A)
where
0
A(u, y; A) =
U1
Un
~~ .. ~ .~. ~~1~
••••••••
~~~1~.
and
1 - Atu
A(A) =
-A~1
- At12
1 - "A.t22
are polynomials in A of at most the (n - 1)-st and n-th
degree. Thus the zeros of A(A) form the reciprocal spectrum of the
form T as defined above, i.e. the totality of values of X for which the
form E - >..T has no reciprocal.
By means of the formula
A(u, y; A) _ ~ E(
)
AA(A)
}.. u, y
the series on the left, which does not converge for al1 X, is continued
analytically into the entire :>..-plane. The reciprocal form R, as well
as the resolvent T, is a rational function of }.. whose poles are given
by the spectrum of the form T.
If we expand the determinants .6.(u, y; :>..) and A(:>..) in powers of
:>.., we obtain the expressions
.6.(u, y; >..)
=
Ai(u, y) - X.6.2(u, y)
+ X2Aa(u, y)
-
•· ·
+
(-1)">..n-IAn(u, y),
A(X) = 1 - "A.Ai+ "A.2A2 - · · • + (-I)nxnAn,
LINEAR TRANSFORMATIONS WITH A LINEAR PARAMETER
21
where
0
U 111
U 11h
Y111
tP1P1
tP1P1a
and
tP1aP1
tP1aP2
• • •
tP1aP1&
The summations here are extended over all integers Pi , P2 , · · · , Ph
from 1 to n with P1 < P2 < · · · < Ph .
It is often advantageous to consider the form KE - T with the
determinant
K -
tn
-t21
-t12
K -
~
Its zeros K1 , K2, • • • , Kn (eigenvalues of T) are the reciprocals of
the zeros of A(X). For the reciprocal form (KE- T)-1, which exists
for all values of " different from K1 , K2, • • • , Kn, one obtains the
Neumann series expansion
(KE - T)-1
= ~ + !2 + T2 +
K
K
KS
which is valid for sufficiently large values of I K I . A noteworthy
conclusion can be drawn from this expansion. It is clear from the
above discussion that the left side is a rational function of K with the
1
denominator <p(K); therefore <p(K)(KE - T)- must be a form which
is ~ntegral and rational in "and its expansion in powers of K can contain no negative powers. Accordingly, if we multiply the above
Cn, all the coefficients of
C1Kn-i
equation !>Y <p(K) = Kn
negative powers of " in the resulting expression on the right must
+
+ ·· · +
22
I. LINEAR TRANSFORMATIONS AND QUADRATIC FORMS
vanish. But the coefficient of K-1 is, as is seen immediately, the ex~
pression Tn + c1Tn-l + · · · + Cn, and we thus arrive at the following
theorem, which is due to Cayley: If the determinant of KE - T is
denoted by cp(K), then the relation
cp(T) = 0
is satisfied by the matrix T.
Another important aspect of the spectrum of the eigenvalues
K1 , K2 , • • • , Kn is expressed by the following theorem:
If the eigenvalues of a matrix T are K1 , K2 , · · · , Kn and if g(x) is any
polynomial in x, then the eigenvalues of the matrix g(T) are g(K1),
g(K2), ''' , g(Kn).
To prove this we start from the relation
n
I KE
- TI = cp(K) =
which is an identity in T.
I KE
II
(K -1
K,,),
We wish to obtain the relation
n
- g(T)
I = II (K
- g(K,,)).
-1
Let h(x) be an arbitrary polynomial of degree r which may be written in terms of its zeros X1 , X2 , · · · , Xn in the form
r
h(x) = a II (x - Xp) .
.:-1
Then the identity
r
h(T) = a II (T - xpE)
p-1
holds for an arbitrary matrix T. By considering the determinants of
the matrices in this equation we obtain
I h(T) I =
r
an II I T - XpE
p-1
r
I = ( -1rran II I XpE
p-1
- T
I
TRANSFORMATION TO PRINCIPAL AXES
23
If we now let h(T) be the function KE - g(T), the desired equation
I KE
- g(T)
" (K
I = II
- g(K,,))
11-1
follows immediately.
§3. Transformation to Principal Axes of Quadratic and
Hermitian Forms
Linear transformations x = Z(y) which reduce a quadratic form
K(x, x) =
L:"
kpqXpX,
p,q-1
to a linear combination of squares
K(x, x) =
L:" tepy;
p-1
are highly important in algebra. We are particularly interested in
reducing K(x, x) to this form by means of an orthogonal transformation
Xp
" l,pyq = L,,(y)
= L:
(p = 1, · · ·, n).
q-1
Transformations of this kind are called transformations to principal
axes.
1. Transformation to Principal Axes on the Basis of a Maximum
Principle. Let us first convince ourselves that a transformation to
principal axes is always possible for any given quadratic form K(x, x).
To do this we use the theorem that a continuous function of several
variables (which are restricted to a finite closed domain) assumes a
1
greatest value somewhere in this domain (Theorem of Weierstrass).
1 The transformation to principal axes may also be accomplished by direct
algebraic methods. An orthogonal matrix L is required, such that L' KL = D
is a diagonal matrix with diagonal elements Kt , K2, • • • , Kn • From the relation
KL = LD we obtain the equations
..
l; kP'l l-'
11-1
= lpi Ki
for the matrix elements z,,, which yield the Ki as roots of equation (30), cf. p. 27.
Then, on the basis of simple algebraic considerations we can construct an
orthogonal system of n 2 quantities z,, . The method used in the text is preferable to the algebraic method in that it may be generalized to a larger class of
transcendental problems.
24
I. LINEAR TRANSFORMATIO NS AND QUADRATIC FORMS
According to this theorem, there exists a unit vector 11 with components lu, l12, · · · , l1,,. such that, for X1 = lu , ·. · · , x,,. = l1n, K(x, x)
assumes its greatest value, say K1 , subject to the subsidiary condition
(25)
Geometrically, the vector 11 represents on the "unit sphere" (25) a
point P so that the surface of the second degree K(x, x) = const.
containing P touches the unit sphere at P.
There exists, moreover,. a unit vector 12, orthogonal to 11, with
components l21, · · · , l2,,. such that, for X1 = ~ 1 , · · · · , x,,. = ~n, K(x, x)
assumes the greatest possible value K2 subject to the condition
(26)
in addition to condition (25). The problem solved by 11 for the whole
unit sphere -is solved by 12 for the manifold formed by the intersection
of the unit sphere and the "plane" (26).
Furthermore, there exists a unit vector 13 , orthogonal to 11 and 12 ,
with components l31 , la2, · · · , la,,. such that, for Xi = lai (i = l, · · · , n),
K(x, x) takes on its greatest value Ka, subject to the subsidiary conditions (25), (26), and
n
(27)
l: "2,;cp = o.
,,_1
Continuing in this manner we obtain a system of n mutually orthogonal vectors 11, 12, · · · , 1,,. , which will be called the "principal axis
vectors" or "eigenvectors." According to (21) their components lqp
define an orthogonal transformation
(28)
(p = 1, · · -. , n);
this transformation, we assert, is the solution of our problem.
Since equations (28) are solved by
(29)
the equation x
(p = 1, · · · , n),
Ip is equivalent to the statement YP = 1, yq = 0
TRANSFORMATION TO PRINCIPAL AXES
25
for q ;;C p. Thus, in particular, the maximum Ki is attained for
Yi = 1, Y2 = 0, · · · , Yn = O; hence, in the transformed form
n
C(y, y)
=
L: Cpqypyq
p,q-1
the first coefficient cu equals
n
H(y, y) =
L: hpq YP yq
P,q-1
K1 •
=
K(x, x)
The form
= C(y, y) - K1(y~
+ ... + y!)
assumes, moreover, no positive values. For, by the maximum
character of Ki, H(y, y) is nonpositive provided L:;...1x! = L:;_1y; =
1; hence it is nonpositive for all Yi with L:;=1 y! ;;C 0. If Yi should
occur in the expression for H(y, y), e.g. if h12 = h21 were different from
zero, we would obtain the value
2h12E
+ ~E = E(2h12 + ~E)
2
for H(y, y) with
Y1
=
Ya= · • ·
1,
=
Yn = 0.
This could be made positive by a suitable choice of E.
It has thus been shown that, after the transformation, K(x, x) is
reduced to
C(y, y) = KlY~
+ C1(y, y),
where C1(Y, y) is a quadratic form in then - 1 variables Y2, Ya, · · · , Yn.
If the subsidiary conditioi;i Y1 = 0 is imposed the transformed form
is equal to C1(y, y). In the same way we may now conclude that
C2(y, y), where C2(y, y) depends only
C1(y, y) is of the form K2Yi
· · , Yn , and so forth.
·
y4,
Ya,
variables
2
n
the
on
Thus we have demonstrated the possibility of a transformation to
principal axes so that
+
n
n
L: kpqXpXq
p,q-1
=
L: Kpy~ ,
P-1
We might note that the corresponding minimum problem would
have served equally well as a starting point for the proof; i.e. we might
have looked for the minimum of K(x, x), subject to the auxiliary
26
I. LINEAR TRANSFORMATIONS AND QUADRATIC FORMS
condition E(x, x) = 1. In that case we would have arrived at the
quantities K1 , K2, • • • , Kn in the reverse order. One could also keep
K(x, x) constant and look for the maxima or minima of E(x, x); then
the minimum values Xi would be the reciprocals of the Ki •
2. Eigenvalues. We shall now show that the values Ki defined in
the previous subsection as successive maxima are identical with the
eigenvalues as introduced in §2.
The equation
satisfied by the numbers
K -
0
dete~nant
may be written in the form
0
K1
0
But this
Ki ,
IC -
K2
0
0
0
0
0
=
0.
0
is just the determinant of the quadratic form
which is obtained by applying an orthogonal transformation to the
form
n
KL
x;
p-1
- K(x, x).
Therefore the relation
K -
0
0
0
K1
K -
0
0
K2
0
K-K,.,
IC -
ku
-k21
TRANSFORMATION TO PRINCIPAL AXES
is an identity in K. Consequently the numbers
the algebraic equation
I k 11
I
(30)
-
K
k1n
k2n
k12
k22 -
k21
K
Ki
27
are the roots of
0
I ............................
knn kn2
I knl
K
for the unknown K; i.e. they are the eigenvalues introduced in §2.
Our method of derivation shows automatically that the roots of
equation (30) are necessarily real if the kpq are arbitrary real quantities subject to the condition kpq = kqp • 1 We may also remark in
passing that the absolute values of the reciprocals of the eigenvalues
are geometrically significant as the squares of the lengths of the
principal axes of the surface K (x, x) = 1 in n-dimensional space.
If at least one eigenvalue is equal to zero the form is said to be "degenerate"; it can then be represented as a form of less than n variables.
It is clear from equation (30) that this is the case if and only if I kpq I
vanishes. For K (x, x) to be positive definite the condition Kp > 0,
p = 1, 2, · · · , n is necessary and sufficient.
Suppose the representation of a form K(x, x) in terms of principal
axes
n
K(x, x) =
LP=l KpY;
is given. Then, using the properties of the orthogonal transformations of products discussed above, the expressions
n
2
K (x, x) =
Lp=l K;Y;,
n
K 3(x, x ) =
""'"' K 3PY 2p'
£...,,
P=l
are easily obtained for the iterated forms. It follows that the eigenvalues of the h-fold iterated form K\x, x) are the h-th powers of the
eigenvalues of K(x, x) (this also follows immediately from the theorem
on page 22); moreover we see that, for even h, the form K"(x, x)
is positive definite.
1 Equation (30) is customarily called
the secular equation because it occurs in the problem of secular perturbations of planetary orbits. For a direct
proof that the eigenvalues are real, see Ch. III, §4, 2.
28
I. LINEAR TRANSFORMATIONS AND QUADRATIC FORMS
3. Generalization to Hermitian Forms. A transformation to principal axes can be carried out in exactly the same way for Hermitian
forms. A Hermitian form
H(x, i)
with the matrix H = H' can always be transformed by a unitary
transformation L, given by
into the form
n
H(x, i) =
L
P=-1
n
Kpyp'jjp =
L
Kp
I
YP
2
1
,
p-1
where all the coefficients Kp are real. These eigenvalues Km reappear
as the maxima of the Hermitian form H(x, i), subject to the auxiliary
conditions
and
(i = 1, · · · , m - 1).
4. Inertial Theorem for Quadratic Fo1"Jlls. If we relinquish the
requirement that the linear transformation be orthogonal, a quadratic
form may be transformed into a sum of squares by many different
transformations. In particular, after the above orthogonal transformation has been carried out, any transformation in which each
variable is simply multiplied by a factor of proportionality leaves the
character of the form as a sum of squares unaltered. Thus it is possible to transform the form in such a way that all the (real) coefficients have the value +1 or -1. The following theorem, known as
the inertial theorem for quadratic forms, holds:
The number of positive and negative coejficients, respectively, in a
quadratic form reduced to an expression
cpz; by means of a nonsingular real linear transformation does not depend on the particular transformation.
Proof: The positive and negative coefficients may be made equal
to + 1 and -1, respectively. Suppose, now, that the quadratic
form K(x, x) is transformed by two different transformations into
L
29
TRANSFORMATION TO PRINCIPAL AXES
y! and zi + · ··+ z! - z!+i - · · · - z!
yi + ·.. + y; with r
< s.
Y;+i - · · · We then have
Yi + ·· · + y; + z!+1 + · ·· + z~
Y;+1 + · ·· + y~ + zi + ·· · + z! .
Let us consider the conditions Y1 = · · · = Yr = Za+i = · · · = Zn =
=
0,
which imply that the remaining Yi also vanish. By imagining the
Zi expressed in terms of the Yi and regarding these conditions as a
system of fewer than n equations in the Yi, we obtain the contradiction that there exists a non-vanishing solution vector.
5. Representation of the Resolvent of a Form. The resolvent of the
quadratic form K(x, x) can be expressed in a suggestive way. According to §2 the resolvent may be defined by the symbolic equation
. '\) = [E(x, x) - XK(x, x)r
K(x, x, I\
X
1
E(x, x)
-
•
We suppose that K(x, x) has been brought into the form
2
n
K(x, x)
=L
YP.
Ap
p=l
y;;xP
must be identical with the resolvent of
The resolvent of I:;-1
1
K(x, x), since [E(x, x) - XK(x, x))- goes over into
[
E(y, y) - X
t y;J-
- L
=
1 [(
~
y; - A L
n
Ap -
2)-1
-
YP
p-1
Ap
A
2)-l -
~1 ~ YP
n
Ap
Now the following relations
p-1
when the transformation is applied.
hold:
1 [( n
p-1
A
1
E(y, y)
E(y, y)
]
] = x1 [ ~l Ap Apn
X
Ln Ap~.
- X
Xp ,
using the notation
n
P_
p=l
If we now transform back to the variables
(19) we obtain the expression
(31)
K(x, x; X)
]
.P=l
A y; - Ly;
l[nL __
Ap - X
= -
2
A YP - E(y, y)
]
=
2
p-1
30
I. LINEAR TRANSFO RMATION S AND QUADRA TIC FORMS
for the resolvent of K(x, x); thus, for the bilinear form, we have
(32)
E L~~u)~~~x).
=
K(u, x; A)
p-1
p
From this representation it is evident, incidentally, that the residue
of the rational function K(u, x; A) of X at the point Ap is equal to
-L~(u)L~(x), assuming that A11 ~ Aq for p ~ q.
6. Solution of Systems of Linear Equation s Associated with Fo"Dls.
In conclusion we shall present, with the help of the eigenvectors, the
solution of the system of linear equations
n
(33)
X 11 -
A
L
= Yv
k 11 qXq
·q=l
(p = 1, · · ·, n)
associated with the quadrat ic form
n
.L:
=
K(x, x)
kpqXpXq •
p ,q-1
If we apply the transfor mation to principal axes
to the variables Xi and Yi, K(x, x) goes over into
and the bilinear form K(x, z) is simi]arly transformed.
system of equations (33) becomes
(34)
Hence, our
(p = 1, · · · , n).
the solution of which is
(35)
Up
=
1 -
Vp
AKp -
Vp
-
Ap
--A- - Ap - A Vp.
1 - -
Ap
In terms of the original variables, we obtain the equival ent formu]a
for the solution
(36)
X=~l
L..J l_J
). p '
p-1
1-
~
Ap
I
.1,
31
MINIMUM-MAXIMUM PROPERTY OF EIGENVALUES
in which the solution appears as a development in terms of the eigenvectors 11, 12, · · · , l" of the form K(x, x). We have here used the
notation y·lp = L::-1 l 11qyq.
The principal axis vector or eigenvector lp is itself the normalized
solution of the homogeneous equations
Xq -
>.,,
L"
r-1
k~Xr
=0
(q = 1, · · · , n) .
or
If, for q ~ p, all the Kq are different from
one normalized solution,
or
u,,
= I,
Uq
=0
x
Kp
= 1/>...P, there exists only
(q
~
p)
= 1,,.
If several characteristic numbers coincide the principal axis vectors
are not uniquely determined.
§4. Minimum-Maximum Property of Eigenvalues
1. Characterization of Eigenvalues by a Minimum-Maximum Problem.
In the above discussion we have obtained the eigenvalues by solving
a series of maximum problems, each one of which depended on the solutions of the previous problems of the series. We shall now show
that each eigenvalue can be directly characterized as the solution of
a somewhat different problem in which all reference to the solutions
of previous problems is avoided.
The problem is to maximize the form
K(x, x) =
L:"
kpqXpx"
p,q-1
if the condition (25)
"
:Ex;=
1
p-1
is imposed and if the h - 1 equations
(37)
(v
I, · · · , h -
1; h ~ n)
32
I. LINEAR TRANSFORMATIO NS AND QUADRATIC FORMS
must be satisfied. This maximum value of K(x, x) is of course a
function of the parameters a..,,p • We now choose the a..,,p in such a
way as to give this maximum its least possible value. We assert
that this minimum value of the maximum is just the h-th eigenvalue
Kh of K(x, x), provided the eigenvalues are ordered in a sequence of
decreasing values, "1 being the greatest eigenvalue, "2 the next, and
so on.
The transformation to principal axes changes K(x, x) into
condition (25) into
"
E
Y! = 1,
p-1
(38)
and equations (37) into
(39)
(v= 1 ···h-l·h<n )
'
' -
where the fJ.,,P are new parameters. If we set
Yh+i =
· · · = Yn = 0
equations (39) become h -1 equations in h unknowns Yi, Y2, · · · , Yh,
which can certainly be satisfied for a set of values Yi also satisfying
(38). For these v.alues we have
+ · ·· +
y: ~ «h{Yi
+ · ·· +
y~) = Kh •
Thus the required maximum of K(x, x) for any set of values flvp is not
less than «h ; but it is just equal to Kh if we take for (39) the equations
K(x, x) =
K1
Yi
Kh
Y1 = · · · =
Yh-1 =
0.
It follows therefore that:
The h-th eigenvalue "h of the quadratic form K(x, x) is the least value
which the maximum of K(x, x) can assume if, in addition to the condition
" x;
I:
p-1
= 1,
h - 1 arbitrary linear homogeneous equations connecting the
prescribed.
Xp
are
MINIMUM-MAXIMUM PROPERTY OF EIGENVALUES
33
2•.Applications. Constraints. This independent minimum-maximum property of the eigenvalues shows how the eigenvalues are
changed if j independent constraints
(40)
(8
=
1, ... ' j)
are imposed on the variables, so that K(x, x) reduces to a quadratic
form K(x, x) of n - j independent variables. The h-th eigenvalue
K:h is obtained from the same minimum-maximum problem as Kh,
in which the totality of sets of admissible values Xi has been narrowed
down by (40). Therefore the maximum, and thus the eigenvalue
of K(x, x), certainly does not exceed the corresponding quantity for
K(x, x).
Furthermore, K;+h is the least maximum which K(x, x) can possess
if, in addition to (25), h + j - 1 linear homogeneous conditions are
imposed on the Xp ; K;+h is therefore certainly not greater than Kh ,
for which j of these conditions are given by the fixed equations (40).
We have thus the theorem: If a quadratic form K(x, x) of n variables
is reduced by j linear homogeneous constraints to a quadratic form
K(x, x) of n - j variables, then the eigenvalues i<1 , i<2, • • • , Kn-; of
K (x, x) are not greater than the corresponding numbers of the sequence
K1 , 1e2 , • • • , Kn-; and not less than the corresponding numbers of the se1
quence K;+i, K;+2, • • • , Kn •
If, in particular, we let j = 1 and take for our constraint the condition Xn = 0, then the quadratic form K goes over into its (n - 1)-st
"section,'' and we obtain the theorem: The h-th eigenvalue of the
(n - I)-st section is at most equal to the h-th eigenvalue of the original
quadratic form, and at least equal to the (h + I)-st eigenvalue.
If this theorem is applied to the (n - 1)-st section of the quadratic
form, there results a corresponding theorem for the (n - 2)-nd sec-
tion, and so forth. In general we note that the eigenvalues of any
two successive sections of a quadratic form are ordered in the indicated manner.
Moreover, we may conclude: If a positive definite form is added
1 This may be illustrated geometrically: Let us consider the ellipse
formed by the intersection of an ellipsoid and a plane passing through its
center. The length of the major axis of this ellipse is between the lengths of
the longest and the second axes of the ellipsoid, while the length of the minor
axis of the ellipse is between those of the second and the shortest axes of the
ellipsoid.
34
I. LINEAR TRANSFORMATIONS AND QUADRATIC FORMS
to K(x, x), the eigenvalues of the sum are not less than the corresponding
eigenvalues of K(x, x).
Instead of utilizing a minimum-maximum problem to characterize
the eigenvalues we may use a maximum-minimum problem. In this
case the eigenvalues will appear in the opposite order.
It may be left to the reader to formulate and prove the minimummaximum character of the eigenvalues of Hermitian forms.
§5. Supplement and Problems
1. Linear Independence and the Gram Determinant. The question
of the linear dependence of m given vectors V1 , V2 , • • • , v m may be
very simply decided in the following way without explicity determining the rank of the component matrix: We consider the quadratic
form
m
G(x, x) = (x1 V1 + · · · +xm Vm) 2 = L (vi·Vk)XiXk.
i,k-1
Clearly G(x, x) ~ 0, and the vectors Vi are linearly dependent if and
only if there exists a set of values x1 , X2 , • • • , Xm with (25')
m
L: x~ =
1,
i-1
for which G(x, x) = 0. Thus if the vectors Vi are linearly dependent
the minimum of the form G(x, x) subject to condition (25') must be
equal to zero. But this minimum is just the smallest-eigenvalue of
the quadratic form G(x, x), i.e. the least root of the equation
vi -
(v1 ·V2)
K
(v1 ·Vm)
(41)
= 0.
(Vm·V1)
(Vm·V2)
v! -
K
The theorem follows:
A necessary and sufficient condition for the linear dependence of the
vectors V1 , V2 , • • • , v m is the vanishing of the "Gram determinant"
vi
(42)
f =
(v1 ·v2)
(v1 ·Vm)
SUPPLEMENT AND PROBLEMS
35
An alternate expression for r follows from (41). If the left side of
equation (41), which is satisfied by the (all non-negative) eigenvalues
K1 , K2, • • • , Km of G(x, x), is developed in powers of K, then the term
independent of K is equal to r, while the coefficient of Km is equal
to ( -l)m. According to a well-known theorem of algebra it follows
that
(43)
r
=
K1K2 • • • Km.
Consequently the Gram determinant of an arbitrary system of vectors
is never negative. Relation
(44)
(i, k = 1, · · · , m),
in which the equality holds only for linearly dependent vectors
V1 ' V2 ' ••• ' v m , is a generalization of the Schwarz inequality (see
page 2)
The value of the Gram determinant or, alternatively, the lowest
eigenvalue Km of the form G(x, x) represents a measure of the linear
independence of the vectors V1 , V2 , • • • , v m • The smaller this number, the "flatter" is them-dimensional polyhedron defined by vectors
v1 , V2 , • • • , v m ; if it is equal to zero the polyhedron collapses into
one of at most m - 1 dimensions. In this connection the Gram determinant has a simple geometrical significance. It is equal to the
square of the m !-fold volume of the m-dimensional polyhedron defined by the vectors V1 , V2 , • • • , Vm . Thus, for m = 2, it is the
square of twice the area of the triangle formed from v1 and V2 •
Gram's criterion for linear dependence must of course be equivalent
to the usual one. The latter states that vectors are linearly dependent if and only if all determinants formed with m columns of the
rectangular component array
36
I. LINEAR TRANSFORMATIONS AND QUADRATIC FORMS
are equal to zero. And indeed, according to a well-known theorem
of the theory of determinants,
(45)
r
where the summation is extended over all integers s1, s2, · · · , Sm
from 1 to n with S1 < S2 < · · · < Sm •
2. Hadamard's Inequality for Determinants. Every determinant
au
with real elements
aik
a12
a1n
satisfies the inequality
(46)
Proof: Let the elements
vary, keeping the sums of squares
aik
n
L a~k
k=l
= c~
(i = 1, · · · , n)
fixed. If A!ax is the greatest value of the function A2 of the elements
aik under these n conditions-the existence of such a maximum follows
immediately from Weierstrass's theorem (see page 23)-then the
elements of Amax in each row must be proportional to the corresponding cofactors. For, if h is fixed, we have .
A
=
ah1Ah1
+ · ·· + ahnAhn ;
thus, by the Schwarz inequality,
n
A ~
2
L
k-1
n
a~k
L
A'fik
k-1
If the ahk are not proportional to the Ahk the inequality holds, and A2
certainly can not have its maximum value. For, in this case, by
SUPPLEMENT AND PROBLEMS
37
suitably changing the n quantities ahk (k = 1, · · · , n), with c~ and
the Ahk held constant, the square of the determinant can be made
equal to the right-hand side.
If we now multiply Amax by itself, we obtain, according to the
multiplication theorem for determinants,
since the inner products of different rows of Amax vanish as a result
of the proportionality just demonstrated and of elementary theorems
on determinants. Therefore the original determinant satisfies
Hadamard's inequality
The geometrical meaning of Hadamard's inequality is that the volume
of the polyhedron formed from n vectors of given lengths in ndimensional space is greatest if the vectors are mutually orthogonal.
Hadamard's inequality is also valid for complex aik if A and ai1c
are replaced by their absolute values.
3. Generalized Treatment of Canonical Transformations. For generalizations and applications to many problems of analysis the
following concise treatment of the simultaneous canonical transformation of two quadratic forms is most appropriate. Again we
consider two quadratic forms in an n-dimensional vector space of
vectors x, y , · · ·
(a)
H (x, x) =
L"
hptfl:r« ,
"',q-1
which we assume positive definite, and
(b)
K(x, x) =
L:"
p,q-1
k 11qX,,Xq,
which is not necessarily definite. By definition we interpret
H (x, x) as the square of the length of the vector x, and the polar form
H(x, y) = (x, y) =
L"
"'. q-1
hpqXpyq
38
I. LINEAR TRANSFORMATIONS AND QUADRATIC FORMS
as the inner product of x and y.
transformation
The problem is to find a linear
(p = 1, · · · , n)
which transforms K and H into the sums
n
K(x, x) =
n
L
PPY!,
H(x, x)
P=l
L:
P=l
y;.
To obtain this transformation explicit expressions for the forms K
and H are not required; our proof is based merely on the properties
that H and K are continuous functions of the vector x, that with
arbitrary constants A and µ equations of the form
(47)
H(Ax
+ µy, AX+ µy)
(48)
K(Ax
+ µy, AX+ µy)
+ 2AµH(x, y) + µ H(y, y),
= A K(x, x) + 2AµK(x, y) + µ K(y, y)
= A2H(x, x)
2
2
2
hold, and that His positive definite, vanishing only for x = 0.
We consider a sequence of maximum problems: First we define
a vector x = x 1 for which the quotient
K(x, x)/H(x, x)
attains its maximum value P1 • Without affecting the value of this
quotient the vector may be normalized, i.e. 'Subjected to the condition H(x, x) = 1.
Then we define another normalized vector x 2 for which the quotient
K(x, x)/H(x, x) attains its maximum value p2 under the orthogonality
condition H(x, x 1) = 0. Proceeding in this way, 've define a sequence of normalized vectors x1, x2 , • · • , xk, such that for x = xk the
quotient K(x, x)/H(x, x) attains its maximum value Pk under the
orthogonality conditions
x
(v = 1, · · · , k - 1).
After n steps we obtain a complete system of vectors x1, x2,
for which the relations
(49)
H(x\ xk)
= 1,
i = k;
i
• · • ,
<
and
(50)
i = k;
i<k
k;
x 11
39
SUPPLEMENT AND PROBLEMS
hold. Relations (49) are merely the orthogonality relations stipulated
in our maximum problems. To prove relations (50) we consider
first x 1• The maximum property of x1 is expressed by the inequality
1
K(x 1 + Ef', X1 + ES) - P1H(x + Es, X1 + Es) ~ 0
valid for an arbitrary constant E and an arbitrary vector (.
of (47) and (48), it yields
Because
where
A = K(x1, t) - P1H(x1, r),
B = K(t, t) - P1H(t, t).
Since this inequality is valid for arbitrarily small positive or negative
E it implies that A = 0 or that
(51)
for arbitrary (.
K(x1, t) - P1H(x1, s) = 0
The maximum problem for xh yields as above
K(x\ t) - p,Jl(x\ t) = 0
for an arbitrary vector ( satisfying the relations
H(t, x") = 0
(v = 1, · · · , h - 1).
Now, for h < k, we may take ( = xk. Since H(x\ xk) = 0, we may
conclude that K(xh, xk) = 0 for h < k, while by definition K(x\ xh) =
Ph·
Since the n orthogonal vectors x" form a complete system in our
vector space, an arbitrary vector x can be expressed in the form
where y.,, = H(x, x"). We substitute these expressions in Hand K
and use the expansions corresponding to (47), (48) for n summands;
because of (49), (50) it follows immediately that
H(x, x)
"
=Ly!,
-1
K(x, x) =
L"
.... 1
p,,y! .
Thus we have accomplished the required transformation.
40
I. I..1INEAR TRANSFORMATIONS AND QUADRATIC FORMS
Exactly as before the values Ph are shown to have the following
minimum-maximum property.
Under the auxiliary conditions
(v
=
1, · · ·, h - 1),
Ph (with Pi ~ · · · ~ Pn) is the least value which the maximum
of K(x, x)/H(x, x) can assume-this maximum is regarded as a function of the parameters a,,P ,
To construct the transformation of which we have proved the
existence we first show that for all integers h the "variational
equation"
K(x", t") -:-- p,Jl(x", t")
=0
holds with an arbitrary vector {. So far the relation has been proved
only under the restriction ({, x") = 0 for v < h. However, if { is
arbitrary the vector n = { - C1X1 -:- • • • - c,,,_1xh-I with c,, = ({, x
satisfies the orthogonality condition H(11, x") = 0, 11 < h, hence
0 = K(x"', 11) - PhH(x"', 11) = K(x"', 5) - p,,,H(x"', 5); here the final
equality sign follows from (49) and (50).
Writing the variational equation for xh = x, Ph = p we obtain for
the components x; of x = xh the system of linear homogeneous equations
11
)
n
I:
(ki;
i-1
- p~;)x;
=
0
(i
=
1, · · ·, n);
hence the values p,,, satisfy the determinant equation II ki; - phi; II = 0
and the vectors x"' are obtained from the linear equations after the
quantities p = Ph have been found. Clearly, these considerations
characterize the· numbers Ph and the vectors x"' as the eigenvalues and
eigenvectors of the matrix (kpq) with respect to the matrix (hpq).
Thus for each eigenvalue Ph there exists a solution in the form of a
vector x"'. The solutions for different eigenvalues are orthogonal; if
two eigenvalues are equal the corresponding solutions are not necessarily orthogonal but may be made so by the orthogonalization process
of page 4. These mutually orthogonal solutions may be normalized to unit length; the resulting vectors are the eigenvectors of the
problem and their components are the coefficients of the required
transformation.
41
SUPPLEMENT AND PROBLEMS
These coefficients l,," are obtained from x = L:;..1 yqX" if we multiply· by the vector e" which defines the original coordinate system.
Thus x,, = (x, eP) = L::-1 yq{x", ep); hence l,," = (x", eP).
4. Bilinear and Quadratic Forms of Infinitely Many Variables.
Under suitable conditions our theory remains valid if the number of
variables increases beyond all bounds. For example, this is the case
if both the sum of the squares of the coefficients of the biJinear or
quadratic forms and the sum of the squares of the variables converge.
This theory of forms of infinitely many variables, developed by
Hilbert, may then be applied to numerous problems of analysis.
However, the theory of forms in vector spaces of infinitely many
dimensions can be more adequately developed on the basis of abstract
concepts as indicated in subsection 3. As we shall see, many topics
in analysis can be illuminated from the viewpoint of such a generalized
theory of quadratic forms.
5. Infinitesimal Linear Transformations. An infinitesimal linear
transformation is defined as a transformation whose matrix is
1
A
+ Ea22
· •• 1
+ Eann
where E denotes an infinitesimal quantity of the first order, i.e. a
quantity whose higher powers are, for the problem at hand, negligible
in comparison with lower powers of E. The product of two such infinitesimal transformations with the matrices A = E + (Eaik) and
B = E + (E/3i1c) has the matrix C = E + (Eai1c + E/3i1c). Thus the
product does not depend on the order of the factors; in other words,
infinitesi,mal transformations commute with each other.
Furthermore, the reciprocal matrix of A = E + {Eai1c) is
A -l = E - ( Eai1c), and the determinant of the matrix A is equal to
1
+ E{an + a22 + · ·· + ann).
If the infinitesimal transformation is to be orthogonal, we have the
condition A'A = E, where A' is the transposed matrix. We must
therefore have ai1c + aki = 0, or, in other words:
A necessary and suffici,ent condition for the orthogonality of an in-
42
I. LINEAR TRANSFORMATIO NS AND QUADRATIC FORMS
finitesimal transformation is that the difference between its matrix and
the unit matrix be skew-symmetric.
Any infinitesimal transformation with the matrix C = E + (E'Yik)
may be represented as the product of an orthogonal transformation
A = E + (Eaik) and a symmetric transformation B = E + (E{jik),
where
{jik = !hik
+ 'Yki).
Consider a symmetric transformation Yi = Lk SikXk whose matrix
is S = (sik), not necessarily infinitesimal. Its geometrical significance is that of a dilatation in n mutually orthogonal directions.
To see this let us transform the quadratic form S(x, x) to principal
axes, transforming the Xi into ui and the Yi into Vi. We then have
n
L: SikXiXk
i,k-1
and the equations Yi
=
Lk SikXk
n
=
L
i=l
~
KiU '
become
These equations evidently represent a dilatation by the factor Ki
in the direction of the i-th principal axis. The ratio of the increase
of volume to the initial volume, known as the volume dilatation, is
evidently given by the difference KiK2 • • • Kn - 1 = I s1k I - 1. If, in
particular, the transformation is infinitesimal, i.e. (sik) = E + (E{31:,..),
we have
K1 • • • Kn -
1 = E(Pn
+ ·· · + Pnn)•
Since an orthogonal transformation represents a rotation we may
summarize by stating:
An infinitesimal trans/ormation whose matrix is E
E'Y ik) may be
represented as the product of a rotation and a dilatation; the volume
dilatation is
'Yii .
6. Perturbations. In the theory of small vibrations and in many
problems of quantum mechanics it is important to determine how
the eigenvalues and eigenvectors of a quadratic form K(x, x) =
bikXiXk are changed if both the form K(x, x) and the unit
+(
EL::=1
L::.k=l
43
SUPPLEMENT AND PROBLEMS
form E(x, x) are altered. Suppose E(x, x) is replaced by E(x, x)
EB(x, x), where
EA(x, x) and K(x, x) by K(x, x)
+
n
n
A (x, x) =
+
L atkXiXk ,
i,k=l
B(x,
L f3ikXiXk,
i,k=l
=
x)
and E is a parameter. The problem is then to tran8form E
EB simultaneously into canonical form. If we put
and K
+
K(x, x)
+ EB(x, x)
E(x, x)
+ EA(x, x)
+
EA
n
L b:kxixk,
i,k=l
n
=
L
a~kXiXk,
i,k=l
the equations for the components of the eigenvectors become
n
~('
bik £..J
' )Xk' =
p 'aik
(i = 1, · · ·, n),
0
k-1
where p' may be obtained from the condition that the determinant of
this system of equations must vanish. Let us denote the eigenvalues
of K(x, x) by Pi, P2, · · · , Pn and assume that they are all different;
let the corresponding values for the varied system be denoted by
p~, p~, · · · , p:. The original form K(x, x) may be assumed to be
a sum of squares:
K(x,
x)
=
pix~
+ p-iX~ + · · · + PnX~.
The quantities p~ , being simple roots of an algebraic equation, are
single-valued analytic functions of E in the neighborhood of E = 0;
the same is, therefore, true of the components x~k of the varied eigenvectors belonging to the eigenvalues p~. Thus the quantities p~
and x~" may be expressed as power series in E, the constant terms of
which are, of course, the original eigenvalues Ph and the components of
the original eigenvectors Xhk, respectively. In order to compute
2
successively the coefficients of E, E , • • • we must substitute these
power series in the equations
(i, h
=
1, · · ·, n)
in which we have b~k = Pik + E{3ik , a~k = Oik + E<Xik , with Pii =
By collecting the terms
Pi, Pik = 0 (i "¢ k), Oii = 1, Oik = 0 (i "¢ k).
44
I. LINEAR TRANSFORMATIONS AND QUADRATIC FORMS
in each power of Ein these equations and then setting the coefficient
of each power of Eequal to zero we obtain an infinite sequence of new
equations. An equivalent procedure, which is often somewhat more
convenient, is given by the following considerations of orders of
magnitude, in which Eis regarded as an infinitesimal quantity. We
first consider the equation with i = h. By setting the coefficient of
the first power of E equal to zero we obtain
+ EfJhh
+
EOlhh
Ph
I
l
Ph =
= Ph -
Ephahh
+ EfJhh,
except for terms of the second or higher orders in E. The same procedure applied to the equations with i ¢ h yields the result
X~h
=
=
1
1,
Xhi
ai11Ph -
fJih
-E - - - -
Ph -
Pi
except for infinitesimal quantities of the second order in E.
By using these values of the components of the eigenvectors we may
easily obtain the eigenvalues up to and including the second order
in E. Again we consider the h-th equation for the components of the
h-th eigenvector:
If we neglect quantities of the third order in E on the left-hand side
and write the term with h = k separately, we obtain
I
b,,,h -
I
I
Phahh
~I
= £..J
(
E
I
b,,,k -
I
OlkhPh -
I )
Ph ahk
Ph -
k-1
_E2
f'
k=-1
(a&h PA - f3k,.)
Ph - Pk
fJkh
Pk
2
•
It follows that
1
Ph
=
Ph -
E
(
Phahh -
fJhh
)
-
E
2
(
ahh flhh -
Phahh
)
+
E
2
~'
£..J
k=l
2
(akhPh Ph -
fJkh)
·
Pk
.
Here we have used the symbol L:~ to denote summation over all
values of k from 1 to n except for k = h.
7. Constraints. Constraints expressed by linear conditions
'Y1X1
+ ••· + 'YnXn = 0,
45
SUPPLEMENT AND PROBLEMS
and the resulting diminution of the number of independent variables
.q-1 kpqXpXq , may be regarded
of the quadratic form K (x, x) =
as the end result of a continuous process. Consider the quadratic
2
form K(x, x) + t(-y1X1 + · · · + 'YnXn) , where tis a positive parameter.
If t increases beyond all bounds, each eigenvalue increases monotonically. The greatest eigenvalue increases beyond all bounds, while
the others approach the eigenvalues of the quadratic form which is
obtained from K(x, x) by elimination of one variable in accordance
with the given constraint.
8. Elementary Divisors of a Matrix or a Bilinear Form. Let A be a
tensor and A = ( aik) the corresponding matrix. Then the polynomial
L;
" -
l"E - Al
au
-~1
" -
-~n
lZ22
may be decomposed according to certain well-known rules into the
product of its "elementary divisors"
(K - r1)4'1, (t< - r 2)• 2 ,
• • • ,
(IC - r,.)8•,
where some of the numbers r1 , r2 , · · · , rh may be equal. For each
divisor (IC - r.)4'" there is a system of e, vectors f~'>, f~">, · · · , f~:>
such that the equations
are valid.
Here the n vectors
If they are introduced as new variables
are linearly independent.
2
1
x~ >, x~ >, · · · , x~~>, the matrix A is transformed into the matrix
0
0
A,.
46
I. LINEAR TRANSFORMATIONS AND QUADRATIC FORMS
in which Ai, A2, · · · , Ah are themselves matrices; A. is a matrix
of order e" :
( r. 0 0
0
0
0
1 r.
0
0
0
0
1 r" J
A.=
0
9. Spectrum of a Unitary Matrix. We shall now show that the
spectrum of a unitary matrix lies on the unit circle, i.e. that all of its
eigenvalues have the absolute value -1.
\Ve note that the elements of a unitary matrix cannot exceed unity
in absolute value. Therefore the absolute values of the coefficients
of the characteristic equations of all unitary matrices of the n-th
degree must lie below a certain bound which is independent of the
particular matrix considered. Since the absolute values of the first
and last coefficients of the characteristic equation are equal to 1,
this means that the absolute values of the eigenvalues must lie between certain positive upper and lower bounds which are independent
of the particular matrix. On the other hand all powers Am of a
unitary matrix A are also unitary, and their eigenvalues are the
m-th powers of the corresponding eigenvalues of A. But the absolute
values of these powers and their reciprocals can remain below a bound
which is independent of m only if the absolute value of each eigenvalue
(and all of its powers) is 1.
Another proof, which can be used for infinite matrices as well,
follows from the convergence of the Neumann series for (E - AA)- 1•
The series
1
(E - XA)- = E
+
XA
+
2
XA
2
+ ··· ,
where A is a unitary matrix, certainly converges if I X I < 1. For the
elements of the matrices Am all have absolute values of at most 1,
and thus the geometric series is a dominating series for the matrix
elements. Thus no zeros of IE - AA I can lie inside the unit circle.
On the other hand we have, in virtue of the relation AA' = E,
(E - XA)-1 =_;!A' (E
X
+~A'+~
A' + · · ·).
X
X
2
2
SUPPLEMENT AND PROBLEMS
47
Here the geometric series on the right converges for 11/A I < 1 since
Thus no zero of IE - AA I can lie outside the unit circle. Therefore all these zeros lie on the unit circle,
and our assertion is proved.
A' is also a unitary matrix.
References
Textbooks
B6cher, M., Introduction to Higher Algebra. Macmillan, New York, 1907.
Kowalewski, G., Einfiihrung in die Determinantentheorie. Veit, Leipzig,
1909.
Wintner, A., Spektraltheorie der unendHchen Matrizen. S. Hirzel, Leipzig,
1929.
Monographs and Articles
Courant, R., Zur Theorie der kleinen Schwingungen. Zts. f. angew. Math. u•
Mech., Vol. 2, 1922, pp. 278-285.
Fischer, E., Uber quadratische Formen mit reelen Koeffizienten. Monatsh.
f. Math. u. Phys., Vol. 16, 1905, pp. 234-249. The maximum-minimum character of the eigenvalues probably was first mentioned in this paper.
Hilbert, D., Grundzuge einer allgemeinen Theorie der linearen Integralgleichungen, especially sections 1 and 4. Fortschritte der mathematischen
Wissenschaften in Monographien, Heft 3. B. G. Teubner, Leipzig and
Berlin, 1912.
CHAPTER II
Series Expansions of Arbitrary Functions
Many of the relations discussed in Chapter I are closely analogous
to theorems in which functions of one or more variables, defined in
some given fundamental domain G, take the place of vectors inn-dimensional space. The problem of expressing a vector in n-dimensional space as a linear combination of n independent but otherwise
arbitrary vectors is analogous to the problem of representing a more
or less arbitrary function in the domain G as a linear combination of
members of some given set of functions. (It will become obvious
that the number of functions in this set must be infinite.) This is
known as the problem of series expansion of an arbitrary function in
terms of a given set of functions. In the present chapter this problem,
which appears in various forms throughout mathematical physics, will
be treated from a general point of view.
We shall restrict ourselves to piecewise continuous functions; i.e.
we consider functions for which the fundamental domain G may be
so subdivided into a finite number of domains that, in the interior
of each domain, the function is continuous and approaches a finite
limit as a point on the boundary of one of these domains is approached
from its interior. To simplify notation we shall at first consider functions of one variable x, whose fundamental domain G is a finite interval on the x-axis. If we are concerned with several variables, say
the two variables x and y, we shall assume that the boundary of the
fundamental domain G is formed by a finite number of segments of
curves with continuously turning tangents. If we consider the
points on the boundary as belonging to the fundamental domain we
shall speak of a "closed domain."
Furthermore we shall in many cases assume that the functions
considered are piecewise smooth, i.e. that they are piecewise continuous and possess piecewise continuous first derivatives. Unless the
contrary is specified explicitly we shall assume that our functions
have real values.
48
49
ORTHOGONAL SYSTEMS OF FUNCTIONS
§I. Orthogonal Systems of Functions
1
1. Definitions. The integral
(1)
(f, g)
=
Jf
g dx
taken over the (finite) fundamental domain is called the inner product
(f, g) or (Jg) of two functions f(x) and g(x). It satisfies the Schwarz
inequality
(f, g)2 ~ (f, f)(g, g),
(2)
where the equality holds if and only if f and g are proportional.
As in the case of vectors, this follows either from the positive definite
character of the function
J('/...f + g) dx of the variable X, or directly
2
from the identity
2
(f, g) = (f, f)(g, g) -
!
JJ (f(x)g(~) -
f(~)g(x)) 2 dx d~.
Two functions f(x) and g(x) for which (f, g) vanishes will be termed
orthogonal. The inner product of a function f (x) with itself is called
the norm of this function and is denoted by NJ:
NJ= (f,f)
(3)
=
f J2 dx.
A function whose norm is unity is said to be normalized. A system
of normalized functions 'P1(x), cp2(x), · · · any two different members
of which are orthogonal is called an orthonormal system and the
relations
(tp,,, cp,,.)
= o,,,,.
(o,,.,
=
1,
o,,,,.
= Ofor v ~ µ.)
expressing this property are the orthogonality relations.
An example of an orthonormal system of functions in the interval
0 ~ x ~ 27r, or more generally in any interval of length 27r, is given
by the functions
1
V2;'
cos x
sin x
y';.'
y';.'
cos 2x
y';,
sin 2x
y';,
....
For functions of area] variable which take on complex values it is
1
If no misunderstanding is possible, limits of integration will be omitted.
,)Q
II. $ERIES EXPANSIONS OF ARBITRARY FUNCTIONS
convenient to extend the concept of orthogonality in the following
way: Two complex functions f(x) and g(x) are said to be orthogonal
if the relations
(j, g) = (J, g) = 0
hold, where J and g denote the complex conjugate functions to f
and g, respectively. The function f(x) is said to be norma1ized if
NJ
=
JIf
2
1
=
dx
The simplest example of a complex ortho-
1.
normal system is given by the exponential functions
1
V2;'
in the interval 0
~
x
~
2r, as is seen immediately from the "ortho-
g<mtility relations"
(4)
_!_ 121r eiCJS-.,,>~ dx = o,,..
2r o
(o.,.,,
= 1, o,,..,
= 0 forµ. ~ v).
We say that r functions f1 , f2 ,
, fr are linearly dependent if a
homogeneous linear relation ~~- 1 cih = 0 with constant coefficients
ci, not all equal to zero, holds for all x. Otherwise the functions are
said to be linearly independent. It is worth noting that the functions
of an orthogonal system are always linearly independent. For, if an
identity
C1~
+
<>$)!
holds, we may multiply by
c.,, = 0.
+ ••• + Cn<Pn = 0
'fJ.,,
and integrate, obtaining the result
2. Orthogonalization of Functions. From a given system of infinitely many functions v1 , v2 , • • • , any r of which are linearly independent for arbitrary r, an orthonormal system 'P1, <P2, • • • may be
obtained by taking 'Pn as a suitable linear combination of V1, v2, • • • , Vn •
The orthogonalization procedure is exactly analogous to the procedure used for obtaining an orthogonal system of vectors from a set
of linearly independent vectors (cf. Chapter I). We start by choosing 'P1 = v1/v'NVi. Next we find a number c~ in such a way that
the function v2 - c~<P1 is orthogonal to <P1 , i.e. we set c~ = (<P1V2) ;
v2 - c~'P1 cannot vanish identically because of the linear independence of v1 and v2 and, therefore, of 'P1 and v2 • We may then divide
51
ORTHOGONAL SYSTEMS OF FUNCTIONS
this function by the square root of its norm, obtaining a normalized
function cp2 which is orthogonal to cp1 • We next find two numbers
. va - c"1<P1 - c2" <,02 is
. orthogonal
c1" , c2"·m sueh a way t hat t hf
e unction
to both cp1 and <Pi , i.e. we set
= (cp1va) and = (cp2Va). This function cannot vanish identically and can, therefore, be normalized;
we divide it by the square root of its norm and obtain the normalized
function <Pa • Continuing this procedure indefinitely we arrive at the
desired orthogonal system of functions.
In speaking of orthogonalization, we shall, in general, mean the
procedure just described in which functions are both normalized
and orthogonalized.
c:
3. Bessel's Inequality.
c:
Completeness Relation. Approximation in
the Mean. Let <P1, <Pi, • • • be an orthonormal system and let f be
any function. The numbers
c. = (Jcp,,)
(5)
(v
=
1, 2, · · ·)
are called the expansion coejfici,ents or components off with respect to
the given orthonormal system.1
From the obvious relation
(6)
we obtain, by writing out the square and integrating term by term,
0
~
/J2
f
dx - 2 ,,_. c,,
/f
cp. dx
+ •-1
f c!
n
NJ -
2
n
:E
c! + :E
c!,
,,_.
11-1
and hence
(7)
Since the number on the right is independent of n, it follows that
(8)
This fundamental inequality, known as "Bessel's inequality," is true
for every orthonormal system. It proves that the sum of the squares
of the expansion coefficients always converges.
i The term "Fourier coefficients" is also used since the expansion considered
is a generalization of the Fourier series expansion.
52
II. SERIES EXPANSIONS OF ARBITRARY FUNCTIONS
For systems of functions with complex values the corresponding
relation
QO
L I c. 1
2
(8')
P-1
~ Nf
=
(!, J)
holds, where c,, is the expansion coefficient c"
may be obtained from the inequality
J f(x) - :f: c,,<P,,
I
2
1
dx = NJ -
11-l
This relation
(f, ;p").
=
:f:
I
2
c,,
1
P=l
~
0.
The significance of the integral in (6) is that it occurs in the problem of approximating the given function f (x) by a linear combination
2::=1 'Yv<P11 with constant coefficients 'Y" and fixed n, in such a way that
the "mean square error" M
J
(f -
=
I::=1 'Y11<P
2
11 )
dx is as small as
possible. For, by simple manipulations of the integral, we obtain
the identity
J(! - :f:11=1 'Y11<P11) dx JJ2 dx + :f:P=l ('Y,, 2
M =
=
c,,)
2
-
1'f=1 c!,
from which it follows immediately that M takes on its least value
for "f,, = c,,.
An approximation of this type is known as an approximation by
the method of least squares, or an approximation "in the mean."
If, for a given orthonormal system <P1, <P2, • • • , any piecewise continuous function f can be approximated in the mean to any desired
degree of accuracy by choosing n large enough, i.e. if n may be so
chosen that the mean square error
J(f -
I::=1
2
cvep,,) dx is less
than a given arbitrarily small positive number, then the system of
functions <Pi , <P2 , • • • is said to be "complete." For a complete orthonormal system of functions Bessel's· inequality becomes an equality
for every function f:
QO
•
•
L~: =NJ.
(9)
•-1
This relation is known as the "completeness relation." It may be
written in the more general form
00
(9')
L c.d. =
11-1
(!, g) with c,,
=
(f, <P11), d. = (g, 1p,,),
53
ORTHOGONAL SYSTEMS OF FUNCTIONS
as can be seen by applying (9) to the function f
N(f
+ g)
+ g:
00
=NJ+ Ng+ 2(f, g)
=
L""
(c!
-1
L (c. + d,,) 2
= ,,_1
+ d! + 2c,, d,,),
and then subtracting the corresponding equations for f and g.
Incidentally, a sufficient condition for the completeness of a system
of functions 'Pi, 'P2, · • · is that the completeness relation (9) be
satisfied for all continuous functions f. For, any piecewise continuous
function g may be appYoximated by a continuous function f in such
a way that the integral
J(f -
g) 2 dx is arbitrarily small.
Let the
points of discontinuity of g be denoted by x" • An approximating
continuous function f is obtained if in each interval x" - o ~ x ~
x" + o we replace the graph of g by the straight line-segment that
1
joins the points (x11 - o, g(x11 - o)) and (x11 + o, g(x,, + o)).
If n is sufficiently large, the mean square error integral
J (f -
I::...1 C#11) 2 dx,
where
c1, c2,
cients off, will be arbitrarily small.
M' =
J(u - t. c;p.)' dx J[
=
·· ·
are the expansion coeffi-
This implies that the integral
(g - f)
+ (!-
t. crp,)]'
dx
may be made arbitrarily close to N(g - f) by a suitable choice of n.
From the Schwarz inequality we have, in fact,
(t - t c#,,) + (u -:- J, ! - t,,_1 c#.)
f)-N (f - :E
~ N(g -f) + N (1 - t. e;p,) +
M' = N(g - f)
+N
11-1
2
2v'N(g
_J
C,<P,).
Moreover,
1 If M is an upper bound of I g(x) I and q the number of discontinuities of
g in the interval of integration, the mean square error is
J
(f - u)2 dz ::5 8M2qa,
which can be made arbitrarily small by choosing
a small enough.
54
1I. SERI~S ~XPANSIONS 01' ARlHTRAR"l'. FUNCTIONS
where the a,, are the expansion coefficients for g, since these coefficients yield the least mean square error for g. Thus the completeness
relation holds for g if it holds for f.
It is important to bear in mind that the completeness of an orthonormal system <P1 , 'P2 , • • • , expressed by the equation
lim
n-oo
J(1 - f
c~,,) dx = 0,
= :L'!.1 C,,<p,, , i.e. that f
2
•-1
. does not necessarily imply that f
can be expanded in a series in the functions 'Pv. The expansion is, however,
valid if the series }::.1 Cv'fJv converges uniformly; the passage to the limit
may then be carried out under the integral sign. ·The completeness of
the system 'PI, 'P2 , · · · is, of course, a necessary condition for the
validity of the expansion in general; for example, if we take f to be
one function of a complete system, then all its components with respect to the incomplete system· (consisting of all the functions 'Pv
except f) will vanish. But, even for a complete system 'P1 , 'P2 , • • • ,
this question of the convergence of the series for an arbitrary function requires a more detailed investigation, which will be carried out
subsequently (Chapters V and VI).
If the above limit equation is satisfied we say that the functions
:L:-1 Cv'fJ,, converge to the function fin the mean.
Another important theorem is that a piecewise continous function
is uniquely determined by its expansion coejficients with respect to a
given complete orthonormal system. That is, if two piecewise continuous
functions have the. same expansion coefficients they are identical.
For, the difference of two functions with equal coefficients has the
coefficients zero; according to the completeness relation its norm is
therefore zero, and this difference vanishes identically. Thus a function is uniquely determined by its expansion in terms of a complete
orthonormal system even if this expansion converges only in the mean.
In fact, convergence in the mean will be all that is needed for many
theorems of mathematical physics.
The concept of -completeness of a system of functions retains its
meaning even if the system is not orthonormal. In general, we call
a system of functions complete if every piecewise continuous function
can be approximated in the mean arbitrarily closely by a linear combination of functions of the system. The completeness of such a system is preserved under the process of orthogonalization.
ORTHOGONAL SYSTEMS OF FUNCTIONS
55
4. Orthogonal and Unitary Transformations with Infinitely Many
Variables. There are many analogies between orthonormal systems
of functions and orthonormal systems of vectors in n-dimensional
space. It is often useful to regard the functions 'Pl, 'P2 , • • • as
"coordinate vectors" or "coordinate functions" in a space of infinitely many dimensions (often called a "Hilbert space"). An arbitrary function f is a vector in this space and its expansion coefficients
c., = (/'(),,) are the components of this vector in terms of the system
of coordinates defined by 'P1, 'P2, • · • •
If 1/11 , 1/12 , • • • is a second orthonormal system of functions with
respect to which the components off are d,, = (/1/t,,), and if both
systems are complete, then -the Ci and di are related by the system
of infinitely many equations
(i = 1, 2, •.. ).
(10)
This is seen by applying the completeness relation (9') to the expansion coefficients of the functions f and 'Pi with respect to the system
1/11 , 1/12 ,
In the same way we obtain the inverse set of equations
(i = 1, 2, ... ).
(10')
The coefficients obey the conditions
00
(11)
L aaa;"
=\
('fJ,"fJ;)
=
ai;
<"1.1/1;)
=.
aii,
k-1
00
(11')
I: aa-ak;
k-1
=
which are simply orthogonality conditions in n-dimensional space
(Ch. I, §1) generalized to the space of infinitely many dimensions.
We therefore call a transformation (10) which fulfills conditions (11)
and (11') an orthogonal transformation of infinitely many variables or
an orthogonal transformation in Hilbert space.
Analogously, the expansion coefficients of a function with respect
to two different complex orthogonal systems are related by a unitary
transformation of infinitely many variables.
56
II. SERIES EXPANSIONS OF ARBITRARY FUNCTIONS
5. Validity of the Results for Several Independent Variables. More
General Assumptions. None of our concepts or results is changed if,
instead of functions of a single variable, we consider functions of
several variables, say x and y. The variables are assumed to lie
within a finite domain G, the volume element of which we denote by
dG. We define the inner product (Jg) of two functions f{x, y) and
g(x, y) in the region G as the integral (Jg) =
i
fgdG.
Then essen-
tially nothing need be changed in the notations and proofs of this
section.
Furthermore, all our concepts and results remain valid if the fundamental domain is infinite, provided we assume that the squares of all
functions occurring in the treatment are integrable over the entire
fundamental domain.
Finally, we remark that our methods are applicable even if the
function f becomes infinite in the fundamental domain, provided
that f and its square are integrable over the fundamental domain.
6. Construction of Complete Systems of Functions of Several Variables. If complete systems of functions of one variable are known,
it is possible to construct complete systems of functions of two or
more variables by the following theorem:
Let
cp1(s), cp.h), · · ·
be a complete orthonormal system of functions in the interval
a
s
b, and suppose for each i (i = 1, 2, ···),that
s s
is a similar system in the interval c ::::; t
wu(s, t)
s
d.
Then the functions
= 'Pi(s)l/lki(t)
form a complete orthonormal system of functions in s and t in the
rectangle a S s b, c t
d. (In particular, the system of functions 'Pi(s)cpk(t) is orthonormal and complete in the square a
s
b,
a S t S b.) Specifically, if f(s, t) is a continuous function in this
rectangle, the completeness relation
s
ff
is satisfied.
s s
f(s, t) ds dt =
,~, (JJ f(s, t)wa(s, I) ds dt)'
s s
ACCUMULATION PRINCIPLE FOR FUNCTIONS
To prove this theorem, we note that
where Ui(t) =
f
f(s, t)'Pi(s) ds.
f
J2cs, t) ds =
57
2:7=
1
g~(t),
This is simply an expression of the
completeness of the system 'Pi. Since the series on the right converges uniformly,1 we may integrate term by term with respect tot,
obtaining
ff J2(s,
t) ds dt =
t. Jg~(t)
dt.
Now to the i-th term on the right we apply the completeness relation
for the system of functions 1/tki(t) (k = 1, 2, · · ·) and immediately
obtain the desired completeness relation.
§2. The Accumulation Principle for Functions
1. Convergence in Function Space. The analogy between functions
and vectors in n-dimensional space breaks down in many ways if
infinite sets of functions and vectors are considered. It follows from
an elementary theorem of analysis (Weierstrass theorem on poin ts
of accumulation) that one can select a convergent sequence out of
any infinite set of vectors v with bounded absolute values I v I or
bounded norms v2 = Nv. Moreover, if for a sequence of vectors
V1, V2, • · • the relation lim,:::: N(vn - Vm) = 0 holds, then a
limit vector v = limn-+co Vn exists; finally, limn . . . co Nvn = 0 implies
limn. . . co Vn = 0. These statements are not, however, generally valid
in function space, i.e. in a space of infinitely many dimensions. Thus
it is not possible to select a convergent sequence of functions from
every infinite set of continuous functions f(x) with bounded norm,
and from the relation limn-+co Nfn = 0 for a sequence of continuous
This follows from Dini's theorem: If a series of positive continuous funcconverges to a continuous function S(t) in a closed domain G,
its convergence is uniform. A brief sketch of the proof follows: Let us write
Sn(t) = ~~- 1 fv(t); S(t) = Sn(t) + Rn(t). If our assertion were false there
would exist a positive number a, a set of numbers n1 , 1L2 , • • • tending to infinity, and associated values t1 , t2 , · • • in G such that Rni(ti) ~ a, i.e.
Sni(ti) ~ S(ti) - a. We may assume that the values ti converge to a limit t
which is in G. Now let N be a fixed integer; then, for
~ N, we
have Sn,(ti) ~ SN(ti), and thus SN(ti) ~ S(ti) - a. We now let i increase beyond all bounds and obtain, because of the assumed continuity of all functions, SN(t) ~ S(t) - a, which is impossible for sufficiently large N.
1
tion~ ~;_ 1 fv(t)
n,
58
II. SERIES EXPANSIONS OF ARBITRARY !''UNCTIONS
functions it does not follow that limn-oac f n = 0.
ple, the functions
2 2
fn(X) = 1 - n X
fn(X)
=
in the interval -1
to the function
for x ~ 1/n ,
2
2
~
=
2
for x ~ 1/n
0
x
~
+1.
2
Every subset of this set converges
f(x) = 0
f(x)
Consider, for exam-
1
for x ¢ 0
for x
=
0,
which is discontinuous at x = 0, even though limn-ooo Nfn = 0.
Nevertheless, to carry through the analogy between vectors and
functions, i.e. to preserve, in function space, both Weierstrass's principle of accumulation and the above convergence theorems, is a task
that is essential for many purposes, especially for proofs of convergence and existence. There are two ways to approach this problem:
First, the set of admissible functions may be widened.by changing the
concepts of integral and convergence; this method is used in the
theory of Lebesgue but is not necessary for this book and will not be
employed here. 1 The other approach, which we shall adopt, is to
restrict the admissible functions to a set for which the principle of
convergence is valid. The restriction we shall impose, beyond the
requirement that each admissible function be continuous, is that the
set of all admissible functions be equicontinuous.
Suppose we are dealing with functions of one independent variable x. The requirement of equicontinuity means that, for every
positive E, there exists a positive number o = o(E), depending only
on E and not on the particular function f (x) of the set, such that, if
I X1 - X2 I < o(E), then I f(x1) - f(x2) I < E, provided X1 and x2 are
within the postulated range of the independent variable. For example, all functionsj(x) for which { j'2(x) dx
~
M, where Mis a fixed
constant, form an equicontinuous set of functions in the interval
a ~ x ~ b. Indeed, for any two values X1 and X2 in this interval,
1
See, however, §10, 11, of this chapter.
ACCUMULATION PRINCIPLE FOR FUNCTIONS
59
therefore, by the Schwarz inequality,
From this inequality we see that the condition for equicontinuity is
satisfied for o(E) = i/M.
For equicontinuous sets of functions the principle of accumulation
is valid: From any set of functions which are uniformly bounded and
equicontinuous in the fundamental domain G, it is possible to se"lect a
sequence qi(x), q2(x), · · · which converges uniformly to a continuous
1
limit function in the domain G. This theorem (Theorem of Arzela)
for sets of continuous functions is analogous to Weierstrass's theorem
on bounded point sets; it therefore fulfills our requirement.
To prove the theorem we consider a denumerable set of points
X1 , X2 , • • • , which is everywhere dense in the interval, for example
the set obtained by successive bisections of the interval and the
resulting subintervals. The set ~f values of the functions at x1 contains as a subset a convergent sequence; this follows from Weierstrass's theorem. Therefore it is possible to select from the given
set of functions an infinite sequence of functions a1(x), a2(x), · · ·
whose values at the point X1 form a convergent sequence. In the
same way we can select from this sequence a subsequence bi(x),
b2 (x), · · · which converges at X2, and so on. We now consider the
"diagonal sequence" a 1 (x) = qi(x), b2(x) = q2(x), · · · and assert that
it converges uniformly in the entire interval.
To show this we select an arbitrarily small positive number E and
a number M so large that for every point x of the interval there
exists a point xh with h ~ M for which I x - Xh I ~ ~( E), ~( E) being
the number defined above in the definition of equicontinuity. We
now choose a number N = N(E), depending only on E, so great that,
for m > N, n > N,
I q,,.(XA)
- qn(XA)
I<
E
(h = 1, 2, • • • , M).
1 It is actually sufficient to assume uniform boundedness for a single point
of G; because of the equicontinuity, the functions would then be uniformly
bounded over the entire domain G.
60
II. SERIES EXPANSIONS OF ARBITRARY FUNCTIONS
Because of the equicontinuity of the functions we now have, for some
particular value of h ~ M,
therefore, for m
>
N,
I qm(X) I qn(X) n > N,
I qm(X) -
I<
qn(Xh) I <
qm(Xh)
qn(X)
I<
E,
E;
3e.
This proves the uniform convergence of the sequence of functions
q1 (x), q2(x), · · · for all x in the interval a ~ x ~ b. The continuity
of the limit function q(x) follows immediately from the uniformity of
the convergence. It may be remarked incidentally that these considerations also show that every convergent subsequence converges
uniformly.
A set of equicontinuous functions posseses, moreover, the following
properties: If the sequence of functions fi(x), f2(x), · · · belongs to an
equicontinuous set and if limn-+oo Nfn = 0, then limn-+oo fn = 0. Furthermore, if fn is bounded and if lim;:::: N(fn - f m) = 0, there exists
a continuous function f(x) such that limn-+oo fn(x) = f(x). In both cases
the convergence to the limit is uniform.
To prove the first assertion we assume that limn-+oo fn(Xo) '¢ 0 for
some xo . Then there exist arbitrarily large values of n such that
f!(xo) > 2ci, where 2cl is some positive bound. Because of the
equicontinuity of the functions fn(x) there must then exist a fixed
interval of width o, containing the point x 0 , such that, within this
interval, f ! > ci for the above values of n. Therefore Nfn > Oa. 2 ,
in contradiction to the premise that limn ..... oo Nfn = 0. The proof of
the second part of our assertion is similar; it may be carried out by
the reader.
Another property of an equicontinuous set of functions with
bounded norms is the following, which will be termed the smoothness1
of the set: Let r be a positive integer, and let c1 , c2 , • · · , Cr be arbitrary numbers whose absolute values lie below a fixed bound, say I.
Then there exists a number o( e) depending only on r and the positive
number e and tending to zero with e, such that the relation
I cif1 + C2f"2 +
••• +
Crfr
I<o
1 This concept, which relates to sets of
functions, is not to be confused with
the concept of a "smooth function" introduced at the beginning of this chapter.
MEASURE OF INDEPENDENCE AND NUMBER OF DIMENSIONS
61
follows from the relation N(cJ1 + cJ2 + · · · + crfr) < t, if f1 ,f2, · · · ,fr
are any r functions of the set.
This follows from the theorems just proved if we note that the
functions of our set remain equicontinuous when the set is extended
+ crfr with
cJ2
by including all linear combinations cif1
fixed r and bounded I Ci I .
The smoothness of the sequence of functions f1, f2, · · · may also
be expressed in the fol1owing way: The sequence has the property
that from each subsequence a uniformly convergent subsequence can
be selected.
The following somewhat more general theorem may easily be
derived from the principle of accumulation: Let
+
+ ·· ·
pu(x), P12(x), · · · , P1r(x),
P21(x), P22(x), · · · , P2r(x),
be a sequence of sets G1 , G2 , · · · of r functions each, such that all of
the functions are equicontinuous and uniformly bounded in the interval
a S x S b. Then a subsequence Gn 1 , Gn 2 , • • • may be selected such
that the functions Pn,,k(x) (k = 1, 2, · · · , r) converge uniformly to r
continuous functions Pi(x), p2(x), · · · , Pr(x) as i increases.
Selecting a suitable subsequence, the desired convergence may
indeed be obtained for the functions of the first column; from this
subsequence a new subsequence can be selected such that the second
column also converges. This process is then repeated r - 2 more
times.
§3. Measure of Independence and Dimension Number
I. Measure of Independence. A simple criterion for the linear
dependence or independence of r functions f 1 ' f 2 ' • • • ' fr follows by
a procedure analogous to that employed for vectors inn-dimensional
space. We begin by considering the quadratic form in r real variables t1 , f2 ,. • • , t'"
K(t, t) = N(tif1
(12)
+ · · • + trfr) = f (tif1 + ·· · + trfr) 2 dx
f'
=
I:
i;
lc-1
(fif1c) ti t1c •
62
II. SERIES EXPANSIONS OF ARBITRARY FUNCTIONS
Since K(t, t) is positive definite, its lowest eigenvalue m (i.e. the
minimum value of K when the ti are varied subject to the restriction
L:~it~ = 1) is certainly not negative. We call this number the
"measure of independence" of the functions fi , f2 , · · · , fr. The
functions f 1 , f2 , · · · , fr are evidently linearly dependent if and only
if the measure of independence m is equal to zero. In the case of
linear independence m indicates the "degree of independence".
The vanishing of the Gram determinant
(fif1)
(13)
r(f1' ... 'fr) =
CJ1/r)
................ .
CJr/1)
CJr/r)
of the system of functions f 1 , f2 , · · · , fr is equivalent to the vanishing
of its measure of independence. This follows from the fact that the
Gram determinant is the product of all the eigenvalues of K. Since,
moreover, the eigenvalues are non-negative, we know that
mr ~ r ~ mMr- 1 , where M is the greatest eigenvalue of K, and
r ~ 0. 1 Thus, the vanishing of the Gram determinant is a necessary
and sufficient condition for the linear dependence of the functions
f1 'f2' '' • , fr•
If we form a linear combination f = L:~ 1 udi of r linearly independent functions f1 , f2 , · · · , fr and if, moreover, f is normalized,
we find that none of the coefficients ui can exceed in absolute value the
bound 1/V m, which depends only on the measure of independence.
For, if we write
we evidently have from the definition of m
therefore L:i..1 u~ ~ l/m. If a system of r functions whose measure
of independence lies above the positive boundµ is orthogonalized, i.e. if
the functions of the set are replaced by suitable normalized linear com1
The latter inequality is a generalization of the Schwarz inequality; in fact
it is the Schwarz inequality if r = 2.
MEASURE OF INDEPENDENCE AND NUMBER OF DIMENSIONS
63
binations of the functions, no coejfici,ents with absolute values in excess
of 1/,YP can occur.
2. Asymptotic Dimension of a Sequence of Functions. The sequence
of normalized functions (or, more generally, functions of bounded
norm) f1, f2, • • • is said to be exactly r-dimensional if any r + 1
functions of the sequence are linearly dependent, and at the same
time there exists at least one set of r functions of the sequence which
are linearly independent. In this case every function of the sequence
may be written as a linear combination t1Y1 + Lty2 + · · · + trYr with
constant coefficients t1 , Lt , • • • , tr of r basis functions Yi , Y2 , · · • , Yr;
the entire sequence of functions is composed of members of the
"linear space" (or "linear family") t1g1 + Lty2 + · · · + trgr •
If the sequence of functions f 1 , f 2 , • • • is not of finite dimension,
two possibilities exist: 1) For every arbitrarily large positive integer
s there exist sets of s functions fn 1 ,fn2 , • • • , f n. with arbitrarily large
indices n1 , nt , · · · , n, such that the measure of independence of these
functions lies above a fixed bound, independent of the indices ni but
possibly depending on s. In this case we ascribe the asymptotic
1
dimension oo to the sequence of functions. 2) For sufficiently
large s the measure of independence of fn 1 , • • • , fn. converges to
zero if all the indices n 1 , nt , · · · , n, increase beyond all bounds. Here
the smallest integer r for which the measure of independence tends
to zero for s > r is said to be the asymptotic dimension of the sequence.
In particular, r = 0 if Nfn tends to zero with increasing n. If the
asymptotic dimension of a sequence is r and if sufficiently many
functions at the beginning of the sequence are omitted, then any
r + 1 of the remaining functions are "almost" linearly dependent.
This terminology, suggested by the analogy to vectors in n-dimensional space, would be meaningful if the functions sufficiently
far out in a sequence of asymptotic dimension r could be approximated with prescribed accuracy by the functions of a linear space
formed from r basis functions. In general, however, this is not true
unless the concepts of function and of integral are extended according to the theory of Lebesgue. Since we wish to remain in the realm
of our more elementary theory, we shall prove the above assertion
by making certain restrictive assumptions as discussed in §2. In
fact, we shall simply assume that the sequence of functions is smooth.
1 The simplest example of a sequence with infinite dimension is a sequence
of orthonormal functions for which the measure of independence of every
nuite subset has the valqe 1,
64
II. SERIES EXPANSIONS OF ARBITRARY FUNCTIONS
In this case, the following theorem holds: If f 1 , f 2 , • • • is a smooth
sequence of functions with asymptotic dimension r, then there exist r
linearly independent functions (which may be taken to be orthonormal)
gi , g2 , · · · , gr such that, for sufficiently large n, each function f n difJers from a function of the linear space t1g1
t2g2
trUr by less
than an arbitrarily small positive quantity E, while no space with fewer
than r basis functions has this property.
This linear space may also be characterized in the following way:
Let Gi , G2 , · · · , Gm , · · · be sets of r functions f mi , f m 2 , • • • , f m,
of the sequence, such that the measure of independence of each set
lies above a fixed positive boundµ and the indices mi (i = 1, · · · , r)
increase beyond all bounds with increasing m. Then the linear
spaces of functions Sm whose basis functions are the functions of
Gm converge, with increasing m, uniformly to a limiting space T,
defined by r linearly independent functions gi , g2 , · · · , gr . By
"converge" we mean here that, for sufficiently large m, every normalized function of Sm differs arbitrarily little from a function of T.
To formulate the proof conveniently, we introduce the concept of
distance bet\veen a function and a space. We say that the distance
between a function f and the linear space of functions S is less than
the positive quantity d, if there is a function of S such that th2
absolute value of the difference between this function and f is everywhere less than d. Similarly we ascribe a distance less than d to two
linear spaces of functions S and S* if every normalized function of
one space differs by an amount whose absolute value is less than d
from some normalized function of the other space.
It now follows immediately that, for sufficiently large m and n,
the distance between the function fn and the space Sm is arbitrarily
small. For, the measure of independence of fn, f mi, · · · , f m,' is
certainly arbitrarily small for sufficiently large m and n. Since our
sequence of functions is smooth, there exist r + 1 numbers
Uo, 'U,1, • • • , Ur, L~=o u~ = 1, for which l Uofn + uifmi + · · · +urfm, I
is arbitrarily small. Note that the absolute value of uo cannot become arbitrarily small with increasing m and n, since otherwise the
measure of independence of f mi ' f m2 ' • • • ' f mr would become arbitrarily small, in contradiction to our assumption that the measure
of independence lies above the bound µ. We may therefore divide
ulfn + UJmi + · · · + Urf mr by Uo, putting Ui/Uo = -ti and Con-
+
+ ·· · +
65
WEIERSTRASS'S APPROXIMATION THEOREM
elude that, for sufficiently large n and m, the function fn differs arbitrarily little from a suitable function tdmi + tdm2 + . . .
trfmr of
the linear space Sm . Thus, for sufficiently large m and n, the distance
between the spaces Sn and Sm is also arbitrarily small. Now, let E
be a positive number, later to be taken sufficiently small, and let
E1 , E2 , · · · be a sequence of positive numbers with L:t.1 Ei = E.
Furthermore, let mi be a positive integer such that, for n > mi and
m > mi, the distance between Sn and Sm is less than Ei. Let us
now take any r normalized functions hn , h12 , · · · , h1r of the space
Sm 1 and determine the normalized functions h21 , h22, · · · , h2r of the
space Sm 2 (m2 > mi) such that I h2i - hli I < E1 (this is possible
because the distance between Sm 1 and Sm 2 is less than E1). In the
same way we determine a set of normalized functions h31, h32, · · · ,h3r
of the space Sm 3 (m3 > m2) such that I h3i - h2i I < E2 , and so
Eq-1 (p < q) the sequence
forth. Since ! hpi - hqi I < Ep
of functions hni (i = 1, · · · , r) with any fixed i converges uniformly
to a limit function gi , and I Ui - h1i I < E. If Eis chosen sufficiently
small, the functions U1 , g2, · · · , Ur will have a nonvanishing measure
of independence if the functions h11 , h12 , · · · , h1r do; thus the gi
are linearly independent. These functions Yi , Y2 , · · · , Yr evidently
fulfill all our requirements.
+
+ · ·· +
§4.
Weierstrass's Approximation Theorem. Completeness of
Powers and of Trigonometric Functions
1. Weierstrass's Approximation Theorem. The most elementary
example of a complete system of functions is given by the powers
They form a complete system of functions in every closed interval
a :::; x :::; b; in fact, the following approximation theorem of Weierstrass1 holds: Any function which is continuous in the interval a :::; x :::; b
may be approximated uniformly by polynomials in this interval.
This theorem asserts more than completeness; it asserts the possibility of uniform convergence rather than just convergence in the
mean.
To prove this, we assume that the interval a :::; x ~ b lies wholly
1 K. T. W. Weierstrass, Uber die analytische Darstellbarkeit sogenannter
willkurlicher Funktionen reeller Argumente, Sitzungsber. K. Akad. Wiss.
Berlin, 1885, pp. 633-639, 789-805; Werke, Vol. 3, pp. 1-37 1 Berlin, 1903.
66
II. SERIES EXPANSIONS OF ARBITRARY FUNCTIONS
in the interior of the interval 0 < x < 1; thus, two numbers a and (3
may be found with 0 < a < a < b < (3 < 1. We may suppose
that the function f(x), which is by assumption continuous in the
interval a ~ x ~ b, has been extended continuous1y to the entire
interval a ~ x ~ (3.
Let us now consider the integral
1
1
J,.
=
2
(1 - v )" dv.
We see immediately that J" converges to zero with increasing n.
Now, if ois a fixed number in the interval 0 < o < 1, and if we set
J!
=
i
1
(1 - v2)" dv,
we assert that
J! - 0
lim
,._.. J,.- - ,
which means that for sufficiently large n the integral from 0 to o
forms the dominant part of the whole integral from 0 to 1. In fact,
for n ~ 1,
J"
>
J! =
1
n
(1
Jo (1 - v) dv = n
1,
+
f,
1
2
(1 - v )" dv
< (n + 1)(1 J:
Jn
<
(1 -
o2)"(1
-
o) <
o2)",
(1 -
o2)"
and hence
. J!
1un
- -- 0
fl-+00
J,.
.
We now assume a
~ x ~
J:
b and form the expressions
f(u)[l - (u - x) 2]" du
Pn(x) = ----,-1- - - - - -
Ll o - u2r
d'Ut
<n
=
1 2 .. ·)
1
'
'
67
WEIERSTRASS'S APPROXIMATION THEOREM
which are polynomials in x of degree 2n whose coefficients are quotients of definite integrals. We shall .show that they afford the
desired approximation.
By making the substitution u = v + x we find for the numerator
11
1
a f(u)[l -
(u - x)
r du
2
ffJ-:J:
a-:J: f(v
=
=
+ x)[l
- v
2
]"
dv
1-· + 1' +alfJ-:;e
-1
a-:r:
+ I2 + 13,
= I1
where the positive number o in the interval 0 < o < 1 will be suitably fixed later. The integral 12 may be transformed to
12
=
2
(1 - v
f(x) {
1
= 2f(x)(J n
J!)
-
r dv + {, [f(v + x) + {, [f(v + x)
2
f(x)](l - v
- f(x)](l - v
2
r dv
dv.
)"
Because of the uniform continuity of f(x) in the interval a ~ x ~ {3
it is possible, for arbitrarily small E > 0, to choose a o = o( E) in the
interval 0 < o < 1, depending only on E, such that, for I v I ~ oand
a ~ x ~ b, 1 f(v + x) - f(x) 1 ~ E. It then follows that
I£:
[f(v
+ x)
- f(x)](l - v')• dv
I :5:• £:
(1 - v')" dv
<
E {
=
2EJn.
2
(1 - v
)"
dv
1
Furthermore, if M is the maximum of I f(x) ! for a ~ x ~ {3 we obtain
I 11 I < M
[ ' (1 - v
I ls I < M
{
r dv =
2
r
2
(1 - v
dv
MJ:,
= MJ:.
Therefore, since the denominator in P n(x) is equal to 2J n
I p n (x)
- f(x)
I < 2M
J: +
J*
E.
,
68
II. SERIES EXPANSIONS OF ARBITRARY FUNCTIONS
Since lim 11...."° (J!/ J 11 ) = 0, the right side may be made less than 2e
by a suitable choice of n; thus f(x) is indeed approximated by P n(x)
uniformly in the interval a ~ x ~ b.
.
2. Extension to Functions of Several Variables. It may be shown
in exactly the same manner that a function f of m variables
Xi, X2, • • • , Xm, which is continuous for ai ~ Xi~ bi (i = 1, 2, · · · , m;
0 < ai < ai < bi < /3i < 1), may be approximated uniformly by
the polynomials
[f,
(1 - u')' du)
J
3. Simultaneous Approximation of Derivatives. A similar argument
leads to the following general result: A function f (xi , x2 , · · · , Xm)
which, along with its derivatives up to the k-th order, is continuous
in the closed region ai ~ Xi ~ bi ·' may be uniformly approximated
by polynomials P(xi , x2 , · · · , Xm) in such a way that the derivatives off up to the k-th order are also approximated uniformly by
the corresponding derivatives of the polynomials.
To prove this, we again assume that 0 < ai < bi < l and that
the function and its derivatives are extended continuously into a
larger rectangular region ai ~ Xi < /3i (O < ai < ai < bi < /3i < 1)
in such a way that the function and its derivatives up to the (k - 1)-st
order vanish on the boundary of the larger region. Then the polynomials p n(X1 , X2 , • • • , Xm) defined in the previous subsection yield
the desired approximation. This may be shown very simply by differentiating with respect to Xi under the integral sign, replacing this
differentiation by the equivalent differentiation with respect to ui,
and finally transforming the integral by integration by parts, making
use of the assumed boundary conditions.
4. Completeness of the Trigonometric Functions. From § 4, 1 we
can deduce the important fact that the orthonormal system of
trigonometric functions
(14)
1
y"2;'
cos x
sin x
y;'
y';'
cos 2x
y;'
sin 2x
.y;'
69
FOURIER SERIES
s s
is complete in the interval -7r
x
7r.
The following more inclusive theorem can also be proved: Every function f(x) which is
continuous in the interval -'Ir
x S 7r and for which f(-7r) = f(7r)
may be approximated uniformly by trigonometric polynomials
s
~0 +~(a,, cos vx + 13,, sin vx)
where a,, and 13,, are constants.
To prove this, we write () instead of :t and consider a ~' 11-plane
with the polar coordinates p and O = p cos 0, 11 = -p sin 0). The
function
a
i.o(~,
17)
=
pf(O)
is then continuous in the. entire ~' 17-plane and coincides with the
2
given function f(O) on the unit circle ~2
17 = 1. According to
Weierstrass's approximation theorem it may be approximated uniformly by polynomials in ~ and 11 in a square containing the unit
circle. If we then set p = 1, we see that f (0) may be approximated
uniformly by polynomials in cos () and sin 8. But, by well-known
formulas of trigonometry, every such polynomial may also be written
in the above form
+
~o + ~ (a" cos vx + 13,, sin vx).
A continuous function f(x) which does not satisfy the periodicity
condition f( -7r) = f(7r) may be replaced by a continuous function
g(x) satisfying this condition, in such a way that the integral
L:
2
(f(x) - g(x)) dx is arbitrarily small.
From this it foHows that
every continuous function can be approximated in the mean by
trigonometric polynomials, and therefore that the trigonometric
functions form a complete set.
§5. Fourier Series
1. Proof of the Fundamental Theorem. It follows from the considerations of §1 and from the orthogonality of the trigonometric
functions that the best approximation in the mean of degree n is
obtained by the so-called Fourier polynomial
n
sn(x) =
! ao
+L
1
(a,, cos vx
+ b,, sin vx),
70
JI. SERIES EXPANSIONS OF ARBITRARY FUNCTIONS
with
a.
= 1111'
f(x) cos vx dx
(v = 1, 2, · · · , n)
ill' f(x) sin vx dx
(v = 1, 2, · · · , n)
1r
b. = -1
(15)
-r
1r
-r
1111' f(x) dx.
ao = -
1r
-r
This polynomial may also be written in the more concise form
!2a. = a. - ib,.,
2ao = ao,
"
.L
sn(x)
•-n
a.,
(15')
a,. ein
2a., = a.,
= _!__ 1,..
2r
+ ib.,,
f(t) e-il't dt
(v
11
>
v
< 0,
0,
= 0, ±1, ±2 · ·-)
-r
+
i sin vx = e il':i:.
in virtue of the relation cos vx
It is not a priori certain that these polynomials, which yield the
b~st approximation in the mean, also yield a uniform approximation
to the function-Le. it is not certain that the infinite series limn-+«' Bn(x)
converges uniformly and represents the functionf(x). This question
is the central problem of the theory of Fourier series.
For the sake of convenience we shall suppose that the function f(x)
is initially defined only in the interval - r < x < r, and then continued periodically beyond this interval by the functional equation
2r) = f(x). Furthermore, at each jump discontinuity, we
f (x
require f(x) to be equal to the arithmetic mean of the "righth) and
0) = limh...o f(x
hand" and "left-hand" limits, f(x
f(x - O) = limh-+O f(x - h) (h ·> O), respectively; i.e. we set
f(x - 0)).
0)
f(x) = lff(x
The following theorem then holds: Every function which is piecewi8e smooth in the interval - r :::; x :::; r and periodic with the period
2r may be expanded in a Fourier series; that is, the Fourier polynomials
+
+
+
+
+
ft
sn(x) =
! ao
+ I: (a., cos vx + b., sin vx)
ir-1
converge to f(x) with increasing n.
Moreover, we shall prove: The
71
l'OURIER SERIES
convergence of the Fourier series is uniform in every dosed interval in
which the function is continuous.
We shall first give the proof for the case of continuous f (x) in which
discontinuities occur only in the derivative f'(x). If the expansion
coefficients of f'(x) are denoted by a" and {j", we have
a,,
=
{j" =
!
T
!
T
ao
1,.. J'(x) cos vx dx = ~ lir f(x) sin PX dx = vb",
T
-T
-T
lir-ir J'(x) sin 11x dx = -~ 111" f(x) cos 11x dx = -11a11,
T
-T
= 0.
Since f'(x) is piecewise continuous, we have the completeness relation
Thus
(a,, cos 11x + b,, sin 11x) I = I:f: ! (11a,, cos 11x + vb,, sin 11x) I
I:f:
-""
-"
~It. ·'<a!+ b!).
v'tJ
~ . I! 111" f'2 dx . • It ; .
. 11 "-""
11 T -ir
This immediately estabJishes the absolute and uniform convergence
of the infinite series
QO
lim sn(x} =
n-ao
! ao +
L
(a,, cos 11x
+
b,, sin PX),
-1
which represents the function f(x), because of the completeness of the
trigonometric functions.
To verify that the Fourier series expansion is also valid for functions which are discontinuous but piecewise smooth, we start by
considering a particular function of this type, defined by
· h(x)
= !(T -
h(O) = 0,
h(x
+ 2T)
= h(x).
x),
0
< x < 2...,
72
II. SERIES EXPANSIONS OF ARBITRARY FUNCTIONS
This function jumps by 7r at the points x = ±2k?r (k = 0, 1, · · · ).
Its Fourier coefficients are
ao
o,
=
1
b,, = -
a,,= 0,
(v
11
=
1, 2, · · · ).
If the Fourier expansion is valid for piecewise continuous functions
we have
=
h(x)
£ sin vx.
,_I
JI
To justify this equation we first form the function
g(x)
=
h(x) (I - cos x)
2h(x) sin ~,
=
2
which is everywhere continuous and piecewise smooth. By the
above reasoning, the Fourier series _L:':..1 {3,, sin vx of this function
converges uniformly and is equal to g(x). The coefficients (3,, are
related to the b,, by the equations
{3,,
=
b,, - !(b,,_1
+ b,,+i)
(v = 2, 3, · · · ),
/31 = b1 - ib2 .
If we set
.L::-1 b,, sin
vx = sn(x) and
_L::=1 {3,, sin
(1 - cos x)sn(x) = <rn(X) - ibn sin (n
vx = u,,(x) we have
+ l)x + !bn+l ~in nx.
With increasing n, bn converges to zero and the sum un(x) converges
uniformly to g(x). It follows that (1 - cos x)sn(x) also converges
uniformly to g(x) in the interval -?r ::; x ::; ?r, and therefore that
sn(x) itself converges uniformly to h(x) in every closed subinterval
not including the point x = 0.
At this excluded point x = 0 all partial sums Sn vanish, so that
limn.... 00 sn(O) = 0. Thus the value of the series at the point of discontinuity is also equal to the value of h(x), namely to the arithmetic
mean of the right-hand and left-hand limits +7r/2 and -?r/2.
As the function h(x) jumps by 7r at x = 0, the function h(x - ~)
jumps by 7r at x = eand is otherwise continuous in the basic interval.
Now, if j(x) is a piecewise continuous function which jumps by
s(~ ..) = j(~i + 0) - f(~ .. - O) at the points x = ~ .. (i = 1, · · ·, r)
of the interval 0 ::; x < 27r, then
F(x)
= f(x)
-
:t s(~i)
i-1
7r
h(x -
~i)
73
FOURIER SERIES
is a function which is everywhere continuous and which, along with
f(x), has a piecewise continuous first derivative. Therefore, F(x)
may be expanded in a Fourier series which converges absolutely and
uniformly. But the function
may also be expanded in a Fourier series, the convergence of which
is uniform in every i~terval not containing any of the points of discontinuity. Thus the theorem formulated at the beginning of this
section is proved completely.
2. Multiple Fourier Series. The trigonometric functions can be
used to construct orthogonal systems over "cubes" in higher dimensional spaces. For simplicity we consider only the "plane" case
(two dimensions). All our arguments wi11, however, be valid for
any number of dimensions.
The functions
=
0, 1, ..
~
; v = 0, 1, ... ),
cos
µs
cos vt
(µ
sin
µs
cos· Pt
(µ = 1, 2, ... ; v = 0, 1, ... ),
cos
µs
sin vt
(µ = O, 1,
; v = 1, 2, ... ),
sin
µs
sin vt
(µ = 1, 2,
; v = 1, 2, ... )
form an orthogonal system in the square 0 ~ s ~ 271", 0 ~ t ~ 271".
The expansion formulas may be written most simply in terms of
complex exponentials. If F(s, t) can be expanded in a uniformly
convergent double Fourier serie;, the series is
00
00
L L
F(s, t) =
a"" ei<1£a+m,
11-00 P-oo
with
1
a"JI' = 4- 2
71"
10
2ll' d 1211' dt F(
s
0
s, l) e-i(µa+vt) .
The completeness of the system of functions and, thus, the completeness relation
oo
"·~
00
Ia,..1
2
=
12r
0
2r
1I
0
F(s,
t)l2 ds dt
74
II. SERIES EXPANSIONS OF ARBITRARY FUNCTIONS
follow by our general theorem on the construction of complete systems in several variables from complete systems in one variable (see
page 56).
Furthermore, the Fourier series for F(s, t) converges absolutely
and uniformly if iJ2F(s, t)/iJs at exists and is piecewise continuous,(cf.
subsection 1).
3. Order of Magnitude of Fourier Coefficients. If a periodic function f(x) is continuous and has continuous derivatives up to the
(h - 1)-st order, if its h-th derivative is piecewise continuous, and
if the Fourier expansion of f(x) is f(x) = L:':..-oo a,,ein:, then the
coefficients a,,, for I J1 I ~ 1, can be estimated by
Va~
2 I a,. I =
+ b~ =:s;
~,
JI
where c is a constant. Thus, the smoother the function, the more
rapidly the coefficients of the series tend to zero.
The above relation may be obtained immediately if expression
(15') for the coefficients is integrated-by parts h times.
4. Change in Length of Basic Interval. If the function f(x) is
periodic with period 2l it may be expanded in the series
f(x)
= !ao +
with
- a,.
t. (a,
f
+ b, sin v j z)
coSP x
1 fH
= 1: .1o 1<t) cos JI it dt,
b, =
l1 121 f(t)
0
r
sin v l t dt,
which may also be written in the form
ao
i•(tr/l)s
~
f( x) -_ £...J
,
a,. e
a.
=~
1"
f(t)
e-"'Ctr/
°' dt.
5. Examples. For simple examples of the application of the
theory of Fourier series the reader is referred to elementary texts. 1
1
For example R. Courant, Differential and Integral Calculus, Vol. 1, pp.
440-446, Blackie and Son, Ltd., London and Glasgow, 1934; 2nd ed. rev., Interscience Publishers, Inc., New York, 1937.
75
FOURlER SERIES
Here we shall employ the Fourier series expansion to derive the
functional equation of the theta function and ~ general formula by
Poisson.
The functional equation of the theta function
00
L: e-rµ2s
= ,._
O(x)
(x
>
O)
lS
8(z)
)x 8 G).
=
To prove this relation, we set
00
L:
'P(y) =
e-r(µ+tf)2s;
11-00
'P(Y) is evidently a periodic function of y with the period 1, which
possesses all derivatives with respect to y and which may therefore
be expanded in the Fourier series
00
I:
=
'P(y}
a.,,e2ri1111
with
Since the orders of summation and integration may be interchanged
for all x > 0, we obtain for the coefficients a.,, :
=
=
for,
1:
1:
e-rt2z-2ri•t
e-(rv2/z)
l
dt
oo
e-n(t+(iv/z))2
-oo
112
e- dz has the same value
dt
=
e-r•2 /:i:
Vx
v; along a line dm t
= v/x parallel
76
II. SERIES EXPANSIONS OF ARBITRARY FUNCTIONS
to the real axis as along the real axis itself. (If the Cauchy theorem
is applied to the function e-z'I. and to the rectangle with vertices -at
-T, +T, +T + iv/x, -T + iv/x, and if Tis then made to tend to
infinity, the integrals over the vertical sides of the rectangle converge
to zero; for, the integrand converges uniformly to zero, while the
length ·of the path of integration is constant, equal to v/x.) Thus
we have
__1_ ~ -rv2/z 2rivu
( ) -_r-_£..Je
cpy
e
v x "-00
and in particular, for y
=
0,
Here the Fourier expansion has been applied to the transformation
of infinite series in one special case. The method thus exempli6ed
has recently proved very useful in the treatment of certain analytic
functions arising in number theory.
This method is based on a very important transformation formula
for infinite series, known as Poisson's summation formula. Let
:E':~oo cp(27rn) be an infinite series in which cp(x) is a continuous
and continuously differentiable function of x such that the series
E~~oo cp(27rn + t) and L~~oo 'P 1 (27rn + t) converge absolutely and
uniformly for all tin the interval 0 ~ t < 211". The second series is
then the derivative of the first, which may therefore be expanded in
a convergent Fourier series in the interval 0 ~ t < 211":
The sum over n in the last expression may be transformed as follows:
77
THE FOURIER INTEGRAL
Therefore
00
J;QQ <P(2rn +
00
i·Loo eivt Loo <P(r) e
1
t) = 11'"
2
-iPT'
dr.
00
= 0, we obtain
If, finally, we sett
l
J;ao <P(2rn) =
oo
11'"
2
oo
,,L
lco <P(r) e_,.,.,.. dr.
_
00
00
This is Poisson's formula. Clearly it is valid if all the integrals
occurring in the formula exist, if L~~oo <P(27rn + t) converges uniformly in t for 0 ~ t < 211", and if this series represents a function
which can be expanded in a Fourier series.
§6.
The Fourier Integral
Consider a function f(x) which is
1. The Fundamental Theorem.
represented by a Fourier series
l
00
f(x) =
L
a,, ei.,,(r/l):i:,
ot..,
=
.!_ 1 f(t) e-iv(r/ l)e dt
2l
•-oo
-l
in the interval -l < x < l. It seems desirable to let l go to oo ,
since then it is no longer necessary to require that f be continued
periodically; thus one may hope to obtain a representation for a nonperiodic function defined for all real x. We shall continue to assume
that f(x) is piecewise smooth in every finite interval and that, at
the discontinuities, the value of the function is the arithmetic mean
of the right-hand and left-hand limits. We now add the further
assumption that the integral
If we set
11'"
/l =
~'
J~ I f (x) I dx exists.
we obtain
f ~ 1 f(t)
211"
1
f(x) = _!_
-00
Letting l
~ oo ,
and therefore
~ ~
f(x) = _!_
2r
00
du
oo
dt.
0, we find that the formula
L L
00
(16)
e-i•Ht-:i:>
l
f(t)
e-iu(t-:i:)
dt
oo
appears plausible; it is correct if the passage to the limit can be
78
II. SERIES EXPANSIONS OF ARBITRARY FUNCTIONS
justified. For real functions f(x) this may also be written in the
form
1
f(x) = -
(17)
'Ir
1'° 1 f(t) cos u(t 00
du
0
x) dt.
00
A strict proof of the validity of this "Fourier integral formula" is
more easily obtained by a direct verification of equation (16) or (17)
than by a justification of the passage to the limit.
We start with the integral formula
. -1
hm
l"
'If'
-G
11-00
) -vt dt = ![f(x + 0) + f(x
+ tsin
t
f(x
- O)]
= f(x),
where a is an arbitrary positive number. This formula, which is
1
valid for every piecewise smooth function, is due to Dirichlet.
From this formula it follows that
11
1rf(x)
=
lim [
f(x
•
11-00
=
=
to
+ t) dt .lo(" cos ut du
1· f"
lim
du
•-oo
0
1cx
a
+ t)
cos ut dt
L'° du L: f(x + t) cos ut dt.
We assert that the integration over t may be extended from oo. For, if A > a, we have
oo
1 LA - 1" lG = 1" i-G + 1" JG(A l-G 1" + JGr 1"
=
v
f:
If L: - f 1: I~ Ic
0
A
whence, since
0
G
0
I /(x) I dx
-A
= C
f(x
..t
0
0
exists by hypothesis,
I I(
+ t)si~ •t dt +
~
HC
~
-a1 Loooo I f(x + t) I dt = -ac .
lf(x
0
+tl I dt
+{
f(x
+ t)"~ ot dt
I
Jf(x +t) I dt)
1 This formula is usually employed as the foundation of the theory of Fourier
series as well. For a proof see textbooks on the calculus, e.g. Courant, Differential and Integral Calculus, Vol. 1, p. 450.
THE FOURIER INTEGRAL
79
If we now keep v fixed and let A tend to infinity, it foJlows that
and, passing to the limit v --+-
oo ,
we obtain
fun 1" lco I·-co
co
7rf(x)
0
I~ a~.
The right-hand side may be made arbitrarily small by a suitable
choice of a, and thus the desired formula (17) is proved.
Since
1:
f(t) cos u(t - x) dt is an even function of u, the above
equation may also be written in the form
7rf(x) =
On the other hand,
!
L:
therefore
0 = !i
L: L:
du
f(t) cos u(t - x) dt.
f(t) sin u(t - x) dt is an odd function of u;
1: l:
du
f(t) sin u(t - x) dt
provided the integral converges. 1 By subtractin g the last equation
from the previous one we obtain formula (16), valid wherever f(x) is
continuous.
2. Extension of the Result to Several Variables. By repeated application of formula (16) anawgous formulas are obtained for piecewise
smooth functions of several variables; these formulas are valid wherever
the functions are continuous. Thus
4r2F(x1, X2) =
Ifff
F(t1, ti)
e-ifu1(t1-z1> +u2(trz2)J
dt1du1d"2d~,
under the assumptio n that the integrals
and
1 This is the case for those
values of x for which /(x) is continuous. The
integral diverges at points of discontinui ty, as may easily be seen from the
function
f(x)
= 1 for I x I ::;
1,
/(x)
=0
for Ix I
>
1.
80
II. SERIES EXPANSiqNS OF ARBITRARY FUNCTIONS
exist. In general, for n variables, we have
1: ... L:
(2r)"F(x1, ··· ,xn)
=
F(t1' ... 'tn)
e-ilui(ti-zi>+· . .+u,.(t,.-x,.)J
dt1du1 ... dtndu,.
under analogous assumptions. The integrations are to be performed
in the order in which tP.e differentials are written in this formula.
3. Reciprocity Formulas. The Fourier integral theorem (16) takes
on an especially elegant form if one sets
g(u) =
1 Leo f (t) e- iut dt ;
.yz;
00
for, it then states that the equations
l
00
1
.yz;
_
g(u ) =
f ( t) e- iut dt,
1 f_co g(u ) eiut du.
.yz;
f ( t) =
00
00
I
follow from each other. If the left-hand sides are assumed to be
known, these equations form a pair of so-called integral equations
each of which is the solution of the other and which are mutually
reciprocal. For even and odd functions we have the sets of real
equations
g(u)
and
I~
{
I! £
f(t) cos ut dt,
f(t)
f(t) sin ut dt,
f(t) =
I! {
I! 1"°
00
g(u) =
g(u) cos ut du
g(u) sin ut du,
respectively.
The corresponding reciprocity formulas for functions of several
variables are
f(x1, · · · , x,.)
1
= (V2;)"
J• • • J (t
g
t;;l.,
. . . , t;;n
t )
e'<hz1+· .. H,.znl
dt1
.... :i._.c;n,
d.t
'i
g(~1, • • • , ~n)
vkr J···J
=(
f(x1, • • · , Xn)
e-i<hxi+· · .H,.x,.>
dx1 • · · dx,.,
and are certainly valid if f and g have derivatives up to the (n +•1)-st
order in the entire space.
81
EXAMPLES OF FOURIER INTEGRALS
§7. Examples of Fourier Integrals
1. The Fourier integral formula (17)
1
f(x) = -
lco du lco f(t) cos u(t 0
11"
1
= -
11"
x) dt
co
1"° cos ux du lco f(t) cos ut dt
+ -1 1"° sin ux du lco f(t) sin ut dt
0
-co
0
11"
-co
reduces to
2
f(x) = -
11"
1
00
1co f(t) cos ut dt,
cos ux du
0
0
if f(x) is an even function, and to
["°
f(x) = 2- . sin ux du
11"
0
["" f(t) sin ut dt
0
if f (x) is odd.
2. We now consider Dirichlet's discontinuous factor: Let the even
function f (x) be defined by
=
1
for Ix
f(x) =
i
for
Ix I =
1
=0
for
Ix I >
1;
f(x)
f(x)
I<1
we may express it as the Fourier integral
f(x)
= 21co
cos ux du
11"
0
11
cos ut dt
0
= ~ lao sin u cos ux du.
11"
0
u
The term on the right, called "Dirichlet's discontinuous factor," is
useful in many problems.
3. If, for x > 0, we take
([j
>
0)
82
II. SERIES EXPANSIONS OF ARBITRARY FUNCTIONS
we have either
2
f(x) = 7r
~
=
7r
00
1 cos
ux
du
0
1"°
e-flt
cos ut dt
e-flt
sin ut dt
0
1"°o 13132cos+ u2 du
ux
or
-l"°
2
f(x) =
7r
sin ux du
0
1"°
0
') 1"°o 132u sm+. u2 du.'
= ::
ux
7r
the former is obtained if, for negative values of x, f(x) is continued
as an even function, the latter if it is continued as an odd function.
In the second case we must put f(O) = 0. The integral
"° cos ux du
1o 132 + u2
=
~
e-fllxl
2
13
(/3
>
0)
is sometimes called the Laplace integral.
4. The function
f(x) =
e-z2/2
provides a particularly instructive example.
ally reciprocal integral equations
g(u) =
and
~ J.~ f(t)
I! £
cos ut dt
=
00
f(t)
=
In this case, the mutu-
I! J.~e-•'
12 cos ut dt =
I~ £ e-u
.~·12
00
g(u) cos ut du =
212
cos ut du = e-'
212
are actually identical.
§8. Legendre Polynomials
1. Construction of the Legendre Polynomials by Orthogonalization of
the Powers 1, x, x2, • • • • If the powers 1, x, x 2, • • • are orthogonalized in a given basic interval, say - 1 ~ x ~ 1, (see §1) we obtain
a complete orthonormal system of functions which is in many ways
83
LEGENDRE POLYNOMIALS
even simpJer than the system of trigonometric functions. This procedµre yields a sequence of orthonormal polynomials which are
uniquely determined to within a factor ±1. They become unique
if we require that the coefficient of the highest power of x in each
polynomial be positive
We assert that these polynomials are, except for constant factors
(depending on n but not on x), identical with the polynomials
Po(x)
= 1,
P n(x)
2
= - 1-
dn(x - lr
dxn
2nn!
= 1, 2, ... ),
(n
1
which are known as the Legendre polyrwmials. Since it is easily
seen that there can exist, apart from constant factors, only one
set of orthogonal polynomials in which every degree is represented,
it suffices to prove that P n is a polynomial of the n-th degree and
that the system of the polynomials Pn(x) is indeed orthogonal. Now
Pn(x) is evidently a polynomial of then-th degree; explicitly,
x2"-n
f (-1r-r (n) (2v(2v)!
- n) !
· 5 · · · (2v - 1) x
f (_1r-" (n 1·3
- v) !(2v - n) !2
Pn(x) = _1_
2 11n!
=
11'-0
v
211_
11
-
11'-o
11 •
..
The terms in negative powers are to be omitted; this comes about
automatically if we put (-r)! = oo for all positive integers r. For
even n, the first term is thus
(-l)n/2 1 ·3·5· · · (n - 1) .
2·4·6· · ·n
•
'
for odd n it is
(-l)Cn-1)/2
x.
1·3·5· · ·n
2·4·6· · ·(n - 1)
The first few Legendre polynomials are
Po(x) = 1,
Pa(x) = jx
3
-
jx,
1 A. M. Legendre, Recherches sur !'attraction des spheroides homogenes,
Mem. math. phys. pres. a l'Acad. sc. par divers sav., Vol. 10, 1785, pp. 411-
434; Recherches sur la figure des planetes, Mem. math. phys., Reg.de l'Acad.
sc., 1784, 1787, pp. 370-389.
84
II. SERIES EXPANSIONS OF ARBITRARY FUNCTIONS
To prove that the P n(x) form an orthogonal system, we shall -denote (x 2 - 1r by un(x). We then have, for eve:ry non-negative
integer m < n,
This may be demonstrated if we remove the factor xm by repeated
partial integration and note that all derivatives of un(x) up to the
(n - 1)-st vanish at the limits of the interval of integration. It
foJlows that
(m
< n),
and thus that any two different Legendre polynomials are ill fact
orthogonal. In order to obtain the necessary normalization factors we now compute
1:
[un <n\x)]2 dx by repeated partial inte-
gration:
(- l t
l:
Un
U~2 n) dx
(2n) !
=
l:
(1 - xt(l
+ xr dx.
Now
r1 (1
L
- xt(l
+ xt dx
1
(1 - xt- 1(1
= _n_ [
n
+1
n(n -
= (n
+
I)(n
f1
1) · · · 1
2) · · · (2n) L1 (l
+
2
2n+l
(n!)
2
(2n) !(2n + 1)
and therefore
fl 2
+ xt+l dx =
...
-1
L1 P nCx) dx = 2n
2
+I.
+
2n
x)
dx
85
LEGENDRE POLYNOMIALS
The desired normalized polynomials are therefore
~n(x)
=
~ Pn(x)
=
~ /2n
V
+ 1 _1_ dn(x
2
2
dxn
2nn!
-
1)"
(n = 0, 1, 2, · · · ).
The Legendre polynomials Pn(x) have the property that
Pn(l) = 1;
this is seen immediately if then-th derivative of (x - I)"(x + 1)"
is evaluated according to the product rule and x is set equal to 1 in
the resulting expression.
2. The Generating Function. The Legendre polynomials are
particularly important in potential theory, where they occur as the
expansion coefficients of a "generating function." Consider two
points in a plane, one at a unit distance from the origin and the other
at a distance u < 1 from the origin, and suppose that the angle between their radius vectors is cos- i x. The distance between the two
points is then V 1 - 2ux + u 2• If we now expand the reciprocal
of this distance in powers of u we find, by the binomial theorem,
(18)
where Qn(x) is a polynomial of the n-th degree in x. The function
on the left is said to "generate" the expansion coefficients Qn(x).
We may show that Qn(x) is identical with the Legendre polynomial
Pn(x) either by evaluating Qn(x) explicitly by the binomial theorem
or by demonstrating that the polynomials Qn(x) obey the same orthogonality relations as the Pn(x). We choose the latter procedure.
From the definition (18) we immediately obtain
1
1
~
Vl - 2xu U 2 Vl - 2xv V2 = n~O Qn(x)Qm(x)unvm.
+
+
Integrating the left-hand side with respect to x from -1 to
obtain, by an elementary calculation,
-1- log
VUv
l+vUV
1 -
v UV =
~
L.J
n=O
2
2n
+
1
u nn
v.
+ 1 we
86
11. s:mRI:mS :mxPANStONS o:r AR:lUTRARY :FUNCTIONS
Integrating the right-hand side term by term and equating the coefficients of unvm to those in the expression for the left-hand side we find
1
[
{
Qn(x)Qm(x) dx =
0
_2_
1
2n
+1
for
n¢m
for
n = m.
By setting x = 1 in equation (18) we find Qn(l)
1. This completes the proof of the identity of Qn(x) with Pn(x).
3. Other Properties of the Legendre Polynomials. (a) Recursion
Formula. Differentiating the generating function with respect to u
we immediately obtain the following recursion formula connecting
three successive Legendre polynomials:
(19)
(n
+
l)Pn+1(x) -
{2n
(b) Differential Equation.
+
l)xPn(x)
+ nPn-1(x)
=
0.
The n-th Legendre polynomial
'
1
dn
2
y(x) = (x 2nn! dxn
1)
n
satisfies the linear homogeneous second order differential equation
(20)
(x
2
-
l)y"
+ 2xy' -
n(n
+ l)y =
0
or
[(x2
(20')
-
l)y']' - n(n
+ l)y
=
0.
+
This may be proved by (n
1)-fold differentiation of the equation
(x 2 - l)u' = 2nxu, with u = (x 2 - l)n and u<n> = 2nn!y. 1
(c) Minimum Property. If the Legendre polynomial Pn(x) is multiplied by the reciprocal C of the coefficient of xn, so that the coefficient of xn in CP n(x) is equal to 1, we obtain polynomials characterized by the following minimum property: they have the smallest
distance in the mean from zero of all polynomials of the n-th degree
with the leading coefficient 1. To prove this we note that, in the
integral
1:
(Xn
+ an-1Xn-l + ··· +
2
tzo) dx, the integrand may be
1 From this differential equation it follows that the zeros of Pn(x) (all of
which by Rolle's theorem are real and lie in the interval -1 <: x < 1) are all
simple, since the second and all higher derivatives of Pn(x) would vanish at
any multiple zero.
87
EXAMPLES OF OTHER ORTHOGONAL SYSTEMS
written in the form (CP n(x)
tegral is therefore equal to
+ Cn-1Pn-1(x) + ... + Co) 2•
The in-
and this expression takes on its least value for
Co = C1 = ' ' · = Cn-1 = 0.
§9. Ex,,mples of Other Orthogonal Systems
1. Generalization of the Problem Leading to Legendre Polynomials.
We shall now generalize the problem which led to the definition of
Legendre polynomials.
Let a non-negative "weight function" p(x) be given in the interval
a ~ x ~ b. The problem is to study the system of functions ob2
tained by orthogonalizing Vp(x), xyp(x), x VP(X}, · · · in the
interval a ~ x ~ b.
These functions are, of course, linearly independent, as are the
powers 1, x, x 2, • • • • In the orthogonal system the factors of p(x)
are polynomials Q0 (x), Q1 (x), · · · of degree 0, 1, · · · which may be
determined uniquely by normalizing conditions and which are termed
1
"orthogonal polynomials belonging to the weight function p(x)."
For example, for
v
a= -1,
b
=
1,
p(x) = 1
we obtain the Legendre polynomials p n(X); for
a= -1,
1
b = 1,
p(x) =
vT="X"2'
1 The polynomials Q 0(x), Q1(x), · · · , when multiplied by suitable factors
C, possess a minimum property similar to that of the Legendre polynomials:
the integral
J
p(x)(x"
+
an-1x11 -
1
+ ... + ao)
2
dx
takes on its least value when the polynomial in the integrand is CQ,.(x). The
polynomial in the integrand may again be written as a linear combination of
the Q,(x), in the form (CQn(x) + Cn-1Q,._1(x) + · · · +co). Since the functions
v'p(x)Qn(x) are orthogonal, and, in fact, orthornormal if the Qi(x) are appropriately defined, the integral is equal to C2 + ].;~:J c! , which assumes its
minimum at c 0 = ci = · · · = c,._ 1 = 0.
88
II. SERIES EXPANSIONS OF ARBITRARY FUNCTIONS
the Tchebycheff polyrwmials
T n (x )
=
-1 )
(
1
2n-l cos n cos x ;
for
a= -1,
=
b
p(x)
1,
the polynomials
( )
QnX
+ 1) cos-
1
= sin [(n
~
x] .
,
for
a = 0,
p(x) = xH(l - x)v-q
b = 1,
(q
>
0, p - q
>
-1),
the Jacobi or hypergeometric polynomials; for
a= -
b =
oo,
oo,
the Hermite polynomials; for
a= 0,
b=
oo,
the Laguerre polynomials.
We shall now study the Tchebycheff, Jacobi, Hermite, and Laguerre polynomials in more detail.
2. Tchebycheff Polynomials. 1 The Tchebycheff polynomials
To(x)
= 1,
T n(x) =
2
1
_1 cos (n cos- 1 x)
n
(n ~ 1)
2
form an orthogonal system of polynomials with the weight function
p(x) = 1/~ in the interval -1 ~ x ~ 1, since
(1
Li
dx
Tn(x)Tm(x) ~
1
= zn-tm-2
lr
0
cos
n(J
cos m1J do = 0
for n
~
m.
1 P. L. Tchebycheff, Sur les questions de minima, qui se rattachent A la
representation approximative des fonctions, Mem. Acad. sc. Petersb., Ser.
6, Vol. 7, 1859, pp. 199-291; Oeuvres, Vol. 1, pp. 271-378, esp. pp. 295-301,
St. Petersburg, 1899.
2 These functions are indeed polynomials, since cos nfJ is a polynomial in
cos fJ:
cos nfJ = cosn fJ - ( ; ) ~osn- 2 fJ sin 2 fJ + (~) cosn-• fJ sin" fJ - • • • •
89
EXAMPLES OF OTHER ORTHOGONAL SYSTEMS
Tn(x) is the polynomial of n-th degree with leading coefficient 1
whose maximum absolute value is smallest in the interval
-1 :::; x :::; 1; i.e. it deviates least from zero. (The coefficient of
xn in Tn is equal to 1, as is easily seen.)
Proof: Let cos-1 x = 8, and consider the points Xk = cos k1r/n
(k = 0, 1, · · · , n), for which Tn(x) attains its greatest deviation
from zero. Clearly, for
{} =
o,
27r
-
7r
n'
n ,
•••
' 7r,
we have
Tn(x)
=
1
-1
1
2n-l'
2n-1>
211-1'
c-1r
'
2n-l
'
and, in general,
T n (xi)
=
(-l)k
n-l •
2
Suppose the deviation from zero of a polynomial Rn(x)
Xn
an-1Xn-l
in the interval -1 :::; X :::; 1 were less than
that of Tn(x). Then we would have
+
+ ···
T n(Xo) - Rn(Xo)
>
0,
>
T n(X2) - Rn(X2)
0, • • • ;
i.e., the entire rational function Tn(x) - Rn(x) would be alternately
positive and negative at successive points x1c • This function would
therefore have to have at least n roots; but this is impossible, since
it is a polynomial of at most the (n - 1)-st degree.
The polynomials T n(x) may be normalized if we divide by
VL:T~~=~The Tchebycheff polynomials also occur as the expansion coefficients of the generating function
(21)
lf;(x, t) =
1
_
1 2
tx
f
+t
2
t;~ T "(x) (2t) .
n
=
Three successive Tchebycheff polynomials are connect€d by the
cursion formula, valid for n ~ 2,
(22)
re~
90
II. SERIES EXPANSIONS OF ARBITRARY FUNCTIONS
For n
<
2 the recursion formula takes a slightly different form:
T2 - xT1 + iTo
T1 - xTo
=
~ l,
= 0.
The Tchebycheff polynomials satisfy the linear homogeneous second
order differential equation
1
(1 - x 2)y" - xy'
(23)
+ n y = 0.
2
3. Jacobi Polynomials. 1 The Jacobi polynomials Gn(p, q, x) are
obtained for a = 0, b = 1 and the weight function
p(x)
=
xq-1(1 - x)p-q
with q
>
0, p - q
>
-1.
They may also be obtained from the hypergeometric series
F(a,
(24)
R
tJ' "'(,
x
)
= 1+~~
1 'Y x
+ a(a1·2
+ 1) {3({3
+ 1) + ...
'Y( 'Y + 1) x
2
by replacing {3 by the negative integer - n, a by p + n, and 'Y by q.
Therefore they satisfy the hypergeometric differential equation
(25)
x(l - x)y"
+ ['Y -
x(l - x)G~(x)
+ [q -
(a+ {3
+ l)x]y'
- a{3y
=
0
or
(25')
(p
+ l)x]G:(x) + (p + n)nGn(x)
=
0.
They represent the sole entire rational solutions of this equation.
The first few of these polynomials are
Go(p, q, x) = 1,
(1)1 -p q+-1x,
1- (2) p +q 2x + (2) (p +q(q2)(p+ + x '
G (p, q, x )
= 1- f3) p + 3 + (3) (p +
+ x2
q,
\1 .
2
q(q + 1)
_(3)3 +q(q + l)(q
+
+
+ 2)
Gi(p, q,x)
=1-
=
2
G3(p
x)
'
1
2
q
1)
x
(p
3)
2
3)(p
4)
3)(p
4)(p
5) xs·
1 C. G. J. Jacobi, Untersuchungen iiber die Differentialgleichung
der hypergeometrischen Reihe, Journ. f. d. reine u. angew. Math., Vol. 56, 1859, pp.
149-165; Werke, Vol. 6, pp. 184-202, Berlin, 1891.
91
EXAMPLES OF OTHER ORTHOGONAL SYSTEMS
or, in general,
G (
)
n p, q, x = q(q
1 11
X - (1
- x)q-p
+ 1) ... (q + n
dn
- [ q+n-1(1 - )p+n-qj
- 1) dxn x
x
.
From this expression it can be seen that the Jacobi polynomials may
also be defined in terms of a generating function by the relation
(l-x) 1-q(l +x)q-p(t-1 + Vl-2tx+ t2 )q- 1 (t+ 1-Vl-2tx+ t2Y-q
tp-
1
v1 - 2tx + t
2
_ ~ (q + n 1) G (
-
£...J
n -
n
n-o
p, q,
~)
tn •
2
For p = q = 1 they reduce to the Legendre polynomials
(26)
P n(x) = Gn ( 1, 1,
for p = 0, q =
(27)
1
+ 1, -n, 1, l
; x) = F ( n
; x);
! we obtain, essentially, the Tchebycheff polynomials
Tn(x) = :_1 Gn ( 0, !,
2
1
; x) =
!_ F ( n, -n, !, 1 ;
2
1
x).
4. Hermite Polynomials. 1 The Hermite polynomials Hn(x) are the
orthogonal polynomials in the interval - oo < x < oo with the
2
weight function p(x) = e-:r: • They are most conveniently defined
by means of a generating function ift(x, t):
(28)
~ Hn(X) tn
.,, x, t) -_ e-t2+2t:r: -_ ez2 e-(t-z)2 -_ £...J
-- •
,_o n!
,/,(
From this equation it follows immediately that
(29)
Hn(x) = (an 1/t(x, t))
= (-1rez2 dn e-:r:2.
atn
e=O
.
dxn
2
The n-th Hermite polynomial H (x) is thus ( -1) V times the n-th
2
derivative of the function e-z • From the relation iJift(x, t)/ax =
2ti/t(x, t) it follows that
11
(30)
(n ~ 1);
C. Hermite, Sur un nouveau developpement en serie de fonctions, C.R.
Acad. sc. Paris, Vol. 58, 1864, pp. 93-100, 266--273; Oeuvres, Vol. 2, pp. 293312, Paris, 1908; Sur quelques developpements en serie de fonctions de plusieurs variables, C. R., Vol. 60, 1865, pp. 370-377, 432-440, 461-466, 512-518;
Oeuvres, pp. 319-346.
1
II. SERIES EXPANSIONS OF ARBITRARY FUNCTIONS
92
(31)
+ 2(t -
x)l/l(x, t)
=
0 we obtain
Hn+1(x) - 2xHn(x)
+
2nHn-1(x)
from ol/l(x, t)/at
By combining (30) and (31) we obtain
order differential equation for Hn(x):
(n
~
1).
a linear homogeneous second
+ 2nHn(x)
H~(x) - 2xH:(x)
(32)
=0
=
(n ~ 0).
0
The first few Hermite polynomials are
Ho(x)
H2(x)
= 1,
= 4x2
H4(x)
-
= 8x3
Ha(x)
2,
4
= 16x
2
-
48x
-
12x,
+ 12.
In general, then-th Hermite polynomial is
H n(x)
=
(2xr - n(n
1~ l) (2xr+ n(n
2
- l)(n ~ 2)(n - 3) (2xr-4 _ ...•
2
The last term is
<-1r' 1 ~
(n/2) 1
for even n and
(-l)<n-1)/2
((n -
n!
2
1)/2) ! x
for odd n.
1
The orthogonality of the Hermite polynomials may be obtained
from
for n > m by repeated partial integration, keeping in mind formula
2
(30) and the fact that e-:1: and all its derivatives vanish for infinite x:
d"-1 -z2
Leo
eo
2
1
Hm-1(x) d n~i dx = · · · ·
Hm(x)Hn(x) e-z dx = (-1)"- ·2m
L co
x
-co
= (-1r- 2m ml
eo
L
d"-m
Ho(x)
-z2
dx": dx = 0.
00
. 1 The orthogonality relations may also be obtained with the help of the
generating function.
93
EXAMPLES OF OTHER ORTHOGONAL SYSTEMS
To obtain the normalization we set n
L:
H~(x) e-:e
2
1:
= 2n n!
dx
m and have
=
2
Ho(x)e-:e dx = 2n n!
Thus the functions of the orthonormal system are
H,,(x)e-:r: 212
<P,,(x) =
(v
=
V 7r.
0, 1, 2, · · · ).
v2"v!v7r
5. Laguerre Polynomials. 1 The Laguerre polynomial Ln(x)(a = 0,
oo, p(x) = e-:e) occurs as the factor of e-:r: in then-th derivative
b=
of the function xne-:r::
dn
Ln(x) = ec -(x" e-:r:)
dxn
+
- 1) • • • (k + l)xk
E (- l)k(n)n(n
k
- 1) · · · (n - k + l)x"- •
E(-1r-k(n)n(n
k
k-0
= t (-1r-k [n(n - 1) .•. (n - k + 1)]2 xn-k
k!
k-0
+ n (n - 1) x - · · · + (- 1)" n.r) ;
( l) x - n
l! x
21
=
k-0
=
n-2
n-1
n
n (
-
2
2
2
=
thus, for example,
Lo(x)
=
1,
L2(x) = x
2
-
4x
L 4(x)
=
+ 2,
x
4
-
3
L3(x) = -x
3
16x
+ 72x
2
-
+ 9x
96x
2
-
18x
+ 6,
+ 24.
In virtue of the relations
'£
'£ t
f(n)k t"
f (-l)kxk n=k
kl
(-l)k(n)_!_ xkt"
Ln(x) t" =
k kI
n-o k-o
n-o n I
=
k-0
=
t
k...O
tk
(-l)kxk
(1 - t)k+l
kl
00
1
E. Laguerre, Sur l'integrale
1
:r:
e-:r:tf<H>
(1 - t)
e-z
- dx, Bull. Soc. math. France, Vol. 7,
x
1879 1 pp. 72-81; Oeuvres, Vol. 1, pp. 428-437, Paris, 1898.
94
II. SERIES EXPANSIONS OF ARBITRARY FUNCTIONS
the Laguerre polynomials possess a simple generating function,
namely
=
i/l(x, t)
e-s:11<1-1>
1- t
The relation
(1
-ae-
- t ) 2 iJ!/t(x, t) = ( 1 - t - x ) i/I (x, t )
leads to the recursion formula
(33)
Ln+1(x) - (2n
+
1 - x)Ln(X)
+ n Ln-1(x)
2
= 0
(n
2: 1).
(n
2: 1),
(n
2: 1)
This, together with relation
(34)
L:(x) - nL:_1(x)
=
-nLn-1(x)
which follows from the relation
(1 - t)iJ!/t(x, t) = -tl/l(x t)
ax
, '
leads to the formula
(35)
and thus to the linear homogeneous second order differential equation
(36)
xy"
+ (1
- x)y'
+ ny = 0
satisfied by the Laguerre polynomial Ln(x).
The orthogonality relation
1'° e-x
Ln(x)Lm(x) dx = O
(n
>
m)
follows from the equation
00
1 e-:e
0
xkLn(x) dx =
1"° xk
0
ddn (X71 e-x) dx
71
X
for n
>
k,
95
EXAMPLES OF OTHER ORTHOGONAL SYSTEMS
and the normalization is given by
0
{
e-xL~(x)
dx =
1«1 (-ltx" :x:(x"e-x) dx
= nt 1«1 x" e-re dz = (n!) 2 ;
therefore the functions
cp.,(x)
(v = 0, 1, 2, · · ·)
constitute the orthonormal system. 1
6. Completeness of the Laguerre and Hermite Functions. The completeness of the Laguerre and Hermite functions remains to be
investigated, since completeness has thus far been proved only for
finite intervals. We caJl a system of functions in the interval 0 S
x < oo complete if every piecewise continuous functionf(x) for which
the integral
1«1 f \x) dx exists can be approximated arbitrarily well
in the mean by a linear combination of the functions.
To demonstrate the completeness of the Laguerre functions 2 we
multiply both sides of the identity
"
~ Ln(x)
L~ n.
1
lf;(x, t) = - - e-tre/(l-t> =
1 - t
n=O
by e-xt 2 , thus obtaining the corresponding identity
g( x,
t) = _·1_
1 - t
e-il{t+t)/(1-t))re
for the orthonormal Laguerre functions
( )
'PnX
-re/2
=e
Ln(X)
-n.1
•
Now the infinite series L':=o t"cpn(x) converges in the mean to the generating function g(x, t) for I t I < 1. This may easily be seen from
the estimate
"° (
[ . g(x, t) 0
L
N
n=O
t"cpn(x)
)2
dx
=
1
1
_ 12 -
Lt
N
2
",
n=O
Once again the orthogonality relation could be obtained with the aid of
the generating function.
2 This proof was suggested in conversation by J. von Neumann.
1
96
II. SERIES EXPANSIONS OF ARBITRARY FUNCTIONS
obtained by means of the relations
('
2
= 1 _1
Jo
g (x, t) dx
f'
g(x, t)<pn(x) dx
t2 ,
and
=
t".
+
+
Since the quantity a = !(1
t)/(1 - t) assumes all values from
0 to oo as t runs from -1 to 1, it follows that all functions of the
form e-az can be approximated arbitrarily well in the mean by combinations of the Laguerre functions in the interval 0 ~ x < oo •
Suppose that the function f(x) is piecewise continuous and squareintegrable in this interval. If we set e-:r: = t, f(x) goes over into a
function kW which is piecewise continuous in the interval 0 < t ~ 1.
The function ka)/0 is, moreover, square-integrable and can
therefore be approximated in the mean by a function
which is
piec~wise continuous in the closed interval 0 ~ x ~ 1. (For example,
can be taken identically equal to zero in a sufficiently
small neighborhood of the origin and equal to kW/ 0 elsewhere in
the interval 0 ~ x ~ 1.) The function
and, hence, k(t)/~
can be approximated in the mean by polynomials
am
am
am
hn(t) =
ao
+ a1t + a2f + ... + ant"·
It follows that f(x) can be approximated in the mean in the interval
0 ~ x < oo by expressions of the form
Vfhn(t) = e-:r:/ 2 (ao
a1e-:r: + a-ie-2:r:
ane-":r:)
+
+ · ·· +
and therefore by a combination of Laguerre functions.
equivalent to the validity of the completeness relation
This fact is
i:
c! = 1 J2(x) dx,
...o
00
0
1
00
where c,, denotes the expansion coefficient c,, =
f(x)<p,,(x) dx.
The completeness of the Hermite polynomials can be proved on
the basis of the completeness of the polynomials of Laguerre. Any
square-integrable function f(x) is the sum fi(x) + f2(x) of an even
function f 1 and an odd function /2 . For each of these functions the
SUPPLEMENT AND PROBLEMS
97
completeness relation can be easily reduced to the completeness
relation for Laguerre functions by substituting x = u 2• Details of
the proof are omitted.
§IO. Supplemen t and Problems
I. Hurwitz's Solution of the lsoperimetric Problem. The "isoperimetric problem" is the problem of finding the simple closed plane
curve of given perimeter with maximum area. Its solution is known
to be the circle. We confine ourselves to piecewise smooth curves.
This problem is solved by Hurwitz1 in the following way:
Let
x = x(s),
y
=
O~s<L
y(s),
be the parametric representatio n of a continuous piecewise smooth
closed curve of perimeter L and area F. The parameter s is the arc
length. We introduce instead the new parameter t = 27rs/L, which
goes from 0 to 27r ass goes from 0 to L, and denote the Fourier coefficients of x and y by a.,, , b.,, and c.,, , dv , respectively; the Fourier
coefficients of dx/dt, dy/dt are then vbv, - vav and vd.,,, - vc,,. The
relations
(27rL ) '
(dx)
dt
+ (dy)'
dt
(~:)' + (7.)' =I,
F
r{(:"12r)'
2
2
=
= f2r x dy dt
lo
dt
and the completeness relations (9) and (9') then lead to
2(;,J
= ;
+ (~m dt = ~ v'(a! + b! + c! + d!),
F
1
1r
1r
- = -
dy
x-d dt =
0
t
L"°
v(a,,d.,, - b,,c.,,).
r-1
It follows that
L2
-
47rF
GO
=
27r 2
L
v-1
[(va.,, - d,,) 2
+
(vb,,
+ c.,,) +
2
(v
2
-
l)(c;
+
d!)J ~ 0.
1 A. Hurwitz, Sur quelques applications geometriques des series de Fourier,
Ann. Ee. Norm., Ser. 3, Vol. 19, 1902, pp. 357-408, esp. pp. 392-397.
98
II. SERIES EXPANSIONS OF ARBITRARY FUNCTIONS
Evidently the equality can hold only if
b1
= 0,
+Ct
a,, = b,, = c,,
d,, = 0
=
for .,, = 2, 3, · · · ,
i.e. if
x =
!ao +
a1 cos t
y = !bo - bi cos t
+
b1 sin t,
+ a1 sin t.
These are the equations of a circle. Thus all continuous piecewise
smooth closed curves satisfy the "isoperimetric inequality"
L2
(37)
47rF ~ 0,
-
where L is the perimeter and F the area. The equality holds if and
only if the curve is a circle. This proves the isoperimetric character
of the circle.
2. Reciprocity Formulas. Prove the equivalence of the two formulas1
f(t) =
-g(u)
f
=
g(u) cot ?r(t -
f
u) du
f(t) cot ?r(u - t) dt,
assuming that
f
g(u) du= 0,
f
f(t) dt
+ 1)
f(t
+ 1)
0,
and
g(u
= g(u),
= f(t).
The integrals are to be taken as the "Cauchy principal values." The
proof may be carried out by using either the theory of Fourier series
or the Cauchy integral theorem.
3. The Fourier Integral and Convergence in the Mean. The theory ·
of the Fourier integral may be developed in the same way as the
theory of Fourier series in §5; again we use the concepts of convergence in the mean and completeness.
1
Compare Hilbert, Integralgleichungen, p. 75.
99
SUPPLEMENT AND PROBLEMS
Let the real or complex function f (x) be piecewise continuous in
every finite interval, and assume that the integrals
1:
1:
I f(x) I dx,
I
f(x)
2
dx
1
exist. We attempt to find the best approximation in the mean of
the function f (x) by integrals of the form
L:
tp(t)
ei:i:t
dt;
i.e. we attempt to make the integral
as small as possible for a fixed value of T.
prove that
f
jfcx) -
=
1:
L:
2
I f(x) 1
,,,(t) em dt
dx
+ 2r
j'
It is not difficult to
dx
L:
I
tp(x) - g(x)
2
1
dx - 2r
L:
I
2
g(x) j dx
with
Therefore our integral assumes its least value for
q;(t) = g(t).
Furthermore, the passage to the limit T
ness relation
L
oo
00
I
g(x)
2
1
1"°
dx = 1r _
2
~ oo
I
f(x)
yields the complete2
1
dx.
00
This fact will not be proved here, nor shall we state the further
considerations leading to the Fourier integral theorem itself.
4. Spectral Decomposition by Fourier Series and Integrals. Fourier
series and Fourier integrals occur wherever one has occasion to represent a given phenomenon or function as a superposition of peri-
100
II. SERIES EXPANSIONS OF ARBITRARY FUNCTIONS
odic phenomena ·or functions. This representation is known as the
spectral decomposition of the given function. If the Fourier series
for the function f(x) in the interval -l ~ x < l is L:_-oo a,,eirvztz,
the function f(x) is said to be "decomposed" into periodic functions
with the "discrete frequencies" v7r/l (v = 0, 1, · · · ) and the "amplitudes"
IJi {: f (x) e '"'" dx I·
I a, I =
If, on the other hand, the infinite interval - oo < x < oo is considered, the function f(x) is said to be decomposed into a "continuous
spectrum," with the "spectral density"
=
g(u)
at
L:
f(x) e-iuz dx
th~ frequency u.
1
An example of great interest in physics is given by the function
f(x) =e
f(x)
=
iw:i:
0
for
Ix I <
for
Ix I > z
l
which corresponds to a finite train of sinusoidal waves consisting of
n = lw/7r waves. Its spectral density is given by
( ) Le
gu =
l
l
i(w-u)x
d
x=
2 sin (w - u)l
(J)
-
u
.
The function I g(u) I has a maximum at u = w, which becomes more
pronounced as the number n of waves in the train becomes greater.
o, the specOutside the arbitrarily small interval w - o ~ u ~ w
tral density becomes, in comparison with its maximum, arbitrarily
small for large n.
5. Dense Systems of Functions. A system of functions will be
2
called dense if it has the property that every function f (x) which
+
1 This example is significant in optics; for, a sinusoidal wave train of finite
length corresponds not to a sharp spectral line but to a spectrum of finite
width which becomes sharper and more intense as the wave train becomes
longer.
2 Here we follow the definition of H. MUntz.
SUPPLEMENT AND PROBLEMS
101
can be approximated arbitrarily well in the mean by a finite number
of functions of the system can also be approximated in the mean by
functions taken from any infinite subset of the original system of
functions. The remarkable fact that nontrivial systems of this kind
exist-a trivial system is, for example, one in which all the functions
are equal-may be illustrated most easily on the basis of the following theorem :1 If A1 , A2 ,. • · · , An , · · · are positive numbers which
tend to (X) with increasing n, then the functions
1
1
1
'x
+An'
form a complete system in every finite positive interval.
From this theorem it follows immediately that the system is dense,
since every partial sequence of the An also meets the specified requirements.
Because of the Weierstrass approximation theorem it suffices to
show that every power xm may be approximated uniformly by the ·
functions 1/(x
An). The rational function
+
converges to xm with increasing p and q (q ~ p), and the convergence is uniform in every finite positive interval. If we always take
q - p ~ m, this rational function may always be decomposed into
partial fractions and written in the form
~+
x
+ Ap
x
AP+1
+ Ap+i
+···+~
x
+ A/
where Ap, AP+l, · · · , A 9 are constants, since all the numbers An
may be assumed to be different from each other. But this is a linear
combination of the functions of the system under consideration.
Other examples of dense systems of functions have been given by
H. Miintz. 2
1 G. Szego, Uber dichte Funktionenfamilien, Berichte
der sachs. Akad. d.
Wiss. zu Leipzig, Vol. 78, 1926, pp. 373-380.
s H. Muntz, Umkehrung bestimmter lntegrale und absolute Approximation, Ch. II, Dichte Funktionensysteme, Math. Zeitschr., Vol. 21, 1924, pp.
104-110.
II. SERIES EXPANSIONS OF ARBITRARY FUNCTIONS
102
6. A Theorem of H. M iintz on the Completeness of Powers.
M ilntz1 has proved the following interesting theorem: The infinite
sequence of powers 1, x>- 1 , x>-2, • · • with positive exponents which
approach infinity is complete in the interval 0 ~ x ~ 1 if and only
if L~=l l/An diverges.
7. Fejer's Summation Theorem. From Weierstrass's approximation
theorem we concluded that every continu.ous periodic function can
be approximated uniformly by trigonometric polynomials. Such approximating polynomials may easily be constructed on the basis of
2
the following theorem, which was discovered by Fejer. If f(x) is a
continuous pen:odi·c function and if sn(x) is the n-th partial sum of its
Fourier series, then the sequence of arithmetic means
s1(x)
+ s2(x) + · · · + sn(x)
n
2
-1
2
=
7rn
11"
-r
f(x
+ t)
. nt )
sm2
--
(2 . t
sm:z
dt
converges uniformly to f(x).
1:
An analogous theorem holds for the Fourier integral: Let f(x) be
continuous in every finite interval, and suppose that
exists.
Let us write
and
sT(x) =
1
I f(x) I dx
L:
g(t) et:ct dt.
H. Muntz, Uber den Approximationss atz von Weierstrass, Festschr. H.
A. Schwarz, p. 303, Julius Springer, Berlin, 1914; 0. Szasz, Vber die Approxi-
mation stetiger Funktionen durch lineare Aggregate von Potenzen, Math.
Ann., Vol. 77, 1916, pp. 482-496.
2
L. Fejer, Untersuchungen uber Fouriersche Reihen, Math. Ann., Vol. 58
1904, pp. 51-69.
SUPPLEMENT AND PROBLEMS
103
Then the sequence of arithmetic means
S.(x)
=
~f
s.{x) dT
= "
2
T
L:
f(x
+
{int
;t)' dt
converges uniformly to f(x) in every finite interval. In particular,
the convergence is uniform in the whole interval - oo < x < oo if
f(x) is uniformly continuous there.
8. The Mellin Inversion Formulas. 1 . THEOREM 1 : Let s = u + ti be
a complex variable. Let the function f(s) be regular in the strip
<
a
u
<
{3
and let
j_: I
f( u
+
ti) I dt converge in this strip.
Fur-
thermore, let the function f(s) tend uniformly to zero with increasing
I t I in every strip a o u {3 - o (o > 0, fixed arbitrarily). If
for real positive x and fixed u we define
+ s s
l
(38)
a+ooi
g(x) = -.
27r1,
then
(39)
f(s) =
1
x-•f(s) ds,
tT-OOi
fo
00
x•- 1g(x) dx
in the strip a < u < {3.
Proof: Since, by assumption, f(s) converges uniformly to zero for
a + o S u S {3 - oas I t I ---+ oo, the path of integration in (38) may
be displaced parallel to itself as long as it stays within the strip
a < u < {3; thus g(x) does not depend on u. If we choose two
abscissas u1 and u2 with a < u 1 < u < u2 < {3, we have
00
1
0
x'- 1g(x) dx =
11
0
1
x•-t dx ~
2
7r1,
1tr1+00 i x-•i f (s1) ds1
tr 1-ooi
1
H. Mellin, Uber den Zusammenhang zwischen den linearen Differentialund Differenzengleich ungen, Acta Math., Vol. 25, 1902, .PP· 139-164, esp. 156162; M. Fujiwara, Uber Abelsche erzeugende Funktionen und Darstellbarkeitsbedingungen von Funktionen durch Dirichletsche Reihen, Tohoku math.
J., Vol. 17, 1920, pp. 363-383, esp. 379-383; H. Hamburger, Uber die Riemannsche Funktionalgleich ung der s-Funktion (Erste Mitteilung), Math.
Zeitschr., Vol. 10, 1921, pp. ~254, esp. pp. 242-247.
104
II.
SERIES EXPANSIONS OF ARBITRARY FUNCTIONS
The order of the integrations in these integrals may be interchanged,
because we have the estimate
I J1 I S
;7r
~
I J2 I S 271"
l:
I f(<T1 + ti) I dt
f
l"° I f(<T2 + ti) I dt J"° x-i+<a-a
<
oo,
2
> dx
<
oo
'
1
00
dx
X-I+(a-ai)
for the interchanged integrals.
We thus obtain
l
ao
O
x•- lg(x) dx = _!__, J0'2+aoi f(s2) ds2 - _!_, 10'1+aoi f(s1) ds1.
27rt cr1-oo i 81 - 8
211'1 cr2-oo i 82 - 8
According to the Cauchy integral formula, the difference on the
right side equals f(s), since the integrals over horizontal segments
connecting the two vertical lines s = <T1 and s = <T2 tend to zero for
It I~ oo (because f(s) ~ O).
THEOREM 2: Let g(x) be piecewise smooth for x > 0, and let
fo
00
xa-1g(x) dx be absolutely convergent for a
inversion formula (38) follows from (39).
Proof: Let us put x = eu. We then have
_!_· la+ooi x-• f(s) ds
21rt
= _!__ loo e-u<cr+ti) dt
27r
a-ooi
-- e-
-UO'
21r
<
l:
<T
<
{3.
Then the
e"<a+ti)g(e") dv
-
-co
1"° dt l"° eit(v-u) e
g ( e" ) dv.
f}O'
-oo
ao
By the Fourier integral theorem (16) the latter expression is equal
to e-"aeuag(eu) = g(x); the theorem is therefore proved.
Examples of the Mellin integral trans!ormation:
(a) Let
1 for 0 < x < 1
g(x) = i for x = 1
{
0 for x > 1;
since the integral
le
x-1g(x) dx converges absolutely for u
>
0,
we have
f(s)
=
ltO ar g(x) dx =
1
0
!8
(<T
> 0),
105
SUPPLEMENT AND PROBLEMS
from which it follows that
1 lcr+ooi x-a
-ds
g(x) = - .
211"1 cr-ooi S
(u
>
0).
(u
>
0)
(u
>
0).
(u
>
1)
(u
>
1).
(u
>
1)
(u
>
1).
This formula is important in the theory of Dirichlet series.
(b) From the definition of the r -function
r(s)
=
fo
00
X
8
-
l
e-1: dx
one obtains
(c) The formula
r(s)r{s) =
dx
1"° ez~
- 1
0
where s(s) denotes the Riemann zeta function, leads to
1 lcr+ooi
1
x-sr(s)r(s) ds
-- = - .
2?r'l. cr-oo i
ez - 1
(d) The inversion of
1"° xs-l t
= 1"° xs-1. O(x) 2
11"-s/2r(s/2)s(s) =
e-r112z
0
dx
11-1
1 dx
0
IS
l
O(x) = 1
cr+ooi
+ 7ri 1cr-ooi
2
x-s 7r-st r(s/2)s(s) ds
The Mellin transformation is an important tool in the analytic
theory of numbers and also occurs frequently in other problems of
analysis.
9. The Gibbs Phenomenon. If we draw the graphs of a piecewise
smooth function f(x) and of its approximations by the partial sums
of its Fourier series, then we find that the graphs of the partial sums
approach the graph of f(x) in every interval that does not contain a
106
II. SERIES EXPANSIONS OF ARBITRARY FUNCTIONS
discontinuity of f(x). However, in the immediate vicinity of a jump
discontinuity, where the convergence of the Fourier series is not uniform, the graphs of the partial sums contain oscillations which become progressively narrower and move closer to the point of discontinuity as the number of terms in the partial sums increases,
but the total oscillation of the approximating curves does not ap1
proach the jump of f(x). This is known as the Gibbs phenomenon.
To investigate it more closely we may, according to §5, confine
ourselves to the particular Fourier series
x _
r -
L sm vx
•
co
(0
-2- - ....1-.,,
< x < 2r).
With the aid of the formula
Sn(x) =
t
+ 1z sin(n + !)t dt
sin vx = _ ~
2
.,,
"=1
o
2 sin
!t
'
we may write the remainder of this series in the form
_
Tn (X ) -
~
4..J
sin vx _
-- -
7r
1:r: sin
-
~
2
.,,
r-11+1
0
(n + !)t dt
• 1
2 sm 2t
or
Tn(x) =
7r
2-
t
l<n+i>:r: sin
-t- dt + Pn(x)
.
0
where
. t
Pn(x)
__ 1:r: 2 sm 2 0
•
2t sm
t
t
sin (n
+ !)t dt.
2
By differentiation we see easily that the approximation is worst at
the points
2k7r
Xk
= 2n
+
1
(k = 1, 2, · · · , n),
1 This fact was originally discovered empirically by Gibbs.
J. W. Gibbs,
Fourier's series, Nature, Vol. 59, 1898-99, pp. 200 and 606; Papers, Vol.2, pp.
258-260, Longmans, Gre1m and Co., London, New York, and Bombay, 1906.
107
SUPPLEMENT AND PROBLEMS
Its value at Xk is
for which the remainder has maxima or minima.
rn(Xk) =
1r ·
2-
( 2k1l' )
sin x
1kr -Xdx + Pn 2n + l ·
0
For increasing n and fixed k, Pn (2k1l' / (2n + 1)) tends to zero. Thus
the remainder rn(xk), i.e. the deviation of the approximation from
!{7r - x) at the point Xk (which approaches the point of discontinuity), tends to the limit
ktr
1l'
.
hm Tn(Xk) = - 2
n-oo
1
0
•
sm x
- - dx.
X
In particular lim rn(x 1) ~ -0.2811; i.e. the approximation curve
overshoots the curve of f(x) by about 9 percent of the amount of
the jump discontinuity .1
It may be pointed out that the Gibbs phenomenon does not occur
when Fejer's arithmetic means are used as the approximation functions.
10. A Theorem on Gram's Determinant. If G' is a subdomain of
the basic domain G, if cp1 , cp2 , • • • , 'Pn are piecewise smooth functions
in G, and if r is their Gram determinant for the domain G and r'
that for the domain G', then
r'
~
r.
The proof follows immediately from the maximum-minimum property of the eigenvalues. For, f is the product of the eigenvalues of
the quadratic form
K(t, t) =
i
(f1cpi
+ ••• + tn'Pn)
2
dG,
and r' is the corresponding product for
1 M. B6cher, Introduction to the theory of Fourier's series, Ann. Math.,
Ser. 2, Vol. 7, 1906, pp. 81-152, esp. 123-132; C. Runge, Theorie und Praxis
der Reihen, pp. 170-182, Goschen, Leipzig, 1904. For generalizations of the
Gibbs phenomenon to other orthogonal systems, in particular systems in
several variables, see H. Weyl, Die Gibbssche Erscheinung in der Theorie
der Kugelfunktionen, Rend. Circ. mat. Palermo, Vol. 29, 1910, pp. 308-323;
-Ober die Gibbssche Erscheinung und verwandte Konvergenzphanomene,
ibid, Vol. 30, 1910, pp. 377-407.
108
II. SERIES EXPANSIONS OF ARBITRARY FUNCTIONS
Since, obviously,
K'(t, t)
~
K(t, t),
it follows immediately that every eigenvalue of K'(t, t) is less than
or equal to the corresponding eigenvalue of K(t, t).
Another proof may be obtained from the following representation
of the Gram determinant, in which we restrict ourselves, for the sake
.of brevity, to a single variable x in the basic domain 0 ::s; x ::s; 1 :
=
_!_
11 11 ... 11
n! o
o
~(x!)
~(x2)
rp2(x1)
rp2(x2)
<,!>n(X1)
<,!>n(X2)
o
•••
<,!>n(x,i)
This representation corresponds exactly to formula (45) of Chap
ter 1.1
11. Application of the Lebesgue Integral. Many topics discussed
in this chapter become much more elegant if we use the Lebesgue
integral instead of the more elementary Riemann integral. The set
of admissible functions or "spaces" must be extended to include all
functions which are integrable in the Lebesgue sense, and the
methods of the Lebesgue theory must be applied.
The Lebesgue theory is based on the concept of the measure of a
set of points ~' all of which may be assumed to lie within a given
finite interval. Let us suppose that the points of~ are all imbedded
in a denumerable set of intervals; these intervals may, in general,
overlap. Let m be the lower bound of the sum of the lengths of
these intervals; moreover let m' be the corresponding lower bound
for the points of the complementary set ~', i.e. for those points of
the given interval that do not belong to ~. If m
m' equals the
length of the given interval, the set ~is said to be measurable and
mis said to be its measure. According to this definition the measure
of every denumerable set is zero; such sets are called "null sets."
Consider a bounded function f(x) defined in the interval
+
i Compare A. Kneser, Zur Theorie der Determinanten, Festschr. H. A.
Schwarz, pp. 177-191, Julius Springer, Berlin, 1914; Gerhard W. H. Kowalewski, Einfiihrung in die Determinantentheorie einschliesslich der unendlichen
und der Fredholmschen Determinanten, Veit, Leipzig, 1909.
SUPPLEMENT AND PROBLEMS
109
G (a ~ x ~ b), the values of which lie in an interval J. Let us subdivide J into subintervals Ji, J2, · · · , Jn. If for every such subinterval J; the set of points of G for which f(x) has values in J; is
measurable, then the function f(x) is said to be measurable in G. In
this case, if m; denotes the meast+re of the set of points of G for
which f(x) has values in J;, then the sum Li-1 m;f; (f; is any
value of f lying in J ;) converges as the partition of J is refined,
provided only that the lengths of the subintervals J; tend to zero
uniformly. The limit of the sum is called the Lebesgue integral or,
simply, the integral of the function j(x). It is a natural generalization of the ordinary Riemann integral and is denoted by the same
symbol. The integral vanishes for a function which differs from
zero only at the points of a null set. Thus a function may be changed
arbitrarily in any null set, e.g. in the set of rational points, without
affecting the value of its integral. Evidently, the domain of integrable functions has been greatly extended by this new definition.
A function which is integrable in the sense of Lebesgue is said to be
summable.
The concept of the Lebesgue integral may also be applied to
functions which are not bounded in the given domain. We simply
extend the integration first over those subdomains in which - M <
f(x) < N, and then let Mand N increase independently beyond all
bounds. If the limit of the integral exists it is called the Lebesgue
integral over the entire domain.
The following important theorems can be derived from these
concepts:
(a) Lebesgue Convergence Theorems. If a sequence f1(x), f2(x), · · ·
of functions summable in the interval a ~ x ~ bis given and if, for
every x in the interval, the functions f n(x) converge with increasing
n to a function F(x), then the equation
lim
ff
nCx) dx =
f
F(x) dx
is valid (even if the convergence is not uniform) provided that all
the functions fn(x) have absolute values below a fixed bound independent of both n and x.
It is, in fact, sufficient that the inequality
I fn(x) I <
cp(x)
hold, where cp(x) is a fixed summable function independent of n.
110
II. SERIES EXPANSIONS OF ARBITRARY FUNCTIONS
These theorems enable us to justify term-by-term integration of
infinite series in many cases of nonuniform convergence.
(b) Convergence in the Mean. Suppose that the functions f1(x),
f2(x), · · · and their squares are summable and that
lim
,,._co
J Un - f,,.)
2
dx = 0.
fl-+00
The sequence of functions fn(x) is then said to converge in the mean.
The following theorem holds: From any such sequence of functions
it is possible to select a subsequence of fn, which converges pointwise
to a summable function f (x) except at points of a set of measure
zero.
1
(c) Theorem of Fischer and Riesz. This theorem may be formulated in two equivalent ways.
Fischer's formulation: Let the functions f1(x), f2(x), · · · and their
J
Un - f -m) 2 dx = 0.
squares be summable, and assume limm-+oo
fl-+00
Then there exists a square-summable function f (x), such that
fun
fl-+00
J (f
n -
2
f) dx = O.
Riesz's formulation: If w1(x), w2(x), · · · is an arbitrarily given
orthogonal system of functions and if a1 , a2 , · · · are arbitrary real
numbers for which ,L::.1 a! converges, then there exists a summable
functionf(x) with summable square for which av = (f, wv).
If the concepts of integral and function are generalized in the sense
considered here then the above theorem establishes a biunique
relationship between square-integrable functions and coordinates av
with a convergent sum of squares.
(d) Completeness and <:Jlosedness of Systems of Functions. We
call a system of functions closed if there exists no normalized
function orthogonal to all the functions of the system; we assume
that these functions and their squares are summable. The following theorem holds: Every closed system of functions is complete and
vice versa. In fact, if e.g. f(x) is different from zero on a set of posi1 F. Riesz, Sur les systemes orthogonaux de fonctions, C. R. Acad. sc.
Paris, Vol. 144, 1907, pp. 615-619; Uber orthogonale Funktionensystem e,
Nachr. Ges. GOttingen (math.-phys. Kl.) 1907, pp. 116-122; E. Fischer, Sur
la convergence en moyenne, C. R. Acad. sc. Paris, Vol. U4, 1907, pp. 10221024.
111
SUPPLEMEN'l' AND PROBLEMS
tive measure and is at the same time orthogonal to all the functions
of the orthogonal system, then 0 =
J
2:~ 1 (f, w,,) 2 < J2
dx; the sys-
tem of functions is, therefore, not complete. On the other hand, if
the system is not complete there exists a function f(x) such that
JJ2 dx -
L::'=1 a! > 0 for a,,
= (f, w,,);
then, by Fischer's formula-
tion of the Fischer-Riesz theorem, the functions fn = f - 2::..1 a,,w"
converge in the mean to a function cp(x) which is orthogonal to all
the functions w". Therefore the system cannot be closed.
References
'1.'extbooks
Borel, E., Legons sur les fonctions de variables reelles et les developpements
en series de polynomes. Gauthier-Villars, Paris, 1905.
Carslaw, H. S., Introduction to the theory of Fourier's series and integrals.
2nd edition. Macmillan, London, 1921.
Heine, E., Handbuch der Kugelfunktionen, 1and2. 2nd edition. G. Reimer,
Berlin, 1878 and 1881.
Hilbert, D., Grundziige einer allgemeinen Theorie der linearen Integralgleichungen. B. G. Teubner, Leipzig, 1912. (Cited as "lntegralgleichungen.")
Hobson, E. W.; The theory of functions of a real variable and the theory of
Fourier's series. Cambridge University Press, Cambridge, 1907.
Lebesgue, H., Legons sur !'integration et la recherche des fonctions primitives. Gauthier-Villars, Paris, 1904. Legons sur les series trigonometriques. Gauthier-Villars, Paris, 1906.
Whittaker, E. T., and Watson, G. N., A course of modern analysis. 3rd
edition. Cambridge University Press, Cambridge, 1920.
M ort-Oyraphs arul Articles
B6cher, M., Introduction to the theory of Fourier's series. Ann. Math.,
ser. 2, Vol. 7, 1906, pp. 81-152.
Courant, R., Uber die LOsungen der Differentialgleichungen der Physik.
Math. Ann., Vol. 85, 1922, pp. 280-325. Zur Theorie der linearen Integralgleichungen. Ibid., Vol. 89, 1923, pp. 161-178.
Hilbert, D., Uber <las Dirichletsche Prinzip. Festschr. Ges. Gottingen 1901,
Berlin 1901; reprinted Math. Ann., Vol. 59, 1904, pp. 161-186.
Montel, P., Surles suites infinies de fonctions. Ann. :Ee. Norm., ser. 3, Vol.
24, 1907' pp. 233-334.
Szeg0, G., Beitrag zur Theorie der Polynome von Laguerre und Jacobi.
Math. Zeitschr., Vol. 1, 1918, pp. 341-356. Uber Orthogonalsysteme von
Polynomen. Ibid., Vol. 4, 1919, pp. 139-157. Uber die Entwicklung einer
willkiirlichen Funktion nach den Polynomen eines Orthogonalsystems.
Ibid., Vol. 12, 1922, pp. 61-94.
CHAPTER III
Linear Integral Equations
§1. Introduction
1. Notation and Basic Concepts. Let K(s, t) be a function of the
two variables sand t defined and continuous in the region a :::; s :::; b,
a :::; t :::; b, and let A be a parameter. Furthermore, let f(s) and cp(s)
be two functions of the variable s continuous in the interval a :::; s :::; b,
which are connected by the functional equation
(1)
f (s) = cp(s) -
A
JK(s, t) cp(t) dt.
(All integrals are to be extended over the "fundamental domain"
as defined above, unless the contrary is explicitly stated.) The
functional equation (1) is called a linear integral equation of the
second kind with the kernel K(s, t). By (1) every continuous function cp(s) is transformed into another continuous function f(s); the
transformation is linear, since to c1cp1 + C'JfP2 there corresponds the
analogous combination cif1
cd2. We shall be primarily concerned with the problem of solving the integral equation, i.e. with
the problem of determining cp(s) when f(s) is given or of inverting
the linear integral transformation (1). Unless the contrary is explicitly stated, we shall assume that all quantities considered are .
real.
If the function f(s) vanishes identically we are dealing with a
homogeneous integral equati·on. If a homogeneous equation possesses
a solution other than the trivial solution cp = 0, the solution may be
multiplied by an arbitrary constant factor and may therefore be
assumed normalized. If cp1 , cp2 , · · · , 'Ph are solutions of the homogeneous equation then all linear combinations C1cp1 + C2cp2 + · · · +
ch'Ph are also solutions. Therefore, if several linearly independent
solutions are given we may assume that they are normalized and
orthogonal; for, if they were not they could be orthogonalized by
the procedure of Ch. II, §1 without ceasing to be solutions. We
+
112
113
INTRODUCTION
shall assume that linearly independent solutions of the same homogeneous integral equation are orthonormal. A value A (possibly
complex) for which the homogeneous equation possesses nonvanish1
ing solutions is called an eigenvalue of the kernel; the corresponding
solutions cp1, cp2, • • • , 'Ph (assumed normalized and mutually orthogonal) are called the eigenfunctions of the kernel for the eigenvalue A.
Their number is finite for each eigenvalue. For, applying the Bessel
inequality (Ch. II, §1) to the kernel K(s, t) and the orthonormal
functions 'P1 , <P2 , • • • 'Ph , we find
A'
f
[K(s, t)]' dt ;::: A'
t. [!
K(s, t) 'l';(t) dt]'
=
t.
<Ph)';
by integrating with respect to s we obtain
'A.
2
ff
2
[K(s, t)] ds dt
~
h.
This establishes an upper bound for h. Thus every eigenvalue possesses a finite multiplicity (i.e. number of linearly independent eigenfunctions).
We shall see in §6 that the integral equation represents a generalization of the problem of linear algebra which was treated in Ch. I, §2.
Its significance lies in the fact that it enables us to consider a variety
of different problems in mathematical analysis from a unified point
of view.
2. Functions in Integral Representation. In connection with equation (1), it is natural to inquire into the properties of functions
that can be represented by an integral of the form
(2)
g(s) =
f
K(s, t)h(t) dt.
In (2), g(s) is called an integral transform of h(s).
If h(t) is piecewise continuous, g(s) is certainly continuous.
over, if
f
2
[h(t)] dt
~
More-
M, where Mis a fixed bound, then the aggregate
of functions defined by (2) is actually equicontinuous; i.e. for every
positive e there exists a positive number Ci(e) independent of the
'YJ) - g(s) I < e whenever
particular function h such that I g(s
+
Such eigenvalues actually correspond to the reciprocals of the quantities
called eigenvalues in Chapter I.
1
114
III. LINEAR INTEGRAL EQUATIONS
l 7J I < o (compare
Ch. II, §2).
From the Schwarz inequality we
have
[g(s
+ 7J)
- g(s)]
2
~
J[K(s +
M
1],
2
t) - K(s, t)] dt
and since the kernel is uniformly continuous our assertion is proved;
for, the inequality
I K(s + 1], t)
- K(s, t)
I<
u
holds independent of t for arbitrarily small u if 1J is sufficiently
small.
Furthermore, if a sequence of kernels Kn is given for which
lim Kn(s, t) = K(s, t)
1'-+00
uniformly, then for a given h(t) the relation
g(s)
= ~~
JKn(s, t)h(t) dt
holds, and the convergence is uniform ins, since the passage to the
limit may be performed under the integral sign. It therefore follows
that all functions of the form
Yn(s) =
JKn(s, t)h(t) dt,
g(s) =
JK(s, t)h(t) dt
for the functions h(t) considered form an equicontinuous set as long
as h is restricted by
Jh dt
~
2
M.
These functions are also uni-
formly bounded, i.e. their absolute values lie below a common bound.
This follows from the Schwarz inequality
[gn(s)] 2
~
M
J
2
[Kn(s, t)] dt;
[g(s)]2
~
M
J
2
[K(s, t) ] dt.
3. Degenerate Kernels. A kernel, which can be written as a finite
sum of products of functions of s and functions of t
p
(3)
A(s, t) = ~ ah)~,(t),
i-1
FREDHOLM'S THEOREMS FOR DEGENERATE KERNELS
115
is called a degenerate kernel. Here we may assume that the functions ai(s) are linearly independent of each other, and that the same
is true of the functions f3i(t). For, otherwise, one of these functions
could be written as a linear combination of the others and A(s, J,)
as a sum of fewer than p terms of the above form. The possibility
of uniform approximation of a continuous function K(s, t) by polynomials (Ch. II, §4) implies that the kernel K(s, t) may be uniformly
approximated by a degenerate kernel, since every polynomial in s
and t evidently represents a degenerate kernel.
A degenerate kernel A(s, Omay be transformed into another form,
which is often more convenient. We suppose that the 2p functions of
s: a1(s), a2(s), · · · , ap(s); f:Ji{s), f32(s), · · · , {3p(s) are represented as
linear combinations of a system of q orthonormal functions
w1(s), W2(s), · · · , wq(s). This can always be done by orthogonalization of the given functions. Then A (s, t) appears in the form of a
double sum
q
(4)
A (s, t)
=
L Ci,"C1J1(s)w;(t).
i,;-1
2
The products wi(s)w;(t) form a system of q functions of s and t in
the square a ~ s ~ b, a ~ t ~ b which are mutually orthogonal and
therefore linearly independent. If A (s, t) is symmetric, i.e. if
A(s, t) = A(t, s) identically, then
q
L
(Ci; - C;i)w.(s)w;(t)
= O;
i,j-1
because of the linear independence of the products wh)w;(t) this
implies C;; = c;i .
A symmetric kernel K(s, t) can always be approximated uniformly
by symmetric degenerate kernels A(s, t). To see this we need only
A(t, s)], which approximates the
replace A(s, t) by ![A(s, t)
symmetric kernel K(s, t) if A(s, t) does. ,
+
§2. Fredholm's Theorems for Degenerate Kernels
The basic theorems of the general theory of integral equations,
1
which were first proved by Fredholm, correspond to the basic
1
I. Fredholm, Sur une classe d'equations fonctionnelles, Acta Math., Vol.
27, 1903, pp.365-390.
116
III. LINEAR INTEGRAL EQUATIONS
theorems of the theory of linear equations.
mulated in the following way:
Either the integral equation (1),
f(s)
= So(s)
They may be for-
JK(s, t)So(t) dt,
- X
with fixed X possesses one and only one continuous solution So(s) for
each arbitrary continuous function f(s)., in particular the solution
So = 0 for f = 0; or the associated homogeneous equation
(5)
tf;(s) = X
J
K(s, t)tf;(t) dt
possesses a finite positive number r of linearly independent solutions
i/11, t/;2, · • · , t/lr. In the first case the "transposed" integral equation
(6)
g(s)
= So(s)
- X
JK(t, s)So(t) dt
associated with (1) also possesses a unique solution for every g.
second case the transposed homogeneous equation
(7)
x(s) = X
In the
J
K(t, s)x(t) dt
also has r linearly independent solutions x1 , x2 , · · · , Xr ; the inhomogeneous integral equation (1) has a solution if and only if the given
function f (s) satisfies the r conditions
(8)
(f, Xi)
=
J
f(s)xh) ds = 0
(i = 1, 2, · · ·, r).
In this case the solution of (1) is determined only up to an additive
linear combination c1i/;1
c2i/;2
Crt/lr ; it may be determined
uniquely by the additional requirements:
+
(So, !/Ii) =
+ · ·· +
J
So(s)i/lh) ds
= 0
(i = 1, 2, · · ·, r).
We shall first prove these theorems for the case of the degenerate
kernel K(s, t) = A(s, t) represented by equation (3). Here the
theory of our integral equation reduces almost immediately to that
of a system of p linear equations in p unknowns. Let us write the
integral equation in the form
(9)
f(s) = So(s) - A ~ ah)
JrMt)So(t) dt.
FREDHOLM'S THEOREMS FOR DEGENERATE KERNELS
117
Now setting Xi = (fti, <P), multiplying (9) by l3i(s), and integrating
with respect to s, we obtain the system of equations
p
(10)
f; = Xi - A L CiiXi
(j
= 1, 2, ... ' p)
i=l
for the quantities Xi, where f i = (l3i, f) and Cii = (l3i, ai). If this
system of equations possesses one and only one solution X1 , X2 , • • • , Xp ,
then the function
p
<P(s) = f(s)
+ XL
Xiah)
i=l
is certainly a solution of the integral equation, as may be verified
by substituting in the integral equation and making use of equations
(10). Then the transposed system of equations
p
(11)
gi = Yi - A
L
i=l
CiiYi
also possesses a unique solution Y1, Y2, • • · , YP, and
p
<P(s) = g(s)
+ XL
Yi{Ms)
i=l
is a solution of the transposed integral equation (6).
If, on the other hand, the homogeneous system of equations
p
(12)
0 = Xi - A
L
CiiXi
(j = 1, 2, ... ' p)
i-1
possesses a nontrivial solution x1 , x 2
• • • , Xp ,
,
then
p
Y,,(s) = X L Xiah)
i=l
is a nontrivial solution of the homogeneous integral equation (5).
Two linearly independent solutions x 1 , x 2 , • • • , Xp and x: , x~ ,
· · · , x~ oi (12) evidently yield two linearly independent solutions
Y,,(s) = X .L:L1xiai(s) and Y,,'(s) = X Lf=1 x~ai(s) of (5) and conversely.
The existence of r linearly independent solutions 1/11, 1/12, • · • , 1/tr
of (5) and therefore of r independent systems of solutions of (12)
is equivalent to the existence of the same number of linearly independent solutions Ykl , Yk2, • · • , Ykp (k = 1, 2, · · · , r) of the transposed system
p
(13)
Ui = Yi -
AL: Ci,Yi
i-1
(j = 1, 2, ... ' p)
118
III. LINEAR IN'.CEGRAL EQUATIONS
for g; = 0, and therefore of r linearly independent solutions
x1(s), X2(s), · · · , xr(s)
of the transposed homogeneous integral equation (7); here
p
(14)
Xk(s) = X
L
Yk;/3;(s).
i-1
According to the theory of linear equations, the equations (10), and
therefore (13) and (6), always have unique solutions for r = O;
however, for r > 0, the inhomogeneous equation (10) and thus the
integral equation (5) can be solved if and only if the conditions
p
(15)
L
i-1
f1Yki
=0
(k = 1, 2, · · · , r)
are satisfied. Because of the definitions of Yki and f; these conditions are equivalent to
(16)
(f, Xk) = 0
(k = 1, 2, · · · , r).
This completes the proof of the Fredholm theorems for the case
of degenerate kernels.
§3. Fredholm's Theorems for Arbitrary Kernels
In order to utilize the results of the previous section for the treatment of the integral equation with an arbitrary kernel K(s, t), we
employ the convergence theorem of Ch. II, §2.
We suppose that K(s, t) is approximated uniformly by a sequence
Ai(s, t), A 2(s, t), · · · , An(s, t), · · · of degenerate kernels and consider, in addition to the integral equation (1), the approximating
integral equations
(17)
f(s) = <P(s) - X
J
An(s, t)<i0(t) dt.
If X is fixed there are two possibilities:
Case I. For infinitely many indices n the equation (17) possesses
a solution Pn(s) for every f(s), and for all these solutions (Pn, Pn) =
c~ ~ M, where M is a bound independent of n. In this case we
may discard all unsuitable values of n and renumber the kernels
An ; without loss of generality we may assume that the condition is
satisfied for all n.
FREDHOLM'S THEOREMS FOR ARBITRARY KERNELS
119
Case II. The above statement is not true. Then, for some
particular f{s), either
(a) a solution Pn(s) exists for infinitely many n (we may assume,
as above, that it exists for all n), but (Pn, Pn) = c!---+ oo as n---+ oo, or
(b) a solution exists only for a finite number of values of n (we
may assume that it exists for non), and therefore-by the Fredholm
theorems for degenerate kernels-the homogeneous integral equation
(18)
0 = <P(s) - X
JAn(s, t)<P(t) dt
possesses a normalized solution CTn(s) for every n.
In Case I the functions Pn(s) - f(s) are integral transforms of
Pn by kernels An(s, t); according to §1 they form an equicontinuous
and uniformly bounded set. Therefore, by the convergence theorem
of Ch. II, §2, there exists a subsequence of the functions Pn(s) which
converges uniformly to a limit function <P(s). By passing to the
limit in the integral equation (17), which is permissible, we see
that this limit function satisfies the integral equation (1). In C~se I
this equation has, therefore, a solution for every f (s).
In Case II (a) we divide the integral equation (17) for <P = Pn by
Cn and set Pn/Cn = CTn, obtaining the equation
f~)
=
CTn(s) - X
JAn(s, t)CTn(t) dt;
in Case II (b) we note that equation (18) holds for <P = CT n • In
both cases CT n is normalized; hence the respective -sequences of integral transforms CTn(s) - f(s)/cn and CTn(s) are equicontinuous and
uniformly bounded. Therefore subsequences from these sets converge uniformly to limit functions 1/;(s), which necessarily satisfy
the homogeneous integral equation (5),
1/;(s) = X
JK(s, t)i/;(t) dt,
and are normalized. Thus in Case II the
possesses nontrivial solutions, which we call
functions.
To derive from these considerations the
formulated in §2 we recall the fact, proved in
homogeneous equation
null solutions or eigenFredholm theorems as
§1, that for every value
120
III. LINEAR INTEGRAL EQUATIONS
of A only a finite number r of linearly independent eigenfunctions
can exist. If r = 0, Case II, which always leads to a normalized
solution of (5), is excluded. We are therefore dealing with Case I;
i.e., the integral equation (1) possesses a solution for every function
f(s). This solution is unique because a nonvanishing difference of
two solutions would be a nontrivial solution of (5), in contradiction
to the hypothesis that r = 0, i.e. that no such solution exists. The
first Fredholm theorem is thus proved.
Now consider r > 0. Let 1/11, 1/12, · · • , i/lr be r mutually orthogonal
normalized solutions of (5). Then, since An => K,1 the functions
(i = 1, 2, · · · , r;
n
=
1, 2, 3, · · ·)
satisfy Oni(s) => 0 for n ~ oo.
We now define A :(s, t) by
I
A n(s, t)
=
+ X1 ~ onh)i/li(t);
T
An(s, t)
then the A :(s, t) are degenerate kernels which approximate the
kernel K(s, t) uniformly. It is easily seen that all these kernels
A :(s, t) possess the r null solutions i/li(s).
There cannot be more than r linearly independent null solutions
for sufficiently large n. For, if i/lr+i,n(s) were a sequence of additional null solutions, which may be assumed to be orthonormal to
i/; 1 , i/; 2 , • • • , i/lr, then, by our convergence principle, there would
exist a null solution of (5) orthogonal to 1/11 , 1/12 , · · · , i/lr and therefore linearly independent of these functions; this would contradict
the hypothesis that (5) has exactly r linearly independent null
solutions.
Because of the validity of the Fredholm theorems for degenerate
kernels the transposed homogeneous integral equations
(19)
x(s) = A
JA :(t,
s)x(t) dt
also possess exactly r linearly independent null solutions Xi,n(s)
(i = 1, 2, · · · , r) for sufficiently large n. These solutions may be
1 The arrow~
is occasionally used to denote convergence. We employ the
double arrow=> to denote uniform convergence.
FREDHOLM'S THEOREMS FOR ARBITRARY KERNELS
121
taken to be orthonormal. Since the degenerate kernels A ~(t, s)
converge uniformly to the kernel K(t, s) we also obtain r mutually
orthogonal null solutions x1(s), x2(s), · · · , xr(s) for this kernel if we
pass to the limit, employ our convergence principle, and utilize the
equicontinuous and uniformly bounded set of functions Xi.n(s).
The transposed integral equation
(20)
x(s)
=
X
J
K(t, s)x(t) dt
cannot have more than r independent solutions, since otherwise we
could, reversing the argument, show that (5) would necessarily also
have more than r solutions.
Finally, we note that the conditions
(21)
J
(f, Xi) =
f(s)xh) ds = 0
(i = 1, 2, · · · , r)
are certainly necessary for the ~olvability of the integral equation
(1) for r > 0. This follows immediately if we multiply (1) by
xi(s), integrate, and reverse the order of integration, taking (20)
into account. To see that conditions (21) are sufficient we confine
ourselves to those indices n for which limn__ Xi.n(s) = xi(s)
(i = 1, 2, · · · , r). The Ein = (f, Xi,n) converge to zero with increasing n because of (21.) We now construct the functions
r
f.(s) = f(s) -
L
Ein
xi:.• (s);
i-1
they satisfy (f., Xi,n)
equation
(22)
=0
f,h) =
(i = 1, 2, · · · , r).
~(s)
- X
J
A ~(s,
Therefore the integral
t)~(t) dt
possesses a solution Pn(s) orthogonal to 1/t1(s), 1/t2(s), · · · , 1/tr(s)
the Fredholm theorems for degenerate kernels are valid. We see
that (p., Pn) is bounded independently of n; otherwise, the argument
for Case II(a) would yield a solution of (5) orthogonal to 1/t1(s),
1/t2(s), · · · , 1/tr(s), which is impossible by hypothesis. Therefore our
convergence principle again permits us to pass to the limit in the
integral equation, and because f n(s) => f(s), we conclude that (1)
has a solution. This completes the proof of all the Fredholm theorems
for our kernel K(s, t).
122
III. LINEAR INTEGRAL EQUATIONS
That there exist cases in which the homogeneous equation possesses
nontrivial solutions will be seen in the next section.
§4. Symmetric Kernels and Their Eigenvalues
The theory of integral equations, like the theory of bilinear forms
considered in Chapter I, can be developed in greater detail if the
kernel K(s, t) is symmetric, i.e. if it satisfies the relation
(23)
K(s, t) = K(t, s).
The integral equation is then identical with the transposed equation. We shall give a theory of symmetric integral equations which
does not depend on the preceding section.
Our main problem is to find values of the parameter X for which
the homogeneous integral equation (5) possesses a nontrivial (normalized) solution. As mentioned before, these values X = Ai and
the corresponding solutions are known as the eigenvalues and eigenfunctions, respectively, of the kernel K (s, t). We shall now prove
the theorem, analogous to that of Ch. I, §3: Every continuous symmetric kernel that does not vanish identically possesses eigenvalues and
eigenfunctions; their number is denumerably infinite if and only if
the kernel is nondegenerate. All eigenvalues of a real symmetric kernel
are real.
1. Existence of an Eigenvalue of a Symmetric Kernel. We begin
by proving that at least one eigenvalue exists. For this purpose we
consider the "quadratic integral form"
(24)
J(<p, <p) =
ff K(s, t)<p(s)<p(t) ds dt
which takes the place of the quadratic form of Chapter I; <p is any
function which is continuous or piecewise continuous in the basic
domain. The Schwarz inequality yields the relation
[J(<p, <p)] 2
~
(<p, <p) 2
ff
2
K (s, t) ds dt.
Therefore J (<p, <p) is bounded in absolute value if we impose the
condition
(25)
(<p, <p)
1.
SYMMETRIC KERNELS AND THEIR EIGENVALU ES
123
The integral form is equal to zero for all admissible functions "' if
and only if the kernel itself vanishes identically. To see this we
introduce the "bilinear integral form"
J('P, 1/1) = J(l/I, 'I') =
(26)
JJ K(s, t)"'(s)1/;(t) ds dt
and note the relation
(27)
J('P
+
1/1,"'
+
1/1)
= J('P, 'I')
+
2J('I', 1/1)
+
J(l/I, 1/1);
then clearly, if the quadratic form vanishes identically the bilinear
form also does so. Taking 1/;(t) = K(s, t)'P(s) ds in (26), we find
J
that
J(f
therefore
JK(s, t)'P(s) ds
K(s, !)¥>(•) ds)' dt = O;
=
0 for
arbitr~ry
'I'·
If, for a particular
value oft, we now take"' to be equal to K(s, t) we obtain the desired
identity K(s, t) = 0.
If a kernel has the property that J('P, 'I') can assume only positive
or only negative values (unless "' vanishes identically) it is said to
be positive definite or negative definite; otherwise it is said to be
indefinite.
Let us now suppose that J ('I', "') can take on positive values.
We consider the problem of finding a normalized function "' for which
J('P, 'I') assumes the greatest possible Yalue. Since J('P, 'I') is bounded,
there certainly exists a least upper bound K1 = l/A1 for the values of
J ('I', "'); we shall now show that this positive least upper bound is
actually attained for a suitable function 'l'(s). We suppose that the
kernel K(s, t) is approxima ted uniformly by a sequence of degenerate
symmetric kernels
of the form described at the end of §1.
the integral form J n('P, 'P) =
The maximum problem for
JJ An(s, t)'P(s)'P(t) ds dt with the sub-
124
III. LINEAR INTEGRAL EQUATIONS
sidiary condition (25) turns out to be equivalent to the corresponding
problem for a quadratic form in qn variables. Indeed, setting
(i = 1, 2, ... ' q,.),
we obtain
(28)
an expression for J n(cp, cp) as a quadratic form in X1 , X2 , · · · , Xqn to
be maximized subject to condition (25). Now the Bessel inequality
of Ch. II, §1, 3, applied to cp(s) and the orthogonal system of functions w1(s), w2(s), · · · , Wq,.(s), gives
qn
(cp, cp) ~ L: x~ ;
i=l
thus the variables in the quadratic form (28) must satisfy the condition :Lt.1 x~ ~ 1. Therefore the maximum of the form is attained
when L:t.1 x~ = 1, since otherwise the value of J ,.(cp, cp) would be
increased if we multiply by a suitable factor. We are thus
confronted with the problem of transformation to principal axes
which was treated in Ch. I, §3. The maximum of the form is accordingly attained for a set of values x1 , x2 , · · · , Xqn for which the
equations
(29)
(i = 1, 2, ... ' q,.)
hold. The factor of proportionality K1n is moreover equal to the
maximum of J ,.(cp, cp ). To verify this we multiply (29) by Xi and
take the sum, noting that the right side becomes K1n (since L x~ = 1),
while the left side becomes J,.(cp, cp). Let us now set
cp,.(s) = X1w1(s)
+ X2w2(s) + ·· · + XqnWqn(s),
where x1 , X2 , • • • , Xqn is the maximizing set of values. Because of
the orthogonality of thew,, and because
1 x~ = 1, this function is
normalized. Equations (29) imply relation
L:'t,
(30)
cp1i(s) = _!._
Kln
f
An(s, t)cpn(t) dt
and conversely. For, (30) follows from (29) if we multiply by wi(s)
and sum, noting that x, = (cp,. , wi); if on the other hand we multiply
SYMMETRIC KERNELS AND THEIR EIGENVALUES
125
(30) by wi(s) and integrate we obtain (29). Thus the function
cpn(s) is an eigenfunction of An(s, t) belonging to the eigenvalue
µin = 1/Ktn ; i.e.,
(31)
We now let n increase beyond all bounds. Then K1n must converge
to a number «i which is the corresponding positive upper bound of
J(cp, cp); for, from the relation
I K(s, t)
-
An(s, t)
I<
E
and the Schwarz inequality it follows that
2
[J(cp, cp) - Jn(cp, cp)] ~ l(b - a.)2,
where a and b are the limits of integration, provided (cp-, cp) ~ 1.
Thus, for sufficiently large n, the range of J n(cp, cp) coincides arbitrarily closely with the range of J(cp, cp); the same must therefore
be true of the upper bounds of the two ranges. Consequently
limn ..... «> K1n = K1 exists, and thus all the K1n lie below a fixed bound.
Therefore, in virtue of (31) and of the considerations of §1, the
functions cpn(s) are uniformly bounded for all n and form an equicontinuous set. By our convergence theorem it is possible to select
a subsequence cpn 1 , cpn 2 , • • • which converges uniformly to a limit
function 1/;1(s). Passing to the limit in equation (30) and in the
relations J n(cpn, cpn) = K1n and (cpn, cpn) = 1, We Obtain the equations
(32)
K11/;1(s) =
JK(s, t)1/;1(t) dt,
and
(33)
Thus the function 1/;1(s) solves the maximum problem for the form
J(cp, cp); it is an eigenfunction of the kernel K(s, t). Since it was
assumed that J(cp, cp) can take on positive values, K1 cannot be equal
to zero. For any arbitrary function 1/; we therefore have the inequality
(34)
as is seen at once by normalizing 1/;.
to Holmgren.)
(This method of proof is due
126
III. LINEAR INTEGRAL EQUATIONS
2. The Totality of Eigenfunctions and Eigenvalues. To obtain the
other eigenvalues and eigenfunctions we proceed as follows:
We consider the problem of maximizing the integral J(tp, 'P) if,
in addition to the condition ('fJ, 'P) = 1, the condition
is imposed. We assume that under these conditions J(tp, 'P) can
still take on positive values. By the second subsidiary condition,
the range of the integral form J('P, 'P) is more restricted in the present
maximum problem than in the first; hence the maximum "2 = l/µ 2
cannot be greater than the previous maximum K1 ; i.e., "2 ~ K1 and
µ2 ~ µ1 . Here as in the first maximum problem the existence of a
solution could be proved by reduction to a quadratic form and
passage to the limit. It is, however, more convenient to reduce the
problem directly to the problem of determining the first eigenvalue
of another kernel.
We form the symmetric function
(35)
Ku>(s, t) = K(s, t)
µ1
According to the above result the maximum problem
(36)
Jco('P, tp) =
JJ K(l)(s, t)tp(s)tp(t) ds dt
=
ma.ic. = "2 = l/µ2
with the condition (<p, 'P) = 1 can be solved by a function 1/12(s) which
satisfies the homogeneous integral equation
(37)
It is assumed here that J c1> ('fJ, 'P) still admits positive values, so that
"2
> 0.
We write equation (37) in the form
multiply by 1/11(s), integrate with respect to s, reverse the order of
integration in the iterated integral, and note that (1/11, 1/;1) = 1.
The right-hand side then vanishes, so that we have
(38)
SYMMETRIC KERNELS AND THEIR EIGENVALUES
127
i.e. the eigenfunction 1/12 is orthogonal to the eigenfunction 1/11.
Therefore we also have
(39)
JK(s, t)i/t2(t) dt JKcl)(s, t)i/t2(t) dt;
=
thus 1/t2(s) is an eigenfunction of K(s, t), and P.2 the corresponding
eigenvalue:
(40)
Because of the relation (1/12, 1/11) = 0, the value K2 can also be
regarded as the maximum of the integral form J(l)('P, 'P) subject to
the condition ('P, 1/11) = O; but under this condition J(l)('P, 'P) =
J('P, 'fJ). Therefore the function 1/12 is the solution of the maximum
problem formulated at the beginning of this section.
We can continue in the same way, constructing the kernel
(41)
= K(s, t) _
t/11(s)t/11(t) _ t/12(s)i/t2(t)
P.2
P.1
and seeking the maximum of the integral form
(42)
provided this form can still assume positive values. As above we
obtain a normalized solution 1/ta(s), orthogonal to 1/11 and 1/12 , and a
1/µ.3 , which satisfy the homogeneous integral
maximal value Ka
equation
This solution could also be obtained from the problem of maximizing the original integral form by a function which is orthogonal
to 1/11 and 1/12. Just as above it is seen that P.2 ~ P.a.
This procedure may be continued indefinitely, provided that all
the kernels K c1> , K c2> , • · · give rise to integral forms which can
128
III. LINEAR INTEGRAL EQUATIONS
take on positive values.
sequence a first kernel
If, on the other hand, there occurs in this
(-13)
s
for which J<m>(cp, cp)
0 for all cp, then we break off the sequence
with the eigenfunction i/lm(s) and the eigenvalue µm.
In any case we have the result: The least positive eigenvalue µi
of the kernel K(s, t) is the reciprocal of the maximum value "1 which
the integral form J(cp, cp) assumes under the subsidiary condition
(cp, cp). = 1. This maximum is attained when cp is the first eigenfunction 1/11 of K(s, t). The eigenvalues µh (h = 2, 3, · · · ), ordered in an
increasing sequence, are defined recursively as the reciprocals of the
maxima "h which J (cp, cp) attains under the subsidiary conditions
(cp,cp)=I,
(cp, i/lv) = 0
(v = 1, 2, 3, · · · , h - I).
The maximum Kh is attained for cp = i/lh , the h-th eigenfunction.
The sequence of positive eigenvalues breaks off when and if, in
this sequence of maximum problems, one occurs for which J (cp, cp)
cannot take on positive values.
In an analogous manner we can now obtain a sequence of negative
eigenvalues and corresponding eigenfunctions µ_1, µ_2, · · · and
i/l-1(s), i/l-2(s), · · · , provided the form J(cp, cp) can assume negative
values. We need only consider the minimum problems corresponding to the above maximum problems. We thus arrive at an infinite
or terminating sequence of negative nonincreasing eigenvalues
(44)
and associated mutually orthogonal eigenfunctions
i/l-1(s), i/l-2(s), · · · .
The eigenfunctions 1/lh(s) (h > 0) are: orthogonal to the eigenfunctions 1/1-k(s) (k > 0). This fact results from the two equations
SYMMETRIC KERNELS AND THEIR EIGENVALUES
129
if we multiply the first by t/1-k(s) and the second by t/lh(s), subtract
one from the other, and integrate; noting that K(s, t) = K(t, s), we
obtain
which proves the orthogonality, since Kh ~ K-k •
The continuation of this procedure leads to a sequence of eigenvalues, which may be alternately positive and negative. .We order
these eigenvalues according to their absolute values and denote
them by X1 , X2 , · · • ; thus I X1 I : : ; I X2 I ::::; · · · . We denote the
corresponding eigenfunctions by cp1 , cp2 , · · · ; they form an orthonormal system of functions.
If the kernel K(s, t) possesses only a finite number of eigenvalues
X1, A.2, • • • , A.n it must be degenerate and can be represented in the
form
(45)
For, according to the considerations of page 123, the kernel
must vanish identically, since both the maximum and the minimum
of the associated integral form
J(cp, cp)
=
JJ K (s, t)cp(s)cp(t) ds dt
are equal to zero. Thus a kernel which possesses only a finite number
of eigenvalues and eigenfunctions is degenerate. Conversely, a degenerate kernel possesses only a finite number of eigenvalues and eigenfunctions. Indeed, as we have seen above, the problem of determining the eigenvalues of such a kernel is equivalent to the eigenvalue
problem for· a quadratic form, where only a finite number of eigenvalues occurs (cf. Chapter I).
According to §1, an eigenvalue is called multiple or degenerate,
specifically r-fol,d degenerate, if it is associated with r but with no
more than r linearly independent eigenfunctions (which may be
taken to be orthogonal).
multiplicity or degeneracy.
Every eigenvalue can have only a finite
This theorem, proved in §1, may also
130
III. LINEAR INTEGRAL EQUATIONS
be obtained in the following way: We apply the Bessel inequality
to the orthogonal system 'Pi, '1'2, • • • , obtaining
(46)
f [K(s,
I)]' dt
~
t. (!
K(s, t),,,;(t) dt)'
or
(47)
f
[K(s, t)] 2 dt
~ ~ ['Pi~~)] •
2
Two conclusions follow: First, the series
T(s)
(48)
=
E['Pi(s) ]2
i-1
A~
all of whose terms are positive, converges absolutely. Second, if we
integrate with respect to s and note that ('Pi , 'Pi) = 1 we find
(49)
2
(( [K(s, t)] ds dt
JJ
~
t
~~·
i-1 " •
Thus the sum of the reciprocals of the squares of the eigenvalues conTherefore the eigenvalues can have no finite point of accumulation; if there are infinitely many eigenvalues, their absolute
values must increase beyond all bounds and only a finite number of
them can be equal to each other.
From this we shall conclude that the eigenvalues Ai and eigenfunctions 'Pi defined aboYe by successive extremum problems constitute the totality of the real eigenvalues and eigenfunctions (it will
be shown below that no complex eigenYalues can occur). Suppose
that x is an eigenfunction linearly independent of the 'Pi, and that
the corresponding eigenvalue is u, which may be taken to be positive.
Then, by the above argument, x must be orthogonal to all eigenfunctions associated with the eigenvalues Ai rf:. u. If, however, u = µh
is one of the eigenvalues defined above and is r-fold, that is if
µh-1 < µh = µh+l = · · · = µh+r-1 < µh+r ("degenerate case"),
then x may be replaced by an eigenfunction = x + Col/th + · · · +
Cr-11/th+r-1 which is orthogonal to the eigenfunctions 1/th , 1/th+t , • • • ,
i/lh+r-1 , since x was assumed to be linearly independent of these
functions. For simplicity we denote this function again by X·
Then in either of the cases considered x is orthogonal to all eigenverges.
x
131
SYMMETRIC KERNELS AND THEIR EIGENVALUES
functions 1/li.
the relation
Therefore, for every n for which
J(x, x)
= JJff K(s, t)x(s)x(t)
ds dt
µn+i
= !<T (x, x) ~
is positive,
1
- -
µn+I
holds, in virtue of the maximum property of the eigenvalues. Hence,
if there are infinitely many positive eigenvalues "'" , it follows from
limn__ "'" = oo that (x, x) = 0, i.e. that x vanishes identically.
On the other hand, if there are only a finite number n of positive
eigenvalues, then J (x, x) cannot assume positive values under the
subsidiary conditions (x, 1/li) = 0 (i = 1, 2, · · · , n); we conclude
once more that (x, x) = 0 and therefore that x = 0.
This proof holds. equally well for negative u. Therefore every
eigenfunction which is orthogonal to all the 1/li must vanish identically,
which proves our assertion that the 1/li represent all the possible
eigenfunctions.
This fact has interesting consequences which will be useful later.
If '71(s), '72(s), ... and s1(s), s2(s), ... are two sequences of continuous (or piecewise continuous) functions whose norms lie below
a fixed bound M, then the relation
(50)
~ J~n>('ln, sn)
=
l~~
ff K~n>(s,
t),,n(s)sn(t) ds dt = 0
with the kernel
K~n>(s, t) =
K(s, t) -
~ 'Pi(s~;i(t)
holds uniformly; i.e., for a given M, the smallness of the left side
depends only on n.
In fact, by the maximum property of the eigenvalues and eigenfunctions, we have
I J~n)(T]n +
Sn, 'In+ Sn)
1
I ~ I An+i I N('ln +
I J~n)(T]n, 'In) I ~I x:1 I'
1
I J~n><sn, Sn)
Sn)
4M
~ I An+i I
I~ I A~1 I'
It follows immediately from Schwarz's inequality that
N(1111
+
t .. ) = ("1n,
"ln)
+ (t.. , r.. ) + 2(11 .. , r..) ~ 4M.
1
132
III. LINEAR INTEGRAL EQUATIONS
and therefore, since I An I --+
J~n>(1J
+ r, ,., + r)
=
oo
and since
J~n)(1}, ,.,)
+ J~n)(r, r) + 2J~n)(7J, r),
the assertion follows at once.
Furthermore, a kernel is positive definite if and only if all its eigenvalues are positive. For, in this case and only in this case is the
minimum of the integral form J (<p, <p) for normalized <p positive.
Thus J (<p, <p) cannot have negative values.
Finally, all eigenvalues of a real symmetric kernel are real. Although this fact will become obvious in §5, we shall prove the theorem
here in another, more direct way. The theorem states that there
is no complex number A = p + iq where q ~ 0, with an associated
complex function <p(s) = 1/;(s)
ix(s) (tf; and x are real functions at
least one of which does not vanish identically) such that <p(s) =
+
A
J
K(s, t)<p(t) dt.
For, the relation cp(s) = X
JK(s, t)1p(t) dt would
then have to hold for the conjugate complex quantities
as above,
0 = (A - X)
J<p(s)cp(s) ds
=
2iq
J(t/1
2
+x
2
)
ip
and X and,
ds;
i.e., q = 0 which proves that A is real.
3. Maximum-Minimum Property of Eigenvalues. As in the case
of quadratic forms (Chapter I) each eigenvalue An of a symmetric
kernel and its associated eigenfunction can be directly characterized
by a maximum-minimum problem without reference to preceding
eigenvalues and eigenfunctions.
Let us consider first the positive eigenvalues µn of the kernel
K(s, t) and assume that there are at least n of these. We pose the
problem to make J(<p, <p) a maximum if <p is subject to the requirement (<p, ~)
1, and, in addition, to then - 1 restrictions
(51)
(i = 1, 2, · · · , n - 1),
where V1 , v2 , • • • , Vn-1 are any arbitrary given continuous functions.
Certainly there exists an upper bound of J(<p, <p) for some admissible
function <p. This upper bound depends somehow on the choice of the
functions V1 , V2 , • • • , Vn-1 ; we may therefore denote it by
Kn {V1, V2, • • • , Vn-d or briefly by Kn {vi}. In particular, for Vi = t/li,
we have Kn {Vi} = Kn by the theorems of the previous section; this
SYMMETRIC KERNELS AND THEIR EIGENVALUES
upper bound is attained for 'P = 1/ln(s).
set of functions V1 , V2 , · · · , Vn-1 ,
133
We now assert that, for any
Kn{V1} ~ Kn.
To prove this we construct an admissible function c,c(s) = c11/;1(s) +
Cnl/ln(s) as a linear combination of the eigenfunctions
c21/12(s)
1/11, 1/12, · · · , ..Yn. The conditions (<P, <P) = 1 and (51) then take the
form
+ ··· +
n
L: c~
i-1
=
1,
"
ci(l/li , vh) = 0
L
i-1
(h
=
1, 2, · · · , n - 1).
This is a system of n - 1 homogeneous linear equations .in n unknowns Ci with a normality condition; such a system always has a
solution. If the resulting function 'P is substituted in J(<P, <P), one
obtains
J(<P, <P) =
L"
cicd(l/li, 1/lk).
i,k-1
Since J(l/li, 1/li) = l/µ.i and J(l/li, 1/lk) = 0 for i ¥= kit follows that
The maximum of J (<P, 'P) is thus certainly at least equal to Kn , and
we obtain the result: The n-th positive eigenvalue An of K(s, t) is the
reciprocal of the least value Kn which Kn {Vi} attains as the functions Vi
are varied; here Kn{vi} is defined as the maximum value (or least upper
bound) of J(<P, <P) for normed functions <P(s) subject to then - 1 further
conditions (51). The minimum of this maximum is attained for
Vn-1 =
1/ln-1 and 'P = 1/ln .
In the same way we define the negative eigenvalues and the associated eigenfunctions 1/1-n (n > 0) in terms of the maximum of
the minimum of J(<P, <P), subject to the corresponding conditions.
Among the many applications of the above result we mention the
following theorem which can be deduced from the maximumminimum property of the eigenvalues: If to a kernel K(s, t) a positive
definite kernel K+(s, t) (or a negative definite kernel X-(s, t)) is added,
then each positive (or negative) reciprocal eigenvalue of the sum K + K+
X-) is not less (or not greater) than the corresponding reciprocal
(or K
+
13-1
III. LINEAR INTEGRAL EQUATIONS
eigenvalue of the kernel K. 1 The proof is analogous to that of the
corresponding theorem in Ch. 1, §-1.
§5.
The Expansion Theorem and Its Applicati.ons
1. Expansion Theorem. If we knew that in analogy with the
transformation to principal axes of a quadratic form, the kernel
could be expanded in a series
(52)
which would converge uniformly in each variable, then it would follow
that eve1-y function g(s) of the form
(53)
g(s) =
J
K(s, t)h(t) dt,
where h(t) is any piecewise continuous function, could be expanded
in the series
00
g(s)
=
L Yi<Pi(s),
i=-1
However the relation (52) is not valid in general; we are thus
forced to adopt a 8omewhat different approach to justify the expansion for g(s). Let g(s) be an integral transform of h(t) with
symmetric kernel K(s, t) as defined in equation (53); let hi = (h, <Pi)
be the expansion coefficients of h with respect to the orthonormal
eigenfunctions <P1 , <P2 , · · · , and let
g, = (g, cp,)
-~
be the expansion coefficients of g; in accordance with the Bessel
inequality, the series Li!1 h~ converges. By equation (47), the
sum
T(s) =
f
i-1
f<ieh) ]2
A~
See H. Weyl, Das asymptotische Verteilungsgesetz der Eigenwerte linearer
partieller Differentialgleichunge n (mit einer Anwendung auf die Theorie der
Hohlraumstrahlung), Math. Ann., Vol. 71, 1912, pp. 441-479.
1
135
EXPANSION THEOREM AND ITS APPLICATIONS
converges and is uniformly bounded in s (by max
in view of the Schwarz inequality, we have a9~b
[
hn<Pn(8)
An
+ ... + ~<Pm(8)]
f
2
[K(s, t)] dt);
2
Am
~ (h~ + ... h!>C"~~)J' + ... + ["';.~)]').
Since the remainder h! + h!+1 + · · · + h! is arbitrarily small pro2
2
vided n is sufficiently large, and since [<Pn(s)] /A~ + · · · + [<Pm(s)] /A!
is always below a bound independent of 8, it follows that the series
QO
QO
L
Yi <Ph) =
i-1
L
(hi/Ai)<Ph)
i=l
converges absolutely and uniformly.
Its sum
n
-y(8)
= lim Lg, <Pi(s) = lim 'Yn(8)
n-oo i-1
n-oo
is a continuous function of 8. It remains to be shown that -y(8) is
identical with g(s). For this purpose we construct the kernel
Kcn>(8, t)
= K(s, t)
-
~ <Pi:(8~~i(t)
1
so that
g(s) - 'Yn(8) =
J
Kcn>(s, t)h(t) dt;
we multiply this equation by an arbitrary continuous function w(8)
and integrate with respect to 8. Because of relation (50) the righthand side in the relation
f
w(8) (g(8) - 'Yn(8)) d8 =
ff
Kcn>(8, t)h(t)w(8) d8 dt
c9nverges to zero and we obtain
J
w(8) (g(8) - -y(s)) ds = 0
1 We shall henceforth use this notation for the functions which were
denoted by K(.> on p. 131.
136
III. LINEAR INTEGRAL EQUATIONS
since 'Yn(s) :=} -y(s). This equation holds for an arbitrary function
w(s), in particular for w(s) = g(s) - ., (s). Since g(s) - -y(s) is
continuous the relation (g - -y, g - -y) = 0 can hold only if g(s) - -y(s)
vanishes identically, as was to be proved. We have thus obtained
the fundamental expansion theorem:
Every continuous function g(s) which, as in (53), is an integral
transform with symmetn:c kernel K(s, t) of a piecewise contimwiis
function h(t), can be expanded in a. series in the eigenfunctions of K(s, t);
this series converges uniformly and absolutely.
2. Solution of the Inhomogeneous Linear Integral Equation. As
an application of this theorem we shall derive the formula for the
solution of the inhomogeneous integral equation (1),
f
f(s) = cp(s) - ).
K(s, t)cp(t) dt.
We assume initially that the parameter ). is not equal to any of the
eigenvalues Xi . If the continuous function cp with the expansion
coefficients (cp, 'Pi) ·were a solution of the integral equation, then, by
the expansion theorem applied to h(t) = "Acp(t), the function cp(s) - f(s)
= g(s) would be given by the uniformly and absolutely convergent
series
where
t
g(s) = cp(s) - f(s) =
(54)
Ci
= (g, 'Pi).
=
f
K(s, t)cp(t) dt,
But, because of (54), we must have
(g, cpi) = >.
Ci =
cicpi(s) = ).
).
~ (cpi, cp)
ff
=
).
K(s, t)cpi(s)cp(t) ds dt
.
~ (cpi, f)
).
+ ~ (cpi, g),
from which it follows that
(55)
).
Ci
=
f i Xi
-
).
We thus obtain the series expansion for cp,
(56)
cp(s) = f(s)
r
+ >. ~ Ai _.:.
co
). <Pi(s),
which must represent the solUtion of (1). That this really is so,
i.e. that (56) actually provides the solution of (1), is easily seen:
EXPANSION THEOREM AND ITS APPLICATIONS
137
The series converges absolutely and uniformly. To prove this we
need only note that, for sufficiently large i and a given arbitrary A,
the relation I Ai - A I > I Ai l/2 is sure to hold. Thus, except for a
finite number of ternis, the series 2 I A I L~1 I fi II 'Pi(s) I/I Ai I is
a majorant of our series; the uniform convergence of this majorant
has already been proved. If we now substitute the series (56) in
(1), we verify immediately that the equation (1) is satisfied.
In conformity with the theory of §3 this solution fails only if
A = Ai is an eigenvalue; it remains valid even in this case if f(s) is
orthogonal to all eigenfunctions 'Pi belonging to the eigenvalue Ai .
Since, by §3, the integral equation (1) fails to have solutions for
certain functions f(s) only if A is an eigenvalue, it follows that there
can be no eigenvalues of our kernel other than the values Ai . The
assertion that all the eigenvalues of a real symmetric kernel are real,
which was proved on page 132, has now become self-evident.
3. Bilinear Formula for Iterated Kernels. Another application
of the expansion theorem is obtained setting h(cr) = K(cr, t). For
the "iterated kernel"
K< 2>(s, t) =
f
K(s, cr)K(cr, t) dcr
we then have the expansion
K<2>(s, t) =
~ cp~:s)
f
K(cr, t)cpi(cr) dcr
or
(57)
In the same way, the subsequent iterated kernels
f
= ff
= f
K(3)(s, t) =
K<n>(s, t)
K< 2>(s, cr)K(cr, t) dcr
K(s, cr1)K(cr1, cr2)K(cr2 , t) dcr1 dcr2,
K(n-l)(s, cr)K(cr, t) dcr
138
III. LINEAR INTEGRAL EQUATIONS
admit the expansions
(58)
(n = 2, 3, · · · ),
all of which converge absolutely and uniformly both in s and in t,
and uniformly ins and t together (see subsection 4).
From (57) it follows that
(2)( ) = ~ [<Pi{s))2.
K ~8
~ '~ '
-·
thus
lim
ft-+00
(K< >(s, s) - t
2
"'
·
[<Pi~~)]2)
i-1 .
=
O.
I\'
But this means that
(59)
lim
n-+oo
J[K(s, t) -
t
i-1
2
<Pi(s)':i(t)] dt = O;
Ai
:E:..1
i.e., the series
<Pi(s)<Pi(t)/Ai converges in the mean to K(s, t).
If, for fixed s, this series converges uniformly in t and therefore
represents a continuous function L(s, t) oft, we must have K = L.
For, in this case we can carry out the passage to the limit in (59)
under the integral sign, obtaining J[K(s, t) - L(s, t)] 2 dt = 0 and,
therefore, K - L = 0.
4. Mercer's Theorem. 1 Formula. (58) for n > 1 may be regarded
as a substitute for equation (52), which in general cannot be proved.
However, for an important special case, we can state the following
theorem: If K (s, t) is a definite continuous symmetric kernel, or if it
has only a finite number of eigenvalues of one sign, the expansion (52) is
valid and converges absolutely and uni/ormly.
In the proof we shall assume initially that K(s, t) is positive definite, i.e. that all eigenvalues Ai are positive. We first remark that,
for every positive definite kernel H(s, t), the inequality H(s, s) ~ 0
holds. For, if H(so, s0 ) < 0, there would be a neighborhood of the
point s = so, t = So, say Is - so I < E, I t - so I < E, such that
H ( s, t) < 0 everywhere in this region. We then define the function
1 T. Mercer, Functions
of positive and negative type and their connection
with the theory of integral equations, Trans. Lond. Phil. Soc. (A), Vol. 209,
1909, pp. 415-446.
139
EXPANSION THEOREM AND ITS APPLICATIONS
'P(s) by cp(s) = 1 for I s - ·so I S
this function we have
ff
and by cp(s) = 0 elsewhere.
E,
H(s, t)'P(s)cp(t) ds dt
For
<0
in contradiction to the hypothesis that H is positive definite.
we now apply this result to the positive definite kernel
If
we obtain
K(s, s) -
t
i~1
2
[c,oh)]
Ai
~ 0.
Therefore the series
all the terms of which are positive, converges for every value of s.
Because of the relation
<Pn(8) <Pn(t)
( VAn VAn
+
•••
<
-
+
([cpn(s)]
An
2
<Pm(s) <Pm(t))
VAm VAm
2
+ ... +-{cpm(s)] )([cpn(t)] + ... + [cpm(t)]
2
2
Am
An
2
)
Am
(Schwarz inequality) the series L:'::1 'Pi(s)cpi(t)/Ai also converges
absolutely; it converges uniformly in t for fixed s and in s for fixed t.
Thus the function defined by this series is continuous in s for fixed·
t and conversely. Ir{ view of subsection 3, it is therefore equal to
the kernel K.
Finally we show that this series also converges uniformly in both
variables together; because of the above inequality it suffices to
verify the uniformity of the convergence of the series
But, according to what we have just proved, this series is equal to
K(s, s), which is a continuous function. Now the following theorem
140
III. LINEAR INTEGRAL EQUATIONS
holds :1 if a series of positive continuous functions of one variable
converges to a continuous function, the convergence of the series is
uniform in the interval in question. This theorem leads immediately to the result stated.
The existence of a finite number of negative eigenvalues cannot
alter the convergence of the series (52), since the kernel can be made
positive definite by subtractio n of the terms 'Pi(s)cpi(t)/'Ai belonging
to negative eigenvalues. Thus our convergence theorem is completely proved.
§6.
Neumann Series and the Reciproca l Kernel
The preceding theory of integral equations implies a prescription
for actually computing solutions with arbitrary accuracy (see also
§8). It does not give the solutions, however, in an elegant closed
form such as was obtained in the theory of equations in Chapter I.
To find comparable explicit solutions we shall use a method analogous
to that of Chapter I. We rewrite the integral equation (I), inserting
in place of cp(t) in the integral the expression obtained from (1), and
repeat this procedure indefinitely. With the aid of the iterated
kernels we thus write (1) in the form
cp(s) = f(s)
=
f(s)
+ A JK(s, t)f(t) dt + A
2
+A
f
K(s, t)j(t) dt
J K< >(s, t)cp(t) dt
2
f
+ A f K<
+
2
A
K< 2>(s, t)f(t) dt
3
3
>(s, t)cp(t) dt
and see, just as in Chapter I, that the solution is given by the infinite
series
(60)
cp(s) = f(s) +A
J
K(s, t)f(t) dt
+ A JK< >(s, t)j(t) dt + ···,
2
provided that this series converges uniformly.
postulate the uniform convergence of
(61)
1
K(s, t) = K(s, t)
+
Compare footnote on p. 57.
"}..K< 2>(s, t)
+
2
If, moreover, we
A. 2K< 3>(s, t)
+
NEUMANN'S SERIES AND RECIPROCAL KERNEL
141
then the solution of the integral equation (1)
f(s) = cp(s) - X
J
K(s, t)cp(t) dt
is represented by the "reciprocal integral equation"
(62)
cp(s) = f(s)
+ X J K(s, t)f(t) dt.
The function K(s, t) = K(s, t; X) is therefore called the reciprocal or
resolvent kernel.
Series (60) and (61) are known as Neumann's series. They converge
uniformly if IX I is sufficiently small, e.g. if IX I < 1/(b - a)M,
where Mis an upper bound of the absolute value of K(s, t). Thus,
for sufficiently small J X I , the reciprocal kernel is an analytic function of X. It satisfies the relations
(63)
K(s, t; >.) = K(s, t)
+ A JK(s, u)K(cr, t; X) dcr,
K(s, t; >.) = K(s, t)
+ A JK(cr, t) K(s, er; X) dcr,
K(s, t; X) - K(s, t; >.') = (X - X')
J K(s,
u; X) K(cr,
t; X') du;
this can be immediately verified by substitution.
If the kernel K(s, t) is symmetric we can find a remarkable form
for the resolvent kernel, which shows how the analytic function K
depends on X. Assuming IX I to be sufficiently small, we make use
of the series expansion (58) for the symmetric kernels K< 2\s, t),
K«3>( s, t), · · · and take the sum of the geometric series occurring in
(61); we immediately obtain
(64)
By proceeding as in §5, 1 and 2, we see that the series on the right
converges for every value of X which is not an eigenvalue, and the
convergence is uniform in s and t.
Relation (64) which has so far been proved only for sufficiently
small I X I provides the analytic continuation of the resolvent
K(s, t; X) into the entire complex A-plane, with the eigenvalues Ai
142
III. LINEAR INTEGRAL EQUATIONS
appearing as simple poles. Thus (64) represents the decomposition
of the resolvent into partial fractions, and we may express the result
as follows: The resolvent of a symmetric kernel is a meromorphic f unction of X which possesses simple poles at the eigenvalues of the integral
equation. Its residues at the poles Ai provide the eigenfunctions
belonging to these eigenvalues. From the Neumann series and the
representation (64) it follows that the radius of convergence of the
Neumann series is equal to the absolute value of the eigenvalue with
the smallest square.
According to the general theory of functions, the resolvent K(s, t; X),
being a meromorphic function, can be represented as the quotient of
two integral transcendental functions; it is to be expected that each
of these functions can be expanded in an everywhere convergent
power series the coefficients of which can be obtained directly from
the given kernel. In the algebraic case we have such a representation in the formulas of Ch. I, §2. It is reasonable to suppose that
very similar formulas can be obtained here. We may expect, furthermore, that these formulas are in no way restricted to symmetric
kernels only, but remain valid for arbitrary continuous unsymmetric
kernels. Such representations have been obtained by Fredholm,
who then used them as the starting point of the theory. In the next
section we shall derive Fredholm's formulas, again approximating
the kernel uniformly by degenerate kernels An(s, t) and then passing
to the limit n ---+ oo •1
§7.
The Fredholm Formulas
Since we shall not make any use of the Fredholm formulas later on,
we omit some intermediate calculations involving determinants. 2
We shall employ essentially the same procedure and terminology
as in Ch. I, §2. For a degenerate kernel
n
K(s, t)
=
A(s, t)
=
L ap(s){3p(t),
P-1
1 This method was first employed by Goursat: Sur un cas elementaire de
!'equation de Fredholm, Bull. Soc. math. France, Vol. 35, 1907, pp. 163-173.
Compare also H. Lebesgue, Sur la methode de M. Goursat pour la resolution
de !'equation de Fredholm, ibid., Vol. 36, 1909, pp. 3-19.
2 See G. Kowalewski, Einfiihrung in die Determinantentheorie, Veit
Leipzig, 1909.
143
THE FREDHOLM FORMULAS
the integral equation (1) goes over into
n
(65)
<P(s) = f(s)
+ XL Xpap(s)
=
f(s)
+ XE(a(s), x)
P=l
if we set Xp = (<P, {Jp) as before. Making use of the notation
(aq , {Jp), we obtain the system of equations
YP = (J, {J 11 ), kpq
n
(66)
YP
=
A
Xp -
L
kpq Xq
Q.=1
for the quantities Xp . The solution of this system is given by
E(u x)
'
= - A(u, '!f_; "A)
A("A)
and, thus, the solution of ( 1) is
(67)
<P(s) = f(s)
+ XE(a(s), x)
= f(s) - X A(a~(xr
~) ;
in (67), we have
A(u, y; X)
(68)
=
A1(u, ·Y) - "AA2(u, y)
A(X) = 1 - XA1
+ ··· +
+ · ·· +
(-1r-1xn-1 An(u, y),
(-l)"X"An;
with
Ah(u,y) =
L
0
Up1
Up2
Up11
YP1
kPlPl
kPlP2
kPlPh
YP2
kP2Pl
kP2P2
kP2P1&
YP11
kPhPl
kPhP2
kPl&PI&
,
.........................
(69)
kPlPl
ll.h =
kPlP2
kPlPh
kP2P1&
L kP2Pl
....................
kP2P2
In (69) the summation indices Pi, P2,
from 1 to n with Pi < P2 < · · · < Ph .
, Ph run independently
144
III. LINEAR INTEGRAL EQUATIONS
The sum of determinants A(a(8), y; >.)may evidently be written in
the form
J A[a(8), {3(t); X]f(t) dt, so that the solution (67) of the integral
equation takes on the form
<P(8) = f(8)
(62')
+ X J K(8, t; X)f(t) dt
with the resolvent
A(a(8),{3(t);X)
A(A.)
K(8, t; >.)
(70)
D(8, t; A.)
-D(xf-.
In the formulas (69) the summation, instead of extending only
over all sets of indices for which Pi < p2 < · · · < Ph, may just as
well extend over all possible combinations, the result being divided
by h!. Making use of this fact and of the definition of kpq, and
employing certain simple theorems concerning determinants, we
obtain the formulas
D(8, t; >.) =
=
-
A(a(8), {3(t); >.)
Do(s, t) -
:!
+ ;, D2(8,-t)>. 2 -
D1(8, t)X
+ (nc-1r_ l) ! Dn-1(8, t)X
1
(71)
n-l,
D xn
+ (-1r
n.I
n
'
where we have used the abbreviations
A (8, t)
Dh(s, t) =
A (8, s1)
· · · A (s, sh)
ff··· f ~.(~1·'· ~)..~:~1.,.~1~ . ."."." . .~:~1.'.~h:
d8 1 d8 2 .. · dsh
A(sh, t) A(8h, s1) · · · A(8h, sh)
(72)
A(s1, s1)
A(s1, 82)
A(s1, sh)
A(82, 81)
A(82, 82)
A Cs2 ' sh) ds1d82 · · · dsh
A(8h, 81)
A(sh, 82)
for h = 1, 2, · · · , n, and Do(8, t) = A(8, t).
145
THE FREDHOLM FORMULAS
Thus the integral rational functions D(s, t; A) and D(A) of A have
been written down explicitly in terms of the given kernel. The
series (71) may be formally continued as jnfinite series since the
quantities Dh for h > n and Dh(s, t) for h > n - 1, as defined by
(72), all vanish for the degenerate kernel A(s, t) = L:;=1 ap(s)/3p(t).
Now, if the arbitrary continuous kernel K(s, t) is approximated
uniformly by a sequence of degenerate kernels, then the corresponding expressions (72) converge to the corresponding determinants of
the kernel K(s, t). The infinite series
D(s, t; A) = Do(s, t) -
(73)
A
D(A) = 1 - - Di
l!
A
! D1(s, t)
1
+
2
+ 2!A- D2 -
where A is replaced by K in (72), represent integral transcendental
functions of A for the nondegenerate kernel K(s, t). To see this it
is only necessary to show that they converge for every value of A.
If I K(s, t) I ~ M for alls and t, then, by Hadamard's inequality for
determinants (Ch. I, §5, 2),
I Dh(s, t) I ~ v' (h + I)h+1 Mh+ 1 (b
I Dh I ~ Vf1i Mh(b - a)h.
- a)\
Now the series
:t
V(h
+ l)h+1 Mh+t(b
h-9
- a)"' Ah'
h!
.converge for every value of A1 and are majorants of the series of the
absolute values of the terms of (73). Thus the assertion is proved.
It follows that, for every value of A, the convergence relations
lim Dn(s, t; A)
n-+OO
= D(s, t; X),
lim Dn(A) = D(A)
n-+OO
hold uniformly ins and t, where the quantities with index n refer to
the n-th approximating degenerate kernel An(s, t) and those without
1 This can be seen from the fact that 1/h ! < e"/h", since the term h"/h ! occurs
in the expansion of e". Therefore the h-th root of the coefficient of X" in the series
on the right is less than M(b - a)e/h 1 t2 and, consequently, converges to zero
for h-+ co; the same is true for the other series.
146
III. LINEAR INTEGRAL EQUATIONS
index to K(s, t). Therefore, as long as A is not one of the zeros Ai
of D(A), we have, for the resolvent of the kernel K(s, t),
x! D1(s, t) + ···
Do(s, t) (74)
1
- - - - - -2 - - - - = lim Kn(s, t; A);
A
A
K(s, t; A)
1 - 1! D1
+
21 D2 -
···
thus, for the solution of an integral equation with arbitrary kernel
K (s, t) we obtain the formula
<p(s) = f(s)
(75)
+ A JK(s, t; A)f(t) dt.
The above formulas are known as the Fredholm formulas in honor
of their discoverer. Evidently the relation
(76)
holds.
Dh =
J
Dh-1(s, s) ds
Note that we also have 1
(77)
D'(A) = -
f
D(s, s; X) ds
-and that the m-th derivative is given by
(78) D(m)(A) = (-1r
rr ... Jn(Sl' 2' : : : 'SmSm IA) ds1ds2 ... dsm'
jj
s.
S1 , S2 ,
,
where
(79)
and
(80)
K(s1, t1) • · · K(s1, tm) K(si, u1) · · · K(si, uh)
=!Ff
K(s2, ti) · · · K(s2, tm) K(s2, u1) · · · K(s2, uh)
K(sm, t1) · • · K(sm,tm) K(sm, u1) · · · K(sm, uh) du1 ···duh.
K(u1,t1) • · · K(u1,tm)K(u1,u1) · · · K(cr1,cr1a)
K(uh, t1) • • · K(uh, tm) K(crh, u1) • · · K(cr1a, u1a)
1
See Fredholm, loc. cit.
ANOTHER DERIVATION OF THE THEORY
147
We add the remark that the null solutions are obtained for the
zeros X = Ai of D(X), in the case of simple poles, by determining the
residues of the resolvent K(s, t; X) at these points. The proof fol1
lows easily from our formulas.
§8. Another Derivation of the Theory
The general -theory of integral equations as formulated above is
based on the fact that from the family of solutions of the approximating integral equations it is possible to select a sequence which
converges uniformly to a solution of the given integral equation.
Nevertheless, the concepts of measure of independence and of
asymptotic dimension of a sequence of functions, introduced in
Ch. II, §3 enable us to base the theory of integral equations on a
somewhat different foundation. Since this procedure provides some
relevant additional insigl].t it will be presented here.
1. A Lemma. Let 1/11 (s), 1/12(s), · · · , be a sequence of functions
whose. norms are all below a fixed bound M and for which the relation
(81)
1/1,h) - X
JK(s, t)l/ln(t) dt ~
0
holds in the sense of uniform convergence. Then the functions .Yn(s)
form a smooth sequence offunctions with finite asymptotic dimension r.
To prove this we note that relation (81) remains valid if the functions .Yn(s) are replaced by any functions xn(s) which are of the form
Xn(s) = X11/tn 1 + X21/tn 2 + · · · + Xpl/lnp. The absolute values of the
coefficients X1 , x 2 , • • • , Xp are assumed to be bounded, and the
·
functions
"1111 ' "1112 ' • • • ' l/lnp
are any p functions of the sequence I/In such that the indices ni tend
to infinity with n. Now if, among the functions .Yn(s), there are
sets of r functions each with arbitrarily large indices n such that
the measure of independence of each of these sets remains above
a fixed bound a, if, in other words, the asymptotic dimension of the
sequence is at least r, then we may orthogonalize the functions of
these sets; the coefficients that arise in this orthogonalization
process will all be less than the bound I/Va (Ch. II, §3, 1). We
1 For further details concerning the formal apparatus of the Fredholm theory
see, for example, G. Kowalewski, Einfiihrung in die Determinantenth eorie,
Veit, Leipzig, 1909.
148
III. LINEAR INTEGRAL EQUATIONS
thus obtain sets of r mutually orthogona l functions wn,i(s)
(i = 1, 2, · · · , r; n = 1, 2, · · ·) for which the limit equation
!~ (•..;(s)
(82)
holds uniformly ins.
leads to the result
J
- A
=
K(s, t)w•. .(t) dt)
0
The usual argument using Bessel's inequality1
fJ [K(s, t)]' ds dt ::". ~ J[! K(s, t)w.,<(t) dt]' ds
for every n; and from this and from equation (82) it follows that
ff [K(s,
~ ; 2•
2
t)] ds dt
We have thus found a finite bound for the asymptoti c dimension of
the sequence; that the sequence is smooth follows immediate ly
from (82). For, if En denotes a number which tends to zero as n
increases, the Schwarz inequality leads to
[1fn(s)]
2
::;
M>..
2
J[K(s, t)] dt + En
2
which means that the functions 1fn(s) are uniformly bounded;
similarly the i:elation
(xi lfn 1
follows from
+ ·· · + Xp lfn/ ::; EA
J (x11fn
1
+ ··· +
2
f [K(s,
2
Xplfnp) ds
<
2
t)] dt
+ En
E. Therefore the se-
quence is smooth.
2. Eigenfunct ions of a Symmetric Kernel. We shall employ the
lemma just proved to obtain the eigenfunctions of a symmetric
kernel K(s, t) which we approxima te uniformly by the degenerate
symmetric kernels An(s, t). As before, let µi 11>, µ~ 11 >, · · · and
µ2._~>, µ2._~>, • · • denote the positive and negative eigenvalues, respectively, of An(s, t), and let 1fi 11 \s), 1f~ 11 >(s), · · · , 1f2,_~>(s), 1f2,_~>(s), · · ·
be the correspond ing eigenfunctions. Multiple eigenvalues are supposed to be written down an appropriat e number of times. Let
Jr.(<P, <P) =
i
ff
An(s, t)<P(B)<P(t) ds dt
See §4, 2 of this chapter (p. 130).
ANOTHER DERIVATION OF THE THEORY
149
and
JJ K(s, t)<p(s)<p(t) ds dt
J(<p, <p) =
be the integral forms belonging to An and K, respectively, and let us
assume, as we may, that J(<p, <p) admits positive values. Then
Kin> = l/µin> is the maximum of Jn('P, <p) for a normalized function;
let Ki = l/µ1 be the upper bound of J(<p, 'P) under ,the same normalization condition. Since, for sufficiently large n, the values of J ('P, <p)
and J n('P, <p) differ by less than any arbitrarily small fixed number,
we must have limn-+oo µf n> = µi. Therefore relation
(83)
t/l~n> (s)
follows from iYfn>(s) -
- µ1
µ~n>
JK(s, t)t/l~n\t) dt
f
An(s, t)t/lf n>(t) dt
=> 0
=
0 since An(s, t)
=>
Consequently, in virtue of our lemma, the functions irin>
form a smooth sequence of finite positive dimension r (the vanishing
of r would contradict the normalization of the functions iYin». According to Ch. II, §3 they therefore determine a linear space of functions with the normalized orthogonal components t/11.1(s), t/11.2(s), · · · ,
t/11As), which are necessarily solutions of the homogeneous integral
equation
K(s, t).
(i = 1, 2, · · · , r);
thus the functions t/l~n> are eigenfunctions of K(s, t) belonging to the
eigenvalue µ1 .
In exactly the same way we may obtain the remaining eigenvalues
and eigenfunctions of K(s, t). Thus, for example, Kin> = l/µ~n> is
the minimum, obtained by a suitable choice of vi(s), v2(s), · · · ,
Vh-1(s), of the maximum of Jn('P, <p) subject to the conditions
(<p, <p) = 1 and ('P, vi) = 0 (i = 1, 2, · · · , h - 1).
If we now define K1i. = 1/µh as the analogous lower bound of the
upper bound of J(<p, <p), then we again have limn-+oo µ~n> = µh because
of the proximity of the values of Jn(<P, <p) to those of J(<p, <p). 'This
leads to the relation
150
III. LINEAR INTEGRAL EQUATIONS
and the rest of the argument proceeds as above. To obtain the
negative eigenvalues and the corresponding eigenfunctions, we must
consider the corresponding minimum or maximum-minimum problems. If only a finite number of eigenvalues of one or the other
sign occur we simply break off the procedure at the appropriate
stage.
3. Unsymmetric Kernels. This method also provides simplification for the unsymmetric integral equation (1). A brief indication
will suffice; we employ the same notation as before. In Case I, suppose the quantities Pn and Cn are such that for all n the norm c~ remains below the bound M. Then the norm of the difference
Pn - Pm = r nm also remains bounded, namely less than 4M. Furthermore
.~~ [i-••(s)
- >..
JK(s, t);-••
(t) dt] = 0
uniformly in s. Therefore, by our lemma, every subsequence of the
double sequence rnm in which n and m both tend to infinity possesses
a bounded asymptotic dimension r, and the bound of r depends only
on K(s, t) and X. Thus our double sequence tnm defines, through a
limiting process, a linear space of functions (see Ch. II, §3), with a
finite number r of orthorgonal components 1/;1(s), 1/;2(s), · · · , 1/lr(s),
unless the asymptotic dimension of every subsequence is zero, i.e.
unless t nm => 0. In the latter case the Pn(s) simply converge uniformly to a solution of the integral equation (1). In the case r > 0
the lf;i(s) are solutions of the homogeneous equation. We replace
Pn by a function
'ln(s) = Pn(s)
+ X11/;1(s) + · · · + Xrl/lr(s)
which is orthogonal to 1/;1(s), 1/;2(s), · · · , lf;r(s).
the relation
[ TJn(s) - A
f K(s, t)11n(t) dt] -
For these functions
f(s) => 0
certainly holds. As above, we may now apply the lemma to the
differences 'ln - '7m = rnm ; we arrive easily at the conclusion that
the dimension of every partial subsequence of this sequence must be
zero, i.e. that the functions '7n(s) converge uniformly to a solution of
the integral equation orthogonal to the functions 1/li(s).
Simil8il'ly, in Case II we obtain, with the aid of the lemma, a linear
ANOTHER DERIVATION OF THE THEORY
151
space of solutions of the homogeneous integral equation as the limit
set of the sequence un(s) = Pn(s)/cn .
In this way the second approach provides more precise information concerning the nature of the convergence relations which obtain
here. In particular, we see that a solution of the homogeneous or
inhomogeneous equation may be approximate d as closely as desired
by considering approximati ng integral equations with the kernels
An(s, t).
4. Continuous Dependence of Eigenvalues and Eigenfunctions on the
Kernel. In considering how the solutions of integral equations
vary with the kernel, we restrict ourselves to the eigenvalue problem
for a symmetric kernel K(s, t). Suppose that the kernel K(s, t) is
the uniform limit of a sequence of other symmetric kernels Kn(s, t)
(n = 1, 2, · · · ). If we consider functions cp for which (cp, cp) ~ M,
then the values of the quadratic integral forms Jn(cp, cp) and J(cp, cp),
with the kernels Kn and K, respectively, will differ arbitrarily little
for sufficiently large n. The same is therefore true of the maxima
~nd minim.a of these forms under the conditions (cp, cp) = 1, (cp, vJ = 0,
and also for the minima of the maxima or the maxima of the minima.
In other words: The h-th positive and h-th negative eigenvalues vary
continuously with the kernel. As far as the eigenfunctions are concerned, we cannot expect actual continuity in view of the arbitrariness
of sign and of the occurrence of multiple eigenvalues. Instead, the
following is true: Let Ah be an r-foUl, eigenvalue of the kernel K(s, t),
i.e. let
(n)
'\Ah = lim ,Cn)
Ah+r-1,
= · · · = l'Im ,(n)
= l'Im '\A h+l
Ah
bu.t assume that this relation does not hold for xt~l and X~~~. Then,
for increasing n, the linear space formed from the eigenfunctions
t/tin'(s), t/t~l(s), · · · , t/ti~~-1(s) of the kernel Kn(s, t) converges1 uniformly
to the linear space of the eigenfunctions of K(s, t) for the eigenvalue >.h.
This theorem is a complete expression of the continuity properties
of the eigenfunctions. It may be proved almost immediately on the
basis of our lemma by noting that, for the sequence of eigenfunctions
tfti~k(s) (O ~ k < r), the limit relation
[>J.~Wsl - x. j
K(B,
t)>J.~~k(t) dt] = o
holds, and that this sequence is certainly of asymptotic dimension r.
1
For the notion of convergence of linear families see Ch. II, §3, 2.
152
III. LINEAR INTEGRAL EQUATIONS
§9. Extensions of the Theory
The developments of §§1 to 6 and of §8 can be significantly generalized in two directions.
In the first place, all the arguments remain valid if we co.nsider
integral equations for functions of several, say m, independent variables. Thus, suppose f(s) and <i0(s) are continuous functions of the
variables s1 , s2 , · · · , Sm in a fixed finite region G; suppose also that
K(s, t) is a continuous function of the variables s1, s2, · · • , Sm and
ti , t2 , · · · , tm , each set of variables ranging over the region G; let ds
denote the volume element ds1ds2 · · · dsm of G and dt the corresponding volume element dt1dt2 · · · dtm with the understanding that all
integrals are to be taken over the region G. Then the integral equation
f(s) = <i0(s) - X
f
K(s, t)<i0(t) dt
is an integral equation for a function <P(s) of m variables, and its
kernel K(s, t) is a function of 2m variables; our whole theory remains
valid word for word.
Secondly, the requirement that the kernel be continuous is far
more stringent than necessary. It can be considerably relaxed without impairing the results. We shall not attempt to obtain the most
general conditions under which the theory is valid; we shall merely
extend the theory sufficiently far to treat the most important applications. Let us first consider kernels K(s, t) which are functions of
only two variables. Our previous arguments, apart from those that
lead to Mercer's theorem (§5, 4), remain valid, with inessential modifications, for piecewise continuous kernels, since, as shown in Chapter
I, any piecewise continuous function can be approximated in the
mean as closely as desired by a continuous function. The arguments
can also be justified, moreover, for kernels which are infinite at some
points, provided that the integrals
ff
2
[K(s, t)] ds dt,
f [K(s, t)]
2
ds,
f
[K(s, t)]2 dt
exist and that the latter two integrals, regarded as functions oft and
of s, respectively, remain below .a fixed bound. The latter assumptions will be satisfied if, for example, the kernel becomes infinite for
s = t of order less that !, i.e. if K(s, t) is of the form K(s, t) =
153
EXTENSIONS OF THE THEORY
H(s, t) Is - t 1-a with 0 ~ a < ! and H(s, t) continuous everywhere. To show that for such a kernel the theorems obtained are
valid, it is sufficient to approximate the kernel by continuous degenerate kernels An(8, t) in such a way that the following conditions
are satisfied:
[K(8, t) - An(8, t)] 2dt becomes arbitrarily small
f
uniformly ins as n--+
oo
and
f
+
[An(8
'Y],
t) - An(s, t)]2 dt becomes
arbitrarily small uniformly in 8 and in n if I 'YJ I is taken sufficiently
small. All our arguments can then be carried out. Likewise, in
the case of two independent variables, all our theorems remain valid
if, for 81 = t1, 82 = t2 , the kernel becomes infinite of less than the
first order, since in this case the convergence of the integral
ff
[K(81, 82, t1, t2)]2 d81d82 is not affected.
Analogously, we may
admit singularities of order less than I in the case of three variables; in general for n variables singularities of order less than n/2
are permissible in K.
We finally note without proof that· it is not difficult to extend our
theory to integral equations for which only the above hypothesis
concerning the integrals of K 2(8, t) are satisfied. No additional requirements such as continuity of the kernel are needed.
§10. Supplement and Problems for Chapter 111
1. Problems. Show that:
(a) The kernel
~ sin ns sin nt
n
£- - - - - - = 21
n-1
I. +
ti .
s 1og sm 2
sm s-- -t
2
I
(0 ~ s, t ~
1r)
has the eigenvalues An = 2n/1r and the eigenfunctions sin nt.
(b) The symmetric kernel
1
1 - h
2
211" 1 - 2h cos (s - t)
(0 ~ s, t ~ 211")
+h
2
with I h I < 1 has the eigenfunctions 1, sin ns, cos ns, with the eigenvalues 1, l/h", 1/hn.
(c) For the symmetric. kernel
K(s , t) _- _,,-e
1 (s2+t2)/2
V1r
18 e
_.,.2
-oo
dr
f"° e
_.,.2
t
dr
(s ~ t)
154
III. LINEAR INTEGRAL EQUATIONS
the Hermite orthogonal functions e• 212 (dne-• 2 /dsn) are eigenfunctions
with the eigenvalues An = 2n + 2.
(d) The eigenfunctions of the symmetric kernel
K(s, t)
=
e<•+012 ["° e-" dr
J,
T
(0
s ss
t)
are the Laguerre orthogonal functions
a" e-•hi (1-h)
e-.12 _
I
<Jh" I - h
•
h--0'
+
n
1. 1
2. Singular Integral Equations. The general theory may no longer
be valid if the kernel possesses singularities of too high an order, or
if, in the case of an infinite fundamental domain, it fails to go to
zero fast enough at infinity (the kernels considered in the previous
subsection converge strongly enough).
We shall now give some examples of integr&.l equations with eigenvalues of infinite multiplicity.
The integral formula
the associated eigenvalues are An
1111
1
o
•
sm st
(
=
'V 2 e-at + a2 +t t2)
•
/;.
d
t
=
'V• / ;2 e-aa + a2 +S s2
holds identically in a. Therefore, in the fundamental domain
0
s, t < oo , the kernel sin st has infinitely many eigenfunctions for
the eigenvalue X = 1.
Hermite's orthogonal functions (see subsection l(c)) are eigenfunctions of the kernel e•,, with the eigenvalues ~"/V27r. Thus each
of the four numbers ±I/-y'2;, ±i/V'};; is an eigenvalue of this
kernel of infinite multiplicity.
An example of an integral equation 2 with infinitely many eigenvalues in a finite interval is the equation
s
~(s)
=
X
1:
e- 1- ' 1 ~(t) dt;
1 See R. Neumann, Die Entwicklung
willkiirlicher Funktionen nach den
Hermiteschen und Laguerreschen Orthogonalfunktionen usw., Dissertation,
Breslau, 1912.
2 Related integral
equations have been treated by E. Hopf: Uber lineare
Integralgleichungen mit positivem Kern, Sitzungsber. Akad. Berlin (math.phys. Kl.), 1928, pp. 233-275. See also the article11 by U. Wegner, G. H. Hardy,
and E. C. Titchmarsh cited there.
155
SUPPLEMENT AND PROBLEMS
2
its solutions are eaill with the eigenvalues A = !(1 + a ). Thus
every A > ! is an eigenvalue.
1
3. E. Schmidt's Derivation of the Fredholm Theorems. Let us take
form
the
A = 1 and let us writ~ the kernel K(s, t) in
n
Kts,
L a.(s)f311(t) + k(s, t),
=
t)
-1
with
ff
2
[k(s, t)} ds dt
<
1.
According to §6, the Neumann series
for k(s, t) converges for :h = 1 and therefore represents the resolvent
K(s, t) of the kernel k(s, t). Writing the integral equation (1) in the
form
f 1(s)
=
<P(s) -
fi(s)
= f(s)
J
k(s, t)<P(t) dt
with
n
+ L x.a..(s)
11-1
we thus have the formula
\O(s) = f(s)
+
t.
x,a,(s)
+
J
.(s, t)[f(t)
+
+ -y,(s)fl,(t))
J
t.
J
x,a,(t) dt
or
f2(s) = f(s)
+
= l"(s) -
f
J[t.
K(s, t)f(t) dt
(a,(s)fl,(t)
l"(t) dt,
with
In this way, the given integral equation is reduced in a single step
to one with a degenerate kernel.
1 E. Schmidt, Zur Theotie der linearen und nicht linearen Integralglei chungen. Part II: Auflosung der allgemeinen linearen Integralglei chung, Math.
Ann., Vol. 64, 1907, pp. 161-174.
156
III. LINEAR INTEGRAL EQUATIONS
4. Enskog's Method for Solving Symmetric Integral Equations. 1
We consider a positive definite kernel K(s, t), the first eigenvalue of
which is greater than 1, i.e. a kernel for which the inequality
f
2
[<p(s)] ds -
ff
K(s, t)<p(s)<p(t) ds dt
>
0
is valid for all <p. The integral equation (1) (assuming A
be written in the form f(s) = J(<t>), where·
J(<t>) = <p(s) -
f
1) may
K(s, t)<t>(t) dt.
From any complete system of functions <t>1, <t>2, • • • , we now construct, by a process analogous to the orthogonalization process described in Chapter II, a system of functions vi(s), v2(s), · · · with the
property that
f
vi J(vk) ds = Oik· 2
Such a system is known as a
"complete system polar with respect to the kernel K(s, t)."
set a,, =
f"' J(v,,) ds
=
f v,, f
ds, we immediately obtain <t>(s) =
I::..1
a,,v,,(s), provided that this series converges uniformly.
dentally, the functions v,, satisfy the "completeness relation"
f
If we
<p(s)J[<p(s)] ds =
Inci-
fa!,
"-l
no matter how the piecewise continuous function <p(s) is chosen.
5. Kellogg's Method for Determination of Eigenfunctions. 3 Starting
with an arbitrary normalized function <t>o(s), we determine the
functions <p,,(s) and the numbers A,, from the relations <t>11+1(s)
A,,+1
f
K(s, t)<p,,(t) dt, N<p,,
=
1.
The passage to the limit can be
carried out and leads to an eigenvalue and associated eigenfunction
of the kernel or its iterated kernel.
The reader may establish the connection between this approach
and the notion of asymptotic dimension and carry out the derivation
from this point of view.
1 D.
Enskog, Kinetische Theorie der Vorgange in massig verdiinnten Gasen,
Dissertation, Uppsala, 1917.
2 011: is the "Kronecker
delta" defined by Oii = 1, Oil: = 0 for i ~ k.
8 0. D. Kellogg, On the existence
and closure of sets of characteristic functions, Math. Ann., Vol. 86, 1922, p. 14-17.
157
SUPPLEMENT AND PROBLEMS
6. Symbolic Functions of a Kernel and Their Eigenvalues. For
the integral operator associated with a given kernel, spectral relations
analogous to those developed for matrices in Chapter I exist. Let
us consider, in particular, an integral rational function f(u) =
a"u" which vanishes for u = O; we replace the powers of u by
the corresponding iterated kernels of the symmetric kernel K(s, t)
and thus obtain the kernel
L::=1
n
H(s, t) = f[K] =
L
a"KM(s, t).
l'=l
The following theorem now holds: The eigenfunctions 'Pi of H are
identical with the eigenfunctions of K, and the corresponding reciprocal
eigenvalues 1/i of H are connected with the reciprocal eigenval1ws Ki of
K by the equation
We may, in fact, verify immediately that each eigenfunction 'Pi of
K belonging to the eigenvalue Ai = 1/Ki is also an eigenfunction of
H with the eigenvalue 1/f(Ki). Since the relation
JJ [H(s, t)]
2
ds dt =
~ [f(Ki)]
2
holds we see that H has no other eigenvalues and eigenfunctions.
7. Example of an Unsymmetric Kernel without Null Solutions. The
kernel
K(s, t) = '£sin 11s sin (11
v2
,,... 1
has, in the region 0 ::; s,
kernels are
K<n> (s, t) _
-
+
l)t
t ::; 271'", no null solutions. For, the iterated
7rn-1
L
,,
2
11 (11
sin vs sin (11
2
• • • (v
+ 1)
+ n)t
+n -
1) 2 '
and therefore the Neumann series converges for all values of "A.
the function
The same result may be obtained by showing that
1
D(A.) associated with K is a constant for this kernel.
1 Similar kernels may be found in Goursat, Cours d'analyse (see bibliography).
158
III. LINEAR INTEGRAL EQUATIONS
8. Volterra Integral Equation. 1 If K(s, t) = 0 for s
tegral equation may be written in the form
= ~(s)
f(s)
<
t, the in-
K(s, t)q,(t) dt.
- X{
Integral equations of this type have been treated in particular by
Volterra. The reader may show that the resolvent for this equation is an integral transcendental function of X, and that therefore
Volterra's integral equation possesses one and only one solution for
every X, c9nsequently, no null solution for any X and its Neuman
series converges.
9. Abel's Integral Equation. 2 Abel studied a particular integral
equation of the Volterra type, which is important in many applicaions, in order to solve the following problem: Let a material point
of mass m move under the influence of gravity on a smooth curve
lying in a vertical plane. Let the time t which is required for the
point to move along the curve from the height x to the lowest point
of the curve be a given function f of x. What is the equation of the
curve? This problem leads to the integral equation
f
(x) -
-
l:r:
o
'P(t) dt
v2ucx -
t)
If we assume that f(x) is a continuously differentiable function vanishing at x = 0, the solution of the equation turns out to be
'P(x) = V2g
l:r:
7r
0
f'(t) dt ,
vx
-t
where g is the acceleration of gravity; the equation of the curve is
y
=
l:r; v I "' (t) 2
i
I
dt.
More generally, we may consider the equation
f(x) =
J:r:
a
'P(s) dt
(s - x)a
(0
<a < 1),
1 V. Volterra,
Le9ons sur les equations integrales et les equations integrodiff erentielles, Chapter II, Gautier-Villars, Paris, 1913.
z Abel, Solution de quelques problemes a l'aide d'integrales definies, Works,
Christiania, 1881, I, pp. 11-27; B6cher, Integral Equations, p. 8, Cambridge
University Press, Cambridge, 1909.
159
SUPPLEMENT AND PROBLEMS
the solution of which, for continuously differentiable j(x), is
f'(s) ds .
sin a?r
f(a)
rp(x) = sin ar
a (s - x)l-a
7r
(x - a)l-a
7r
l:i:
+
10. Adjoint Orthogonal Systems belonging to an Unsymmetric
Kernel. 1 From an unsymmetri c kernel K(s, t) we may con-
=
struct the two symmetric kernels K'(s, t)
and K" (s, t)
=
JK(u, s)K(u, t) du.
J K(s, u)K(t, u) du
Then there exists a sequence
of pairs of functions and associated values Xsuch that
f K(s, t)Y,,,,(t) dt,
x! f K'(s, t)rp,,(t) dt,
rp..(s) = X.,
y;,,(s)
= X.,
JK(t, s)rp,,(t) dt,
rp.(s) =
y;,,(s)
= x!
f K"
Every function which can be written in the form
(s, t)l/l"(t) dt.
JK(s, t)h(t) dt
admits of an absolutely and uniformly convergent expansion in a
series of functions of the orthogonal system "°" ; similarly, any function of the form
JK(t, s)h(t) dt can be expanded in a series of the
L::..1
rp,,(s)1f;.,(t)/X. holds, profunctions 1f;,,. The relation K(s, t) =
in each variable.
uniformly
converges
right
the
vided the series on
X,, and by the
values
the
by
uniquely
determined
The kernel K is
systems.
orthogonal
two independent
11. Integral Equations of the First Kind. An integral equation of
the first kind is an equation of the form
(84)
f (s)
=
JK(s, t)rp(t) dt.
We have seen, for example, that a function can be expanded in a
series of the eigenfunctions of a kernel if a solution of the integral
equation of the first kind (53) exists. Other examples are given by
the Fourier integral and the Mellin integral transformati on (Ch. II,
§10, 8). The specific difficulty of the theory of integral equations
of the first kind is due to the following fact: for a continuous kernel
K(s, t), the equation (84) transforms the manifold of all piecewis<'
1 E. Schmidt, Zur Theorie der linearen und nichtlinearen Integralgleich ungen. I. Entwicklung willki.irlicher Funktionen na.ch Systemen vorgeschriehener. Math. Ann. Vol. 63, 1907, pp. 433-476.
160
III. LINEAR INTEGRAL EQUATIONS
continuous functions cp(s) into a more restrictive manifold, since all
functions f(s) obtained this way are certainly continuous. If K(s, t)
is differentiable every piecewise continuous function, and in fact
every integrable function, is transformed into a differentiable function; Thus, in general, the integral equation with continuous f(s)
cannot be solved by a continuous futiction cp. For more general classes
of functions j(s) we may expect (84) to be solvable only if K(s, t)
deviates from regular behavior in some way. The reader may consider the preceding and subsequen t examples of such integral equations from this point of view; an infinite fundamen tal domain has
the same effect as a singularity of the kernel.
For a symmetric kernel, one may attempt a solution of the form
<i0(s) = .L:~1 AvXllCf'v(s), where Xv = (f, 'iOv) are the Fourier expansion
coefficients off with respect to the system <i01, <i02, • • • of eigenfunctions of the kernel. If this series converges uniformly -a condition
that restricts f(s) in view of the increase in Av with increasing v-then
it actually provides a solution of (84).
The general case is covered by a theorem of Picard1 which gives
necessary and sufficient conditions for the existence of a square integrable solution <i0(s) of an equation of the first kind f(s) =
J K(s, t)<i0(t) dt with an arbitrary (possibly unsymmetric) kernel K:
If 'Pi, i/li are the pairs of adjoint functions belonging to K(s, t), as
defined in subsection 10, and Ai the corresponding eigenvalues, then
the above integral equation can be solved if and only if the series
t. Al (J
f(•)<P;(s) ds )'
converges.
12. Method of Infinitely Many Variables. If w1(s), w2(s), · · · is
a complete orthogonal system in the fundamen tal domain, and if
we define Xi= (<iO,·Wi), fi = (f, wi), and kPti =
K(s, t)w 11 (s)wq{t) ds dt,
JJ
then the integral equation (1) leads immediately to the system
co
fi
= Xi -
A
.L: ki;X;
(i = 1, 2, ... )
j-1
of infinitely many linear algebraic equations for the infinitely
1
E. Picard, Sur un theoreme general relatif aux equations integrales de
premiere espece et sur quelques problemes de physique mathematiq ue. Rend.
Circ. math. Palermo, V. 29, 1910, pp. 79-97.
161
SUPPLEMENT AND PROBLEMS
many unknowns X1 , X2 , · · · . The sums 2:':..1 x~ , I:i..1 f~ , and
L~;-1 k~; converge, as may be deduced from Bessel's inequality.
The theory of the solutions of this system of equations then leads to
the theorems about the integral equation (1).
13. Minimum Properties of Eigenfunctions. The eigenfunctions
~i , ~2 , • • • of a symmetric kernel, or the two orthogonal systems ~i
and 1/;i for an unsymmetric kernel, and the corresponding eigenvalues
Ai, may be obtained from the following minimum problem: The kernel K(s, t) is to be approximated by a degenerate kernel An(s, t) =
I::.1 4Ji(s)'Yi(t)//\i in such a way that
ff (K -
2
An) ds dt be-
comes as small as possible. The reader may prove that the solution
is given by 4Ji = ~i, 'Yi = 1/li , /\i = Ai.
14. Polar Integral Equations. For kernels of the form K(s, t) =
A(s!S(s, t), where S(s, t) is symmetric and A(s) is continuous except
for a finite number of jumps, it is possible to derive results similar
to those obtained for symmetric kernels. The case where S(s, t) is
definite, i.e. where it has only positive (or only negative) eigenvalues,
has been studied extensively. In this case, which has been treated
by Hilbert1 and Garbe, 2 the integral equation is said to be polar or
of the third kind. As was the case for symmetric kernels, the resolvent
has only simple and real poles. For the corresponding residues,
which yield the "polar eigenfunctions," an expansion development
theorem holds which is analogous to that formulated by Hilbert for
symmetric kernels. In particular, if the iterated kernel K< 2>(s, t)
does not vanish identically at least one eigenvalue exists. Incidentally, the theorem that the resolvent has only real simple poles
remains valid if S(s, t) is simply assumed to be positive; the additional
theorem that there is at least one eigenvalue holds if S(s, t) is positive
and K< 2>(s, t) does not vanish identically. 3
15. Symmetrizable Kernels. 4 Kernels for which the resolvent
has only real simple poles may be characterized in the following
1 D. Hilbert, Integralgleichungen, Chapter
15. Here a somewhat different
form is taken for the polar integral equation.
2 E. Garbe, Zur Theorie der Integralgleichung dritter Art. Math. Ann.,
Vol. 76, 1915, pp. 527-547.
3 J. Marty, Sur une equation integrale. c. R. Acad. SC. Paris, Vol.150, 1910,
pp. 515-518. Developpements suivant certaines solutions singulieres, ibid.,
pp. 603-606. Existence de solutions singulieres pour certaines equations de
Fredholm, ibid., pp. 1031-1033 .
• J. Marty, Valeurs singulieres d'une equation de Fredholm, c. R. Acad. SC.
Paris, Vol. 150, 1910, pp. 1499-1502.
162
III. LINEAR INTEGRAL EQUATIONS
simple manner: In order that a kernel have this property it is
necessary that there exist a kernel S(s, t) such that the kernels
J S(s, T)K(T,
t) dT and
JK(s, T)S(T,
t) dT are symmetric.
Such
kernels K(s, t) are said to be symmetrizable. Conversely, if for a
suitable positive definite symmetric kernel S(s, t) at least one of the
above integrals represents a symmetric kernel, then all the poles of
the resolvent of K(s, t) are real and simple.
16. Determination of the Resolvent Kernel by Functional Equations.
Prove that the resolvent of K(s, t) is uniquely determined by the
equations (63).
17. Continuity of Definite Kernels. Prove that, if a definite symmetric kernel K (s, t) is piecewise continuous for 0 S s, t S 1 and if
it is continuous at all points s = t and has continuous eigenfunctions,
then it is continuous everywhere in the region 0 ::; s, t ::; 1.
18. Hammerstein's Theorem. If the kernel K(s, t) is continuous
in the fundamental domain 0 S s, t S I and has uniformly bounded
first derivatives in this domain, then the bilinear formula holds for
the kernel itself and not only for the iterated kernel K< 2>(s, t). The
requirement of bounded differentiability can actually be replaced by
considerably less restrictive conditions. 1
References
We refer particularly to the article by E. Hellinger and 0. Toeplitz in the
Enzyklopadie der mathematischen Wissenschaften, Vol. 2. This article
contains a unified exposition of the theory of integral equations and deals in
detail with the connection between this theory and other fields of analysis.
We may further refer to the review article by H. Hahn, Bericht iiber die Theorie
der linearen lntegralgleichungen, Jahresber. d. deutschen .Mathematiker-Vcreinigung, Vol. 20, 1911, pp. 69-117.
Textbooks
Bocher, .M., An introduction to the study of integral equations. Cambridge
tracts, Vol. 10, Cambridge University Press, Cambridge 1909.
Goursat, E., Cours d'analyse mathematique, Vol. 3, 3rd ed., GauthierVillars, Paris 1923, pp. 323-544.
Kneser, A.: Die Integralgleichungen und ihre Anwendungen in der mathematischen Physik. 2nd ed., Vieweg, Braunschweig 1922.
1 A. Hammerstein, Vber die Entwicklung des Kernes linearer Integralgleichungen und Eigenfunktionen. Sitzungsber: Akad. Berlin (phys.-math. Kl.),
1923, pp. 181-184.
REFERENCES
163
Kowalewski, G., Einfiihrung in die Determinantentheorie, einschliesslich der
unendlichen und der Fredholmschen Determinanten. De Gruyter, Leipzig,
1909.
Lalesco, T., Introduction a la theorie des equations integral es. Hermann,
Paris, 1912. (Containing a detailed bibliography up to 1912.)
Vivanti, G., Elementi della teoria delle equazioni integrali lineare. Mailand,
1916. (German edition F. Schwank, Hannover 1929.)
Volterra, V., Le9ons sur les equations integrales et les equations integrodifferentielles. Gauthier-Villars, Paris, 1913.
Monographs and Articles
Carleman, T., Surles equations integrales singulieres a noyau reel et symmetrique. Uppsala Univ. Arsskrift 1923.
Courant, R., Zur Theorie der linearen Integralgleichungen. Math. Ann.,
Vol. 89, 1923, pp. 161-178.
Fredholm, I., Sur une classe d'equations fonctionnelles. Acta math., Vol. 27,
1903, pp. 365-390.
Goursat, E., Recherches sur les equations integrales lineaires. Ann. Fae. SC.
Toulouse, Series 2, Vol. 10, 1908, pp. 5-98.
Hilbert, D., Grundzuge einer allgemeinen Theorie der linearen Integralgleichungen. Leipzig and Berlin, 1912. (Reprint of six communications from the
Nachrichten der K. Gesellschaft der Wissenschaften zu Gottingen, 19041910.)
Holmgren, E., Das Dirichletsche Prinzip und die Theorie der linearen
Integralgleichungen. Math. Ann., Vol. 69, 1910, pp. 498-513.
Landsberg, G., Theorie der Elementarteiler lnearer Integralgleichungen.
Math. Ann., Vol. 69, 1910, p. 227-265.
Schmidt, E., Zur Theorie der linearen und nichtlinearen Integralgleichungen.
Math. Ann., Vol. 63, 1907, pp. 433-476. Ibid., Vol. 64, 1907, pp. 161-174.
Schur, I., tJber die charakteristischen Wurzeln einer linearen Substitution
mit einer Anwendung auf die Theorie der Integralgleichungen. Math.
Ann., Vol. 66, 1909, pp. 488-510. ·
Weyl, H., Singulare Integralgleichurigen mit besonderer Berucksichtigung des
Fourierschen Integraltheorems. Dissertation, Gottingen, 1908.
CHAPTER IV
The Calculus of Variations
Many problems of mathematical physics are connected with the
calculus of variations, one of the very central fields of analysis. In
this chapter we shall develop certain fundamental concepts of the
variational calculus in order to obtain the differential equations of
mathematical physics and methods for their solution.
§1. Problems of the Calculus of Variations
1. Maxima and Minima of Functions. The calculus of variations
originates from a generalization of the elementary theory of maxima
and minima. For a better understanding we shall first examine the
well-known elementary theory. It is concerned with the problem
of finding, for a given continuous function f(x, y, · · ·) in a given
closed region G, a point x0 , Yo · · · in G at which the function
f(x, y, ···)has a maximum or minimum (extremum) with respect to
all points of G in a vicinity of xo , Yo , · · · . This problem always has
a solution, according to the theorem of Weierstrass: Every function
which is continuous in a closed domain G of the variables possesses a
largest and a smallest value in the interior or on the boundary of the
domain. If the function f(x, y, · · ·) is differentiable in G and if an
extreme value is attained at an interior point xo, Yo, · · · , then the
derivatives of f(x, y, · · ·) with respect to each variable vanish at
Xo, Yo, · · · ; in other words, the gradient of f is equal to zero. But
this necessary condition is in no way sufficient, as can be seen from
the existence of points of inflection or saddle points (examples are
f(x) = x3 at Xo = O;f(x, y) = xy at Xo = 0, Yo = 0). In general, points
at which all the derivatives of the function vanish, so that df = 0,
are known as stationary points. Stationary points which furnish a
maximum or minimum relative to a suitable vicinity are called
''extrema.''
If the variables are not independent but are subject to the restrictions g1(x, y, · · ·) = 0, g2(x, y, · · ·) = 0, · · · , g1i.(x, y, · · ·) = 0,
164
PROBLEMS OF THE CALCULUS OF VARIATIONS
165
we obtain necessary conditions for an extremum or stationary point
by means of Lagrange's method of multipliers. This method consists
in the following procedure: In order to find a point in the interior of
the domain of the independent variables at which f(x, y, ···)has an
extremum or is merely stationary, we introduce h + 1 new parameters, the "multipliers," Ao, A1, · · · , Ah and construct the function F =
Xof + A1U1 + A2(J2 + · · · + AhUh . We now determine the quantities
xo , Yo , · · · and the ratios of Ao , A1 , • · · , Ah from equations
aF
(1)
ax
= 0
,
aF
ay
=0
,
aF
OAh =Uh = 0,
the number of which is equal to the number of unknowns. These
equations represent the desired conditions for the stationary character of f(x, y, · · ·) or the extremum off under the given restrictions.
If Ao'¢ 0 we may (and shall) put Ao = 1 because Fis homogeneous
in the quantities Ai. The Lagrange method is simply a device which,
preserving sy.mmetry, avoids explicit elimination of h of the variables
from the function f(x, y, · · ·) by means of the subsidiary restrictions.
We now consider a few instructive though elementary examples.
(a) Of all triangles with given base line and given perimeter the isosceles
triangle has the greatest area. Of all triangles with given base line and
area the isosceles triangle has the least perimeter. These statements
may easily be verified without calculation by considering the ellipses
for which the given base line is the line connecting the two foci. Even
this simple example exhibits a characteristic reciprocity, which we
shall encounter again later (§12, 2).
(b) Refraction and Reflection of Light. According to Fermat's
principle of least time a ray of light requires less time along its actual
path between two points than it would require along any other conceivable ("virtual") path satisfying the given conditions. From
this it follows immediately that light travels in straight lines in any
homogeneous medium. If the light ray is required to meet a given
curve (mirror) without crossing it, then a simple analysis of the derivative of the light time shows that the path must consist of two
straight line-segments making equal angles with the tangent to the
given curve at the point where they meet it (law of reflection). If,
on the other hand, the given curve forms the boundary between two
IGG
IV. THE CALCULUS OF VARIATIONS
regions in which the velocity of light has the values c1 and c2 , respectively, and if the ray is to pass from one region to the other,
then it must consist of two straight line-segments which satisfy the
well-known law of refraction (Snell's law), sin a1/sin a2 = c1/c2,
where a1 and a2 are the angles between the normal to the boundary
curve and the two line-segments at the point of intersection.
(c) A Problem of Steiner. Given three points At, A2, Aa which
form an acute triangle, a fourth point Pis to be found such that the
sum of the distances PA.1
P A2
P Aa is as small as possible. We
consider a circle through P with center at Aa; then P must be placed
in this circle in such a way that PA 1 + P A2 is as small as possible.
According to the law of reflection this means that the lines PA 1 and
P A2 make equal angles with the radius P Aa . The same must be
true if the indices 1, 2, 3 are interchanged; all three angles A1PA2,
A2PAa, AaPA1 must, therefore, be equal and equal to 211"/3. Thus
t.he problem is solved.
(d) The Isoperimetric Problem for Polygons. Among all polygons
which are not self-intersecting and have an even number 2n of sides
and a given perimeter 2l find the one with the greatest area. The
desired polygon Il(A 1 , A 2 , · · · , A2n) is the regular 2n-gon. To
prove this we first convince ourselves that Il is convex. If II were
not convex, a straight line lying entirely outside IT could be drawn
through two non-adjacent vertices, say Ak and A z ; we could then
reflect the polygonal sequence AkAk+l · · · A z_1A z in this straight line
and obtain a polygon with the same perimeter and with an area
greater than that of the original polygon by twice the area of the
triangle A1A2A 3 • We may thus confine ourselves to convex polygons.
Next we show that all the sides of the polygon are equal in length.
For, if the lengths of two successive sides AiA2 and A2Aa were different,
we could, according to the result of (a), replace the vertex A2 by a
vertex A~ such that AiA~
A~Aa = AiA2
A2Aa while the area of
the triangle AiA~A 3 is greater than that of A1A2Aa. Thus the area
of the polygon with the vertex A~ is greater than that of the polygon
with A2, in contradiction to the hypothesis that II is the maximal
polygon. Finally, in order to show that the polygon II can be inscribed in a circle, \Ve divide IT into two polygons by means of the
diagonal connecting opposite vertices Ai and An+1. The perimeters
of these polygons IT1 and IT2 are evidently equal; their areas are also
equal, for if that of II 1 were greater than that of Il2 we could replace II~
by the reflection of II 1 in the diagonal, obtaining a new polygon II*
+
+
+
+
PROBLEMS OF THE CALCULUS OF VARIATIONS
167
whose perimeter is still 2l but whose area is twice that of II1 and therefore greater than that of II. We now show that for every vertex Ah
the angle A1A~11+1 must be a right angle. For, if this angle were not
right for the vertex A1i of II1, we could decompose II1 into the triangle
AiA~n+1 and two polygons H1 and H2 which join the triangle at the
sides AiAh and A~n+l · Consider a right triangle A~AhA ~+1 whose
sides A~Ah and A~-~+i are equal to A1Ah and AhA"+1, respectively.
The area of this triangle is greater than that of A1AhAn+i. If we affix
the polygons Hi and H 2 to the new triangle at its sides and reflect
the resulting polygon in the line A :A
we obtain a polygon of
perimeter 2l and area greater than that of II, contrary to hypothesis.
Therefore A 1AhA"+i is a right triangle, and consequently all vertices
lie on a circle. This compJetes the proof of the extremum property
of the regular polygon.1
The preceding treatment, based on a classical idea of Steiner,
shows that in individual cases a direct geometrical approach may lead
more readily to the result than general analytic procedures.
(e) Other Examples, Maximum of a Minimum. We have already
encountered other typical examples of stationary values which are
neither maxima nor minima but maxima of a minimum or minima of
a maximum. For instance, we defined the eigenvalues of quadratic
forms as maxima of minima and the Tchebycheff polynomials by a
similar property.
2. Functionals. The calculus of variations likewise originates from
the quest for extrema or stationary values. Its object is, however,
to find extrema of functionals rather than extrema of functions of
a finite number of independent variables. By a "functional" we
mean a quantity or function which depends on the entire course of
one or more functions rather than on a number of discrete variables.
In other words, the domain of a functional is a set or "space" of admissible functions rather than a region of a coordinate space. A
simple example is the length L of a curve y = y(x) between the values
x = Xo and x = x1 • This length is given by the integral
:+i
Zt
L =
1 VI +
zo
y' 2 dx;
i Note that the existence of the extremum follows from Weierstrass' theorem.
For, if one vertex is at the origin the others are confined to a bounded region,
since the total perimeter is given and the area of the polygon is a continuous
function of the coordinates of the vertices.
168
IV. THE CALCULUS OF VARIATIONS
thus the value of L depends on the course of the "argument function"
y(x), which may be taken to be an arbitrary continuous function
with a piecewise continuous derivative.
Such functionals occur
throughout analysis and its applications, and many important problems of analysis are concerned with functional dependence of this kind.
Another example is the area of a surface z = z(x, y) lying above the
region G in the xy-plane. This is given by the integral
fl Vl + z! + z;
dx dy1
and is a functional of the argument function z(i, y).
Other examples of functionals occurred in the previous chapter.
Thus for fixed K (x, y) the function
g(x) =
f
K(x, y)h(y) dy
is a functional of h(y) depending on x as a parameter, 2 and the integral form
ff
K(x, y)cp(x)cp(y) dx dy
is a functional of cp(x). In this chapter we shall be concerned principally with functionals which are integrals of known functions of
the argument function, its derivatives, and independent variables,
such as the arc length of a curve.
A region of definition is needed for a finite number of variables;
here we must define the domain or "space" of admissible functions
from which the argument functions can be selected. This space may,
for example, be the set of all functions which are continuous and
have piecewise continuous first derivatives (compare the following
section).
Although a functional cannot be expressed as a function of a finite
number of variables, it may be regarded as a function of infinitely
many variables. Suppose that the argument functions are expanded
1 Throughout this book we shall employ the subscript notation for partial
derivatives, i.e.
fa: = of(x, y, ···)/ox;
2
fzv = a2f/oxoy, etc.
Often the words "function of a function" or "functional transform" are
used for functionals which themselves depend on variables.
PROBLEMS OF THE CALCULUS OF VARIATIONS
169
in power series or Fourier series; the functional then depends on the
expansion coefficients, which constitute the infinitely many variables.
Of course the domain of these variables must be restricted so as to
conform to the conditions imposed on the argument functions.
3. Typical Problems of the Calculus of Variations. The calculus
of variations is concerned with the problem of determining maxima
or minima or, in general, stationary values1 of functionals by seeking
that argument function in the given domain of admissible functions
for which the functional assumes the stationary value or extremum
in question. Ordinary extremum problems of the differential calculus do not usually involve absolute maxima or minima but only
an extremum with respect to values of the function in the vicinity of
the extremum point; in analogy, we shall seek here the extremum of
the functional only with respect to argument functions in a certain
2
neighborhood of the extremal argument function. For this purpose
it is necessary to define the concept of the neighborhood of a function
f(x, y, · · · ). If h is a positive quantity, a function fi(x, y, · · ·) is
said to lie in the neighborhood (h) of the function f(x, y, · · ·) if
If - f1 I < h throughout the region of definition.3
We are now in a position to formulate the fundamental problem
of the calculus of variations: In a given domain of admissible argument functions, find an argument function (or functions) of a given
functional for which the latter is an extremum with respect to all
argument functions of the domain lying in a sufficiently smalJ neighborhood (h) of the extremal argument function. If the functional
to be made an extremum depends explicitly on variable parameters
x, y, · · · in addition to the argument functions, i.e. if the functional
is not a number but a function of these parameters, then we must
determine these parameters as well as the argument functions, in
order to obtain the extremum. We shall now illustrate the problems
of the variational calculus by a p.umber of simple examples.
In §3, 1 the "stationary value of a functional" will be defined precisely.
An extremal function is a function which makes a given functional
stationary.
a For certain problems it is convenient to refine the concept of neighborhood
in successive steps. The functionfi(x, y, ···)is said to be in the neighborhood
of first order (h) of f(x, y, · · ·) if the relations If,, - f1z I < h, I fv - fiv I <
h, · · · are satisfied in addition to If - f1 I < h. More generally, we speak of
a neighborhood of order n + l if these inequalities hold for f1 and all its
derivatives up to and including the n-th order.
1
2
170
IV. THE CALCULUS OF VARIATIONS
(a) Geodesic Curves. On a given surface, find the shortest curve
between two given points. If the surface is given by the parametric
representation x = x(u, v), y = y(u, v), z = z(u, v) in rectangular
coordinates x, y, z, and if we employ the customary notatious
2h
2
2
2
2
2
Z11 t en
y,,
ZuZv' g = x,,
Yu.Yv
Zu 'f = XuXv
Yu
e = Xu
the length L of a curve on the surface defined by the equation
v = v(u) between Uo and u1 is given by the integral
+
1 ve +
"1
L =
+ +
+
+
+
2fv'
uo
+ gtl
2
du.
The problem is thus to find functions v(u) which yield extrema of this
integral.
(b) Light Rays; Brachistochrone. According to Fermat's principle
(page 165) the path of a light ray in an inhomogeneous two-dimensional medium in which the velocity of light is c(x, y) solves the variational problem
T =
z1
1
zo
vi+ Y
c(x, y)
'2
dx =min.
In this problem, as in the previous one, all continuous curves are admissible which have piecewise continuous derivatives and join the
given end points of the path. Closely related to the problem of the
light ray is the brachistochrone problem, with which Jakob Bernoulli
in 1696 gave the initial impetus to the development of the calculus of
variations. Two points A (xo , O) and B(x1, Yi) with Yi > 0 are to
be connected by a curve along which a frictionless mass point moves
in the shortest possible time from A to B under gravity acting in the
y-direction. The initial velocity of the mass point is zero. After
falling a distance y the point has the velocity y2gy, according to
elementary mechanics, where g is the acceleration due to gravity. It
follows that the time of transit is the integral
(2)
T =
1 11- +
z1
zo
y'2
--dx.
2gy
The class of admissible functions consists of all the positive functions
y(x) with continuous second derivatives for which y(x0 ) = 0 and
y(x1) = Y1.
(c) Minimal Surface of Revolution. Suppose that the curve y =
y(x) ~ 0 is rotated about the x-axis. The resulting surface, bounded
z1
by planes x = xo and x = x1 7 has the area F = 27r
1 YV 1 + y' dx.
2
zo
171
PROBLEMS OF THE CALCULUS OF VARIATIONS
The curve y = y(x) which leads to the smallest surface of revolution
is thus characterized by the variational problem
z1
1
zo
yyl
+ y'
2
dx =min.
(d) Isoperimetric Problem. The original formulation of this problem is: Find a closed curve of given circumference and greatest
area. Assuming the curve to be convex and divided into two parts of
equal area by the x-axis (see subsection l(d)) we arrive at the following problem: The integral
f
y(x) dx
is to be made a maximum by a suitable choice of
the intergal
fV
1
+ y'
2
dx
=
~
and y(x), while
l
has a given value. Here y(x) may be any function which is continuous and has a piecewise continuous first derivative in the interval
0 ~ x ~ ~and for which y(O) = y(~) = 0.
An analogous problem may be formulated with the upper limit ~
fixed.
This problem, known as the special isoperimetric problem, may be
reduced to a simpler type of variational problem by introducing
as the independent variable the arc length s
=
lz v
1
+
y' 2 dx,
ranging through the interval 0 ~ s ~ l. Since ds2 = dx + dy ,
the problem now becomes: Find the function y( s) which maximizes
the integral
2
f
2
yyl - (dy/ds) 2 ds,
where y(s) is a continuous function of s with a piecewise continuous
derivative. After determining y(s.) we find
(3)
and thus obtain a parametric representation of the desired curve.
(Compare also Hurwitz's solution of the isoperimetric problem, Ch.
II, §10, 1.)
172
IV. THE CALCULUS OF VARIATIONS
In general, all problems in which one integral expression is to be
made an extremum while another has a given value are termed isoperimetric problems. An example is given by the problem of the
catenary: Determine the location of a uniform string of given length
with fixed end points under the influence of gravity. Suppose that
gravity acts in the negative y-direction. Since the equilibrium position demands that the center of gravity be as low as possible, we
arrive at the following variational problem: Find the function y(x)
for which
z1
1
YVl
zo
+ y'
2
dx
is as small as possible while
z1
=
l
1 vi+
zo
y' 2 ax
has a given value, and the boundary values y(xo)
are given.
Another problem is to solve
Jz
1
(y") 2 dx = min.
zo
with the subsidiary condition
r1 y2 dx
Jzo
=
1,
where the function y(x) is to vanish at the end points of the interval
and is to be continuous everywhere along with its derivatives up to
the second order. Another very important isoperimetric problem is:
Find a function u of x and y, continuous along with its first derivatives in the region G, which satisfies
Jl
(4)
(u!
Jl
2
u dx dy = 1 and for which
+ u!) dx dy + J;
2
CTU
ds
is as small as possible1 (here r is the boundary of the region G and CT
is a fi."{ed function of the arc length s of r).
The minimum problem of Chapter III for the eigenfunctions
1
Cf. §3, 2.
PROBLEMS OF THE CALCULUS OF VARIATIONS
173
of a symmetric kernel is another example of an isoperimetric
problem.
It goes without saying that in all these problems the space of admissible argument functions must be specified by continuity conditions
such that all the functionals occurring in the problem are meaningful.
4. Characteristic Difficulties of the Calculus of Variations. In the
theory of ordinary maxima and minima the existence of a solution is
ensured by the fundamental theorem of Weierstrass.. In contrast,
the characteristic difficulty of the calculus of variations is that problems which can be meaningfully formulated may not have solutionsbecause it is not in general possible to choose the domain of admissible
functions as a "compact set" in which a principle of points of accumulation is valid. A simple geometric example is the following: Two
points on the x-axis are to be connected by the shortest possible
line of continuous curvature which is perpendicular to the x-axis at
the end points. This problem has no solution. For, the length
of such a line is always greater than. that of the straight line connecting the two points, but it may approximate this length as closely as
desired. Thus there exists a greatest lower bound but no minimum
for admissible curves.
Another example of an insoluble variational problem is the following: Minimize the integral
(5)
1: xV
2
dx
by means of a continuous function y(x) which has a piecewise continuous derivative and for which y(-1) = -1, y(l) = 1. It is _easily
seen that the integral can be made arbitrarily small by suitable
functions (i.e. y = -1 for x < -E, y = x/E for Ix I ~ E, y = 1 for
x > E), but does not vanish for any admissible function.
Thus, in the calculus of variations the existence of an extremum in
a parf.icular problem cannot be taken for granted. A special existence proof is needed for the solution of each problem or class of
problems. As we shall see later, this fact gives rise to essential difficulties in many of the problems of the calculus of variations. However, in this chapter we shall deal primarily with the formulation of
necessary conditions for the attainment of an extremum, while the
question whether an extremum is actually attained when the conditions are satisfied may be left open.
174
IV. THE CALCULUS OF VARIATIONS
Before formulating these necessary conditions we shall consider
possible methods for the direct solution of variational problems.
§2. Direct Solution 1
Direct and complete solutions of variational problems can sometimes be obtained by a general procedure consisting of two steps:
first one formulates a suitable approximate ordinary extremum problem in which only a finite number n of parametets is to be determined
and then one passes to the limit n -+ oo in the solution of the approximate problem.
1. The Isoperimetric Problem. One example is the isoperimetric
problem of §1, 3(d): find a closed curve K with perimeter 2l and maximum area; the curve is to be piecewise smooth, i.e. it is to have a continuous tangent except at a finite number of corners. If we assume
that the solution is given by the curve K we may conclude that K
is a circle. For, just as in §1, I(d), it may be proved that K is convex
and that every secant AB which divides K into two parts with equal
circumference also divides it into two parts of equal area; furthermore,
for every point Pon K the angle APB must be a right angle, since
otherwise, by the construction of §I, l(d), one could obtain a curve K'
with the same perimeter but greater area. But this argument is
based on the tacit assumption that the problem actually possesses a
solution, and this assumption requires proof; therefore we shall solve
the problem by a different method which furnishes the necessary
existence proof at the same time. We consider the set of values of
the areas of admissible curves. Since these numbers all lie below the
bound Z27r (the curve can certainly be placed inside a circle of radius l),
this set of numbers has, by the elementary rules of analysis, a least
upper bound M, above which there is no number of the set, but such
that numbers of the set occur in any neighborhood ( E) of M for arbitrarily small E. In other words, there exists a "maximizing sequence"
of admissible curves K1, K2, · · · such that the area F,,, of K,,, converges to M with increasing n. Now every curve K,,, can be approximated as exactly as desired by an inscribed polygon II,,, with sufficiently
many sides such that area and perimeter of II,,, differ arbitrarily
little from those of Kn. We may, without destroying its approximating character, dilate each polygon in such a way that its
1 Direct methods of the calculus of variations will be treated more fully in
the second volume.
DIRECT SOLUTIONS
175
perimeter becomes exactly equal to 2l. Thus the maxmuzmg sequence K1 , K2 , · · · may be replaced by a maximizing sequence of
polygons II~ , II~ , · · · . The number of sides of these polygons may
be assumed to be even, since a (2m - 1)-gon may be regarded as a
2m-gon with two adjacent sides forming part of the same straight
line. We know from §1, l(d) that of all 2m-gons of perimeter 2l
the regular 2m-gon has the greatest area. We therefore obtain an
even better maximizing sequence of our problem by replacing each
polygon II: by a corresponding regular polygon. But as the number
of sides increases these polygons converge to the circle of perimeter
2l, and since the areas of the polygons converge to M the circle has the
area M and is actually the solution of our variational problem.
2. The Rayleigh-Ritz Method. Minimizing Sequences. The above
considerations are based on the following general idea: We consider any variational problem of the form D[~] = min., where the
functional D[~] is an integral of a given expression containing the function ~ and its derivatives up to the k-th order, and where the region
of integration and the domain of admissible functions ~ are given.
It does not :matter whether D[~] is a simple or a multiple integral.
We assume that the set of values of D[~] for the admissible argument
functions~ possesses a greatest lower bound d (whether dis a rp.inimum which is actually attained for a function ~ = u remains an open
question). Then there exist sequences ~1 , ~2, • • • of admissible
functions such that limn-+ao D[~n] = d, while the relation D[~] ~ d
holds for every admissible function ~- Such sequences of functions
will be called minimizing sequences of the variational problem. A
direct solution of the variational problem always consists in the construction of minimizing sequences and the attempt to secure the solution
by a limiting process based on this sequence.
The method which W. Ritz 1 used with great success, especially
for numerical solutions, consists in the following steps: We start with
a fixed complete system of "coordinate" functions w1 , w2 , · · • ,
defined in the region of integration, which has the property2 that all
linear combinations ~"' = c1w1
C2W2
Cn.Wn of a finite number
+
+ ··· +
W. Ritz, tJber eine neue Methode zur Losung gewisser Variationsprobleme
der mathematischen Physik, Journ. f. d. reine u. angew. Math., Vol. 135, 1909,
pp. 1-61; Gesammelte Werke, pp. 192-250, Gauthier-Villars, Paris, 1911.
Even before Ritz, similar ideas were successfully employed by Lord Rayleigh.
2
The question of the existence of such functions will be discussed more fully
in the second volume.
1
176
IV. THE CALCULUS OF VARIATIONS
of the functions are admissible comparison functions for the problem,
and such that for every admissible function <P there is a suitable combination <Pn of this kind for which D[<P] differs arbitrarily little from
D[<Pn]. Then there exist minimizing sequences <P1, <P2, · · · in which
<Pn is a linear combination C1w1
C2w2
CnWn of the functions
w1 , w2 , · • • , wn . We therefore obtain an even better minimizing
sequence if for every n we determine the function <Pn, i.e. the parameters C1, C2, • • • , Cn, by the requirement that D['Pnl = d,,, be a minimum. This requirement represents an ordinary minimum problem
for D[<P,,,] as a function of the n parameters c1 , c2 , · · · , Cn and can
always be fulfilled, according to Weierstrass's theorem (provided
D[<Pn] is a continuously differentiable function of c1 , c2 , · · · Cn ; we
shall assume here that this is the case). The values Ci are determined
by then simultaneous equations aD[<Pnl!aci = 0 (i = 1, 2, · · · , n).
We may now expect the resulting minimizing sequence to converge
to the desired solution. Unfortunately things are not so simple,
as we shall see in §4. In general we can only state that the values
D[<P,,,] = d,,, obtained in this way converge to the desired greatest
lower bound or minimum. Whether the minimizing sequence itself
converges to the solution is a difficult theoretical question which has
to be investigated separately. We shall return to this question on
several subsequent occasions.
In some cases this method may prove useful for numerical calculations even though its convergence is unproved. Its success in
any particular case will depend on the proper choice of coordinate
functions wi. The method will be illustrated by examples in the
next subsection.
3. Other Direct Methods. Method of Finite Differences. Infinitely
Many Variables. One can often obtain minimizing sequences by extending the domain of admissible functions, for example by admitting
not only continuously differentiable functions, but also continuous
functions with piecewise continuous derivatives. We shall consider the problem of the minimum of a simple integral of the form
+
+ ·· · +
:1:1
D[y] =
1
:i:o
F(x, y, y') dx.
If y = <P1(x), y = <P2(x), • · • is a mini-
mizing sequence, and if F(x, y, y') satisfies the necessary continuity
requirements, then the curve given by y = <P,,,(x) can certainly be
approximated by a polygonal arc y = Pn(x) in such a way that the
integral D[<P,,,] differs arbitrarily little from D[p,,,]. It is therefore
DIRECT SOLUTIONS
177
possible to construct mm1m1zmg sequences consisting of piecewise
linear functions, for which the difference quotient and the differential quotient are identical in each subinterval. Thus if we divide
the interval of integration by m points into equal parts of length Ax,
and if we confine ourselves to functions which are linear in each subinterval, the variational problem goes over into, or may be approximately replaced by, the ordinary minimum problem
~ (
~ F xi, Yi ,
i=O
Yi+i A
X
.
Yi) Ax = mm.
for the values Yo, Y1, · · · , Ym+1 of the function at the ends of the subintervals. The resulting functions for m = 0, 1, 2, · · · once again
1
form a minimizing sequence.
This method may be regarded as a special case of the Ritz method,
with suitable piecewise linear coordinate functions.
Under what circumstance s the solution of the finite difference
problem converges to the solution of the minimum problem will be
discussed in the second volume.
It is possible to proceed in a similar manner when the integrand
contains derivatives of higher order, say the second.· The second
differential quotients will then be replaced in the approximati ng
problem by the second difference quotients (YH2 - 2Yi+1 + Yi)/(llx)2.
Our variational problems may also be regarded as problems in the
theory of functions of infinitely many variables. For example, in
Hurwitz's solution of the isoperimetric problem (Ch. "II, p. 97)
the variables in question were the Fourier coefficients and the ana2
lytic expression for L - 47rF exhibits the solution. The Ritz method
also admits of this interpretatio n if we may assume the function
to be developed in a Fourier series c1w1 + c2w2 + · · · and regard the
method as a prescription for obtaining the' infinitely many coefficients
c1 , c2 , • • • • The difficult question of convergence would have to
be investigated separately.
We shall now illustrate these general considerations by some examples.
1 The method described here is essentially the method which led Euler to
the "Euler differential equations" in his work: Methodus inveniendi lineas
curvas maximi minimive proprietate gaudentes, M. Bousquet, La'usanne and
Geneva, 1744.
178
IV. THE CALCULUS OF VARIATIONS
(a) The integral
(6)
fl
D['P] =
+ 'P!) dx dy,
('P!
which extends over the rectang]e R given by 0 ~ x ~ a, 0 ~ y ~ b,
is to be minimized; here the admissible oomparison functions are all
piecewise smooth1 in R, vanish on its boundary, and satisfy the subsidiary condition
(7)
H['P]
=
fl
2
<p
dx dy = 1.
We suppose that the function 'P is expanded in a Fourier series 'P =
(m?rx/a) sin (n7ry/b); this is certainly possible according to Chapter II. The problem is to determine the infinitely many
parameters Cmn from the minimum requirement. Since the functions 'Px and cp 11 are piecewise continuous we may apply the completeness relation of the trigonometric functions to these functions with
the expansion coefficients (7rm/a)cmn, (7rn/b)Cmn, obtaining for the
two integrals the expressions
.L::.n-1 Cmn sin
(8)
D
=
2
7r
ab
-4
L Cmn (m2a + n
b2
m,n-1
2
00
2
2
)
;
ab ~
H = - £..J
4 m,n-1
2
Cmn
in the parameters Cmn. Because of the condition H = 1 it is now evident that the solution of the problem is given by Cmn = 0 except for
the coefficient cu, which must equal 2/v'{ib. Thus our variational
problem is solved by the function
2
. 7rX . 1rY
u=--sm-smVab
a
b'
and the value of the minimum is
d =
7r2 (~2 + ~2).
This may be expressed by relation
(9)
1
D[cp]
~ 7r 2 C~ + ;2)nr"°1
See the. definition given in Ch. II, p. 48.
179
DIRECT SOLUTIONS
for every piecewise smooth function vanishing on the boundary of the
rectangle, because this relation is equivalent to D[tP] ~ d for the nor.:
malized function tP = 'P/VH[ip].
(b) Dirichlet's Problem1 for the Circle. The integral
D[<,0]
=
JL
+ 'P!) dx dy,
(<,0!
extended over the circle K (x2 + y 2 ~ 1) in the x, y-plane, is to be
made a minimum if the admissible comparison functions are smooth
in K and assume given boundary values on the boundary of K. We
introduce polar coordinates r, 8 in K, transforming the integral into
D[<,0] =
2r11
1
1o o ('P; + r2 'Pi)r dr d8.
The boundary values may now be defined by a Fourier series f( 8)
! ao + L:':-1 (an cos n8 + bn sin n8). Suppose further that this
bouI).dary function f has a smooth derivative; according to Ch. II,
ni
2
§5, 3, this means that
an I and n 1 bn l are bounded.
assume the functl.on 'P written in the form
CIO
'P
=!
fo(r)
+L
[f,h)
n=-1
cos n8
+ Yn(r)
We may now
sin n8],
where fn(r), Yn(r) must satisfy the relations fn(l) = an, Yn(l) = bn .
Because of the completeness relation for the trigonometric functions
we obtain the equation
1 [f~(r)]2r dr + f 1 [1~2 (r) + ~r f ~(r)]r dr
+ f 1 [·g~2 (r) + ~ g~(r)]r dr.
1
D[rp]
=
7r
1
2
1
2
?r
0
n==l
0
?r
n-1
r
O
Thus in order to solve the original minimum problem we treat the
series of minimum problems
f (/:• + ~· !)
f
r dr = min.,
l
0
l (
gIn
2+ n 2g2) r dr = mm.
.
T2
n
(n = 0, 1, 2, · · ·) ,
i This nomenclature has become customary since Riemann's time even
though it does not correspond to the historical facts.
180
IV. THE CALCULUS OF VARIATIONS
in which f n and Un are smooth functions that take on the values an
and bn , respectively, at r = 1. In view of Weierstrass's approximation theorem the functions 1, r, r2, • • • certainly satisfy the conditions
for the coordinate functions of the Ritz method for these minimum
problems. Let us take for fn (or Un) a polynomial Cn,o + Cn,1 r + · · ·
+ Cn,mrm with m 2:, n and with Cn,O + Cn,l + · · · + Cn,m = an (or bn).
The reader may easily verify that the solutions, independent of m,
are fn = anrn (or gn = bnrn). Thus all the functions of the resulting
minimizing sequence are equal to each other, and therefore equal to
the solution of the variational problem.
The solution fn or Un of the variational problem may also be found
directly, without employing the Ritz procedure, in the following way:
1j~ r
1
For n = 0 we have to minimize
2
dr; this is evidently accom-
plished with f~ = 0, fo = const. = ao . For n > 0 we must. have
fn(O) = Un(O) = O; otherwise the second part of the integral would be
infinite, since fn is differentiable and may therefore be written in the
formfn(O) + rhn(r) with a continuous function hn(r). We now write
the first integral in the form
J.' v~
-;
1.)'r dr + 2n J.' f.!~ dr
=
J.' (1~ - ; 1.)'r dr + nl~(I).
The value of fn(l) = an is fixed, and we see immediately that we obtain a minimum value, namely nf!(1) = na~, if we set f ~ - nfn/r =
0, obtaining fn = Cnrn; since fn(l) = Un We have Cn = an or fn =
anrn. Corresponding results are obtained for gn(r). Thus the solution of the original minimum problem is
co
(10)
u(r, 8) =
! ao
+ L:
rn(an cos n(}
n=l
+ bn sin n(})
and the minimum value is
co
7r
L:
n(a! +
n=l
b!) .
It should be emphasized that the minimum problem solved here
may cease to be meaningful if less stringent assumptions are made
181
DIRECT SOLUTIONS
concerning the boundary function, for example if continuity alone is
required. Thus, if we take the contjnuous boundary function defined
by the uniformly convergent series p(O) =
\
L':-=1
n
cos (n!O), the
~ (n!) is infinite. In trus case it can
n
easily be shown that there exists no admissible comparison function
with finite D[cp].
(c) Let g(x, y) be a smooth function in the rectangle R defined by
0 ::; x ::; a, 0 ::; y ::; b. We seek a function cp, continuous in the
rectangle along with its first derivatives and vanishing on the boundary, which minimizes the integral
sum
7r L::=1
ma!
=
7r L':=1
fL
J[cp] =
(11)
(cp!
+ "'; -
2cpg) dx dy.
In the interior of R, let us write
~
L...J
.
g( x, y ) =
7rX
•
amn
m,n=l
1rY
•
sm m - sm n -b
a
and
00
=
cp
~
L...J
•
Cmn Sln
m,n=l
1rX
m -a
•
Slil
1rY
n -b ,
where the parameters Cmn are to be determined. Because of the completeness relation for the trigonometric functions the variational
problem then goes over into the problem of determining the ·quantities Cmn such that
4
-b J[cp] =
a
2
71"
L
00
m,n=-1
(2~ + nb22) C~n
a
becomes as small as possible.
mum js furnished by
(12)
u(x, y)
=
1
~
2 L...J
71" m,n=l
L
00
-
2
m,n=l
Umn Cmn
We see immediately that the mjni-
1rY
•
- sm n -b .
+ n 2/b2 sm• m 7rX
a
Umn
2/ 2
m a
This function does indeed satisfy all requirements, since the series,
as well as the series obtained by term-by-term differentiation, con-
182
IV. THE CALCULUS OF VARIATIONS
verges uniformly.
gent series
To see this we note that the absolutely conver-
represent majorants. We shall see subsequently (page 192) that the
function u satisfies the differential equation Uzz
u 1111 = g(x, y);
compare examples (a) and (b).
+
4. General Remarks on Direct Methods of the Calculus of Variations.
We have already noted that the main difficulty in justifying the
direct methods of the calculus of variations is that minimizing sequences do not necessarily converge to a limit function, even if the
existence of a solution is not in question.
A simple example is given by the variational problem of minimal
surfaces, in which the integral
Ji VI + z; + z!
dx dy
is to be minimized; here we admit all surfaces z = z(x, y) with piecewise continuous derivatives which pass through a given space curve
whose projection is the boundary of G. If, in particular, we take for
this curve a curve in the x, y-plane, for example a circle of unit area,
the minimum is given by the function z = 0, i.e. by the x, y-plane
itself. Every sequence of surfaces which passes through the periphery
of the circle and the areas of which converge to 1 is a minimizing
sequence. Now \Ve may at once construct admissible comparison
surfaces whose areas are arbitrarily close to 1 and for which z(x, y)
is arbitrarily large at isolated points. For example, consider a
circular cone of height 1 and base radius E placed perpendicularly
on the plane z = 0. We now take as a comparison surface the surface
consisting of this cone and of the portion of the original circle outside
the base of the cone. A minimizing sequence of such surfaces does
not, for e ~ 0, converge to the solution. It is even possible, as is
easily seen, to construct minimizing sequences in which the points
of divergence are everywhere dense in the circle.
Another example is given by the Dirichlet problem of minimizing
the integral D[<,0] =
J£
(<,0!
+ <P!) dx dy if we admit for comparison all
functions continuous and piecewise smooth in G which vanish on the
boundary. Evidently the unique solution of the problem is <P = 0,
183
THE EULER EQUATIONS
since every other admissible function leads to a positive value of the
integral. Introducing polar coordinates r, 8 about an arbitrary point
P of G, we write the integral in the form
Ji
('P;
+ 'P:/r )r dr d8.
2
We
now consider a circler ~ a about P, lying entirely in the interior of
= 0 outside this circle, =
G, with radius a < 1. Let us set
2
r = a and r = a , and
between
ring
log (r/a)/log a in the circular
2
'P = log a/log a = 1 in the circler~ a • By definition, 'Pis an admissible comparison function; therefore the integral is equal'to
"°
"°
2r
(log a) 2
la T2
2r
log a·
1
a2
r dr = -
If we now let a assume a sequence of values a1 , ~ , · · · which tend
to zero and consider the corresponding functions 'Pi , 'P2 , • • • , we see
that D['Pn] converges to zero; thus these functions form a minimizing
sequence of the problem. But at the point P all the functions have
the value 1 and therefore do not converge toward the solution 'P = 0
of the problem.
L
1
In the case of the variational problem
2
y' dx = min., where y(x)
is a continuous piecewise smooth function of x vanishing at the end
points, all minimizing sequences must indeed converge to the function y = 0, as is easily seen. But the derivatives of the functions of
the minimizing sequence do not necessarily converge to zero, as is
shown by the example of the sequence Yn = x for x .< En , Yn = 2En - x
for En ~ X ~ 2En, Yn = 0 for X > 2En, (limn-+ac> En = 0).
In Volume II we shall see how these difficulties may be overcome;
indeed, the direct methods of the variational calculus have led to
important developments in analysis.
The following sections deal with classical "indirect" methods
based on the reduction of variational problems to differential equations. These methods, which have been emphasized by mathematicians since Euler and Lagrange, attack minimum problems by
means of a general variational formalism which in itself has bec~me
an important tool of analysis.
§3. The Euler Equations
The differential equations of a variational problem, first derived
by Euler, represent necessary but by no means always sufficient conditions which a function must satisfy if it is to furnish the extremum
184
IV. THE CALCULUS OF VARIATIONS
of a given integral. We obtain these Euler equations by reducing the
variational problem to a problem in the differential calculus. Once
and for all we assume that all functions and their derivatives which
occur explicitly in the problems are continuous unless the contrary is
explicitly stated.
1. "Simplest Problem" of the Variational Calculus. Let us first consider the simplest problem of the calculus of variations, namely that
of determining the minimum of the integral
(13)
l
J[y] =
:i:i
F(x, y, y') dx,
zo
where the values Xo, x 1 , y(xo), y(x1) are given. The· function F is
to be twice continuously differentiable with respect to its three arguments x, y, y'. The second derivative y" of the function y is also
assumed continuous. Suppose y = y(x) = f(x) is the desired extremal
function yielding the minimum, i.e. suppose that, in a sufficiently
small neighborhood (h) of the function j(x), the integral J[y] is smallest ·when y = j(x). Now consider a function ?J(X) which is defined in
the interval x 0 ~ x ~ x1 , possesses a continuous second derivative,
and vanishes at the end points, but is otherwise arbitrary. We
construct the function fj = y + E?J(X) = y + oy, where e is a parameter.
The quantity oy = e11(x) is known as the variation of the function
y = j(x). If the parameter e has a sufficiently small absolute value
all the varied functions fj lie in an arbitrarily small neighborhood of
the extremal function y = j(x). Therefore the integral J[fj] =
J[y + E?J], which may be regarded as a function <I>( e) of e, must have
a minimum at e = 0 relative to all values of e in a sufficiently small
neighborhood of 0, and therefore <I>'(O) = 0. Now if we differentiate the integral <I>( e) =
Jx
zo
1
F(x, y
+ e71,
y'
+ E?J') dx
with respect
to e under the integral sign (this is permissible) we obtain as a necessary condition the equation
which must hold for all functions 11(x) which satisfy the above requirements. We transform the second part of the integral by partial
integration, noting that ?J(Xo) = ?J(X1) = 0, and obtain the equation
185
THE EULER EQUATIONS
valid for every one of our functions 'YI·
This equation immediately leads to the desired differential equation in virtue of the following fundamental lemma of the calculus of
variations: If the relation
rl 77(x)cp(x) dx = 0, with cp(x) a continuous
Jzo
function of x, holds for all functions 'Yl(x) which vanish on the boundary
and are continuous together with their first two derivatives, it follows that cp(x) = 0 identically. This lemma, which holds equally
well for multiple integrals, is easily proved indirectly. Let us suppose
that cp(x) is different from zero, say positive, at x = ~· Then there
must exist a neighborhood G, given by ~o < x < ~1, in which cp(x)
4
is positive. We take the function 'Y/(X) = (x - ~o) (x - ~i/ in G
and .,,(x) = 0 outside this interval.
Then we have
i
ZJ
77cp
dx
>
0 in
zo
contradiction to the hypothesis. The assertion cp = 0 remains valid
if we require that the first k derivatives of the function21 77 be continu2
ous; in this case we simply set 'Y/ = (x - ~0 ) z(x - ~1 ) with 2l > k.
From the fundamental lemma it follows immediately that the
function dF,//dx - F11 , which we shall henceforth denote by -[Flu,
vanishes identically in x, i.e. that the function y(x) satisfies the differential equation
d
dx
(14)
F11' -
F11
= 0
or, written out in detail,
(14')
This is the fundamental differential equation of Euler, which occurs
time and again throughout analysis and its applications. Its validity
is a necessary condition for the existence of an extremum. It is a differential equation of the second order, in the general solution of which
there occur two arbitrary constants, the number generally required
in order to satisfy the boundary conditions. We now1 define: every
solution of Euler's differential equation is an extremal of the minimum problem. The differential expression [F] 11 is called the
1
Cf. definition in footnote 2 on page 169.
186
IV. THE CALCULUS OF VARIATION S
variational derivative of F with respect toy, Its role here is analogous
to that of the differential quotient or of the gradient in ordinary minimum problems.
If, as is customary in the theory of differential equations, we want
to solve the Euler differential equation for the highest derivative , we
have to postulate
F,,·u•
¢.
0.
This inequality is known as the Legendre condition; it is of great importance in the problem of investigati ng whether an extremal actually
gives an extremum (compare also §6, pages 214-216).
The essential idea in the above considerations is to imbed the ex- ·
tremal y(x) in· a family of functions y(x)
E7J(X) with the parameter
E. The fact that this parameter occurs linearly is not important . It
is often convenien t to imbed the function in a more general family
of functions y(x, E); the above considerations remain valid if we set
+
11(x) =
y(x,
iJiJ
E
E7J
E)I·
e- 0
.
We now introduce a useful terminology. We called the function
= oy the variation of y(x); similarly the expression
1
OJ
=
E cI>'(O)
=
E {
zo
(15)
= •
(1JF 11
+ 7J'F
11 ,)
dx
J.:' (F, - :x F,.) ~ dx + .F,.~ I~' - .F,.~ 1-·•
1
= [:
zo
[F]y oy dx
+ FY' oylZI '
zo
even if 11 does not vanish at the boundary, is called the variation
or more precisely the first variation of the integral J. This is analogous to the notation of the differential calculus, where the expression
Ej'(x) = df with an arbitrary parameter Eis called the differential of
the function f(x). Thus a necessary condition for a minimum is the
vanishing of the first variation for all y + oy which satisfy the boundary
conditions.
In general, functions or, geometrically speaking, curves for which
oJ vanishes, i.e. extremals, are sometimes called stationary junctions
or curves. This terminology indicates that we may be dealing with
187
THE EULER EQUATIONS
circumstances other than actual extrema, just as in the corresponding
problems of the differential calculus. There are, in fact, many cases
in which one is primarily interested in the vanishing of the variation
rather than in the question of an extremum. Problems in which
one is merely interested in obtaining stationary values are likewise
called variational problems.
The examples considered above (pages 170-173) lead to the following variational derivatives:
(a)
(b)
F
F
= v'e + 2/v'
+ gv'
d
du
f
v'e +
_
gv'
2/v' + gv' 2
v'r-+72
= 1/l(x
y)Vl
=
~(~~
2
,
+
'
+
y' 2
+ 2f,.v' + g"v'
2v'e + 2/v' + gv'2
'
1/ly"
(c)
(d)
.F = YVI
F
+ y'
= YVI
2
yy" = 1
,
2
e"
= (1/111 - 1/l:r:y') (1
_
O·
-
+ y'2);
+ y' 2 (a special case of (b));
yy" = y'2 - 1.
- y' 2 ,
We shall return to the integration of these differential equations
in §4.
2. Several Unknown Functions. A straightforward generalization
of the problem discussed in subsection 1 is the problem of determining several functions y(x), z(x), · · · of x such that the integral
:11:1
J
(16)
=
1
F(x, y, z, · · · , y', z', · · ·) dx
zo
is an extremum (or stationary value}, where the values of the functions at the boundary points may again be given. Once again we
introduce arbitrary functions .,,(x), t(x), · · · , vanishing on the boundary, and assume that the system of functions y = y(x) = f(x),
z = z(x) = g(x), · · · yields an extremum. As above, this leads to
the conclusion that the function
<l>(Ei,E2, ···)
=
l
z1
:r:o
F(x, y
of the variables
E1 ,
+
E1'1],
z
E2, • • •
+ E2t,
· · • , y'
+ Ei'1
1
,
z'
+ E2t',
must have an extremum at
• • ·) dx
E1
=
0,
188
IV. THE CALCULUS OF VARIATIONS
1
E2 = 0, ···.Thus we must have (a<I>/8E1)o = 0, (a<I>/8E2)0 = 0, · · ·
or
. ·.
<>J = Et(a<I>) + E2(a<I>) +
dE2 0
8E1 0
Again it may be
This expression is called the first variation of J.
brought into the form
i.z1 (
lz1
z1
lzo + E2Fz'r zo + E1 zo 11 F11
U = EtF11 1J
1
(17)
+ E21:
)
d
- -d F11 dx
1
x
:x F.) dx + ... ;
1
r(F. -
in the present case the boundary terms vanish. Since we must have
oJ = 0 if one of the functions 7/, r, ... is chosen arbitrarily while all
the others are zero, the arguments employed above show that the
following Euler differential equations hold:
- [Flu
d
= dx
=
Fu' - F11
F11'11'Y
11
+ Fu'z'z'' + ·· · + F11 11Y + F11 •z' +
+ F11 z - F11 = 0,
1
1
1
1
(18)
- [F],
=
dxd Fz' - F.
=
Fz 11'Y
1
11
+ Fz z + · · • + Fz'uY + F,,,z' +
+
1
11
1
1
Z
Fz'z - F,
= 0,
Thus, as a necessary condition for the extremum or stationary
character of the "space curve" y = f(x), z = g(x), · · · , we obtain the
system (18) of differential equations of the second order; the number
of equations in this system equals the number of unknown functions
y, z, · · · to be determined.
All considerations of subsection 1 remain valid. It may be noted
1 The
subscript 0 means that we set
EI
=
E2
= · · · = 0.
189
THE EULER EQUATIONS
that the vanishing of the first variation is a necessary condition
not only in the case of an extremum, but for example, also in the
case of an integral which is to be minimized with respect to variation
of the function y(x) and simultaneously maximized with respect
to variation of the function z(x).
Once again every curve which represents a solution of the system
of differential equations is called an extremal.
A simple example of the Euler equations (18) is given by the problem of determining the shortest lines in ordinary Euclidean space, or
more generally in a non-Euclidean space with the line element
2
ds = gu dx
2
+ g22 dy + gaa dz + 2g12 dx dy + 2g1a dx dz + 2g23 dy dz.
2
2
In this case we have
F = (gn
+
+
2g12Y'
+
2g1ai'
g22Y'
2
+ 2g2aY'z' + gaai' )~
2
and we obtain for the "geodesic curves" of this space the two differential equations
.!!__(g12
dx
+ g22 y' + Y2a z')
_ _!_(iJgn
2F ay
+ 2 iJg12 , +
···) = 0,
.!!__(' g13
+ g2a y' + Ya3 z')
_ _!_(iJgn
+ 2 iJg12 y' +
.. -) = 0.
~
F
F
~
~
In Euclidean space, where we have
2
2
2
ds = dx + dy +di,
iJy Y
F =
~
V 1 + y' 2 + z' 2,
these equations become
d
dx
V1 +
y'
y'2
+
-0
zi2 -
-dxd v' 1 + z'y' + z' -o
2
'
2
-
and are satisfied by all straight lines in space.
The propagation of light in a three-dimensional medium with light
velocity ~(x, y, z) is characterized by the variational problem
T=
1 vi +
z1
zo
y'2
~(x,
+ z'2 dx=mm.
.
y, z)
More generally, we may suppose that the velocity of light depends on
the direction of the ray as well, so that it is represented by an expression ~(x, y, z, y', z'). Then the problem of finding a light ray,
190
IV. THE CALCULUS OF VARIATIONS
a problem in geometrical optics, is equivalent to our general problem, with
F =
v1 + y'2 + z'2
cp
(
x, y, z, y I ' z ' )
•
3. Higher Derivatives. Euler's differential equation is obtained in
an analogous manner for the variational problem of finding stationary values of the integral
(19)
J =
lzi
F(x, y, y', y", · · ·, y<n» dx,
zo
where F is a given function of the arguments x, y, y', · · · , y<n> and
where all those functions are admitted for comparison which have
continuous derivatives up to the 2n-th order and for which the
derivatives up to the (n - 1)-st order have prescribed values at the
boundary. Again we take 11(x) to be an arbitrary function, continuous
with its derivatives up to the 2n-th order and with 11(x) = 11'(x) =
· · · = ,,cn-1\x) = 0 at the boundary points x = Xo and x = Xi. Just
d
as before we obtain for the first variation OJ
Ea/[Y + EfJllE=O the
expression
fJJ
=
E
lzi
(Fy'f/
+
Fy•'fl'
xo
+ · · · + Fy(n)'f/(n» dx.
By repeated integration by parts we can eliminate all the derivatives
of the function .,, from the integral, transforming it into the form
(20) OJ =
2
1
Xl
E
[
zo 1J
d
d
Fy -dx FY'+ dx 2 FY" - · · ·
an Fycn>
+ (-1) dxn
n
J
dx.
Thus, by the fundamental lemma (see subsection 1), we obtain the
differential equation of order 2n,
(21) [F]y = F11
-
d
dx F11 ,
2
+ dxd
2
F11
11
as a necessary condition for an extremum. This equation is again
called Euler's equation. The 2n constants of integration occurring in
the general solution of (21) are to be determined from the 2n boundary conditions.
To determine several functions y, z, · · · · which solve the variational
problem
z1
1
XO
I
I
II
"
F(x,y,z, ···,y,z,
···, y II ,z,
··-)dx =mm.,
191
THE EULER EQUATIONS
we have to consider the corresponding systems of Euler differential
equations in the same way.
4. Several Independent Variables. The problem of determining
extrema of multiple integrals leads to one or more partial differential
equations for the required functions, just as the problems considered
so far lead to ordinary differential equations. Let us consider, for
example, the problem of finding an extremum of the double integral
J
(22)
=
Jl
F(x, y, u,
U:i:,
u 11 ) dx dy
over a given region of integration G by determining a suitable function u which is continuous, has continuous derivatives up to the
second order, and takes on prescribed values on the boundary of G.
We introduce an arbitrary function 11(x, y), on which we shall later
impose the boundary condition 11 = 0, and obtain as a necessary
condition for an extremum the vanishing of the first variation
fJJ
= E(d!!E ~ (E)) e=O = E(d!!:..E J[u
+ E'll) ;
e=O
this is equivalent to equation
which we may again transform by integration by parts. We assume-as usual-that the boundary curve r of G has a tangent
which turns piecewise continuously. Then according to Gauss's
1
integral theorem we have
fl
(T/:r:Fu"'
+ 77 Fu) dx dy
11
=
1~(F
••
dy - F., dx) -
JL ~ (:x
F ••
+ :y F • .)ax dy.
We thus obtain
{JJ =
=
E
rr '11 {Fu
JJo
Ji
dx dy +
- .!.__Full}
- ~Fus
iJy
iJx
5u[F]u dx dy
+
i
E
111(Fus dy - Fu11 dx)
r
5u(Fu"' dy - F u 11 dx) = 0,
1 See, for example, R. Courant, Differential and Integral Calculus, Vol. II,
p. 360, 2nd ed. rev., Interscience Publishers, Inc., New York, 1947.
192
IV. THE CALCULUS OF VARIATIONS
and if, corresponding to the assumption of fixed boundary values of
u, we impose the condition 11 = 0 on the bound,ary, we obtain
(24)
l>J =
E
rr 1/.{Fu
Jlo
- !__ Fu,,, - !__ FU11} dx dy
ax
ay
= 0.
The equation oJ = d must be valid for any arbitrary continuously
differentiable function .,,. The lemma stated in subsection 1 for
simple integrals also holds for multiple integrals and can be proved
in the same way; we therefore conclude that u(x, y) must satisfy the
Euler differential equation ·
(25)
-[Flu
a
a
=ax Fu,,,+ ay Fu 11
- Fu
= 0,
or more explicitly
From the manifold of all solutions of this equation a particular solution must be determined by means of the given boundary condition
(boundary value problem).
Similarly, we obtain a system of such differential equations if there
are several unknown functions to be determined, and a differential
equation of order 2n,
a
(26)
a
+ -axa
2
[Flu = Fu - -ax Fu :t: - -ay Fu
II
2
a Fu
+ax ay
2
Fu :t::t:
a
+ ay
-2Fu1111 + ''' +
:t:ll
n an
2
(-1) -ayn Fu1111'"'11 = 0,
if the function F contains the derivatives uz, u 11 , • • • , u 1111 ... 11 up to
then-th order.
We may consider the example F = !(u! + u!) (cf. page 172).
Here the Euler equation is the same as the "potential equation"
Au
The function F
Euler equation
=
Uxz
2
+ U1111
!(Au) = ! u!z
AAu = Urzzz
+
+
2Uzz1111
= 0.
UzzU 1111
+
U11111111
+!
= 0;
u!11 leads to the
193
THE EULER EQUATIONS
the same Euler equation can be obtained from the integrand
2
(~u) - c(u:u:U1111 - u!11 ) with constant c.
The problem of minimal surfaces, i.e. the integrand
F =
V 1 + z! + z!,
leads to the Euler differential equation
Zzx(l
+ z!) -
2Zx11ZxZ 11
+
Z1111 (1
+
z!)
=
0.
5. Identical Vanishing of the Euler Differential Expression. The
Euler differential expression for an integrand F(x, y, y', · · · ) may
vanish identically for every admissible argument function. Since
admissible argument functions can be constructed for which the
quantities y, y', · · · take on prescribed values at an arbitrary point x,
the identical vanishing of the Euler expression [F] 11 for all functions
y is equivalent to the identical vanishing of this expression if
x, y, y', · · · are regarded as independent parameters. The same
statement holds if the argument function depends on several independent variables.
The simplest case is that of the integrand F(x, y, y'). If the expres11
sion F 11 - Fv'z - F 11 11 y' - F 11 11 y vanishes, it follows that· F 11 'y' = 0
and therefore that Fis of the form F = A(x, y) + y' B(x, y). Then
the Euler differential equation becomes simply the integrability condition
1
1
1
aA _an=
ay
ax
and the integral
L:
F dx =
L:
(A
+ By') dx
=
0
'
c::,
(A dx
+ B dy)
is indeed independent of the curve of integration, according to a well1
known theorem of integral calculus. If the upper limit x is regarded as a variable the integral becomes a function G(x, y) of the
upper limit; we obtain
F(x, y, y') =
Jx G(x, y).
1
We shall see presently that this independence also follows directly from
the identical vanishing of the Euler expression.
194
IV. THE CALCULUS OF VARIATIONS
Thus the Euler differential expression for F vanishes identically if
and only if this relation holds.
The situation is similar for the integrand F(x, y, y', · · · , y<n>).
In this case also a necessary and sufficient condition for the identical
vanishing of the Euler differential expression [F]u is that F be representable in the form
F
=
dG
dx
where G(x, y, y', · · · , y<n-I>) contains only derivatives of yup to the
(n - 1)-st order.
We may verify by a simple calcu1ation that this condition is sufficient or we may see. this in the following way: The integral
l:c
1
F dx
zo
is assumed to depend only on the values of the function y and its
first n - 1 derivatives at the end points; therefore it is not changed
if the function is varied in the interior, as long as these boundary
values are retained. Therefore the first variation and consequent1y
the Euler expression vanish identically.
To show that the condition is necessary we consider a family of
functions y(x, a) with the parameter a and fixed (independent of
a) boundary values of y, y', · · · , yen-I). If we denote the integral
with the argument function y(x, a) by J (a), the formulas for the
first variation yield
aJ
- =
aa
1z1 [F]u -ay dx = 0
aa
zo
since [F]u vanishes identically. Thus J does not depend on a and
is therefore a function only of the coordinates xo and X1 and of the
values of y and its first n - 1 derivatives at the end points. If we
suppose the initial point xo fixed and the upper limit x1 variable, we
obtain an equation of the form
z1
1
zo
1
F(x, y, y, · · ·, y
(n)
) dx
=
/
G(x1, y, y, • · ·, y
(n-1)
),
from which the rest of our assertion follows by differentiation with
respect to the upper limit.
The case of integrals with argument functions of several variables
is analogous, a.s Jong as the integrand F(x, y, u, ux, Uy) contains only
195
THE EULER EQUATIONS
derivatives of the first order. We then obtain, in the same manner
as before, the following theorem: A necessary and sujficient condition
for the Euler expression [F]u to vanish identically in the argument function u is that F be representable in the form
F =
A:e
+ B 11
where A and B are functions of x, y, and u. An expression of this form
is known as a divergence expression.
A divergence expression may also be characterized by the requirement
that the value of the double integral
Ji
F dx dy remain unchanged if the
function u is so varied that the variation is confined to a subregion interior
to G.
According to the Gauss integral theorem we have
fl
F dx dy
=
[(Ady - B dx),
where the integral on the right is a line integral extended in the positive sense around the boundary curve r of G.
The situation may be somewhat more complicated if the integrand
F contains partial derivatives of orders higher than the first. The
theorem that the Euler expression vanishes identically if and only if
F can be represented in the form
F
=
Az
+ B11
(i.e. if F is a divergence expression) remains valid; however, it is not
in general possible to choose A and B in such a way that the· derivatives occurring in these functions are of lower order than those occurring in F.
The simplest example of a divergence expression of the second
order is F = U:e:eU 1111 - u! 11 • We have here
F =
(UzU:e11)11
(UzU1111)z -
=
-H(u!) 1111 -
= - (UyUzy)z
2(UzU11 )z11
+ (U11Uzz)11
+ (u!)zz].
Another example is given by the identity
U:ezU1111 -
(1
u!11
+ u! + u!)
312
iJ [
= ay
J
+ u! + u!
UzzUy
(u! + l)v1
a[
- ax
u%1/u11
(u!
+ 1)v1 + u!-+-u;
J
·
196
IV. '£HE CALCULUS OF VARIATIONS
The expression
2
U.1:zUu11 -
(1
Uzy
+ u! + u!) 372
is the Gaussian curvature of the surface z = u(x, y) multiplied by
Vl + ui + ui; its divergence character expresses the well-known
fact that the integral of the curvature over a segment of a surface,
i.e. the total curvature of the segment, depends only on the tangent
planes of the surface along the boundary of the segment.
Our considerations lead to the following theorem: If the difference
between the integrands of two variational problems is a divergence expression, then the Euler equations and therefore the families of extremals
are identical for the two variational problems. (See footnote, page
211.)
6. Euler Equations in Homogeneous Form. In geometrical problems, where we are concerned with determining curves or surfaces by minimum conditions, it is often appropriate to refrain from
arbitrarily designating coordinates as independent variables. Instead we may employ a parametric representation x = x(t), y = y(t)
for the curve (or x = x(u, v), y = y(u, v), z = z(u, v) for the surface)
where t( or u and v) is the independent variable, and where the equations
x=y=O
(or
XuYv -
XvYu
=
YuZv -
are not satisfied simultaneously.
YvZu
=
ZuXv -
ZvXu
= 0)
Here differentiation with respect to
t is indicated by a dot. We begin by considering the simplest variational problem, which takes the form
(27)
J =
1
z1
zo
(
F x,
y, dd~)
dx =
J['1
to l}(x, y, x, y)
dt = min.,
where
This function l1 is "homogeneous" of degree 1 in the derivatives
which means that for all k it satisfies the homogeneity relation
(28)
l}(x, y, kx, ky)
= kl}(x, y, x, y)
x, y.
197
THE EULER EQUATIONS
and consequently equation
(29)
which follows from (28) by differentiating with respect to k and set1
ting k = 1. If, conversely, lY is any homogeneous function of the
first degree in x and y, i.e. if lY satisfies equation (28), then the variational problem
JlY
dt = min. determines a curve independent of the
choice of parameter. For, if we perform the parametric transformation t = t( r) with dt/ dr > 0, the interval to ~ t ~ ti goes over into
ro ~ r ~ r1 and we obtain, making use of (28),
1
('
J,o lY
(
('
dx dy)
x, y, dr' dr dr = J,o
1
lY
1
Tl
(
x, y,
•
X
dr
dt
lY(X, y, X, fJ) d- dr
T
TO
j
dt)
dt .
y J:,.
a,-;.'
·t1
lY(X, y, X, fJ) dt.
to
Thus the variational problem is invariant with respect to a parametric
transformation that does not alter the sense of traversal; the extremal
curves do not depend on the choice of parameter.
The homogeneous problem leads to the two Euler equations
(30)
lYx - ~% = 0,
lYY - ~y
= 0,
which, together with relation (29), must essentially be equivalent to
the original differential equation (14), and therefore cannot be independent. We find the interdependence by deriv.ing the following
identities from (29) by differentiation:
lYx = XlYxx + YlYx1i ,
XlYH + YlYx1i = 0,
lY:rz:lYr1i:lY1i1i
lYY = XlYyx + YlYY1i ;
XlYx1i + YlY1i1i = 0;
= '[/:
-xfJ:il.
In geometrical examples the function is often only positively homogeneous (i.e. (28) holds fork > 0) not homogeneous in the complete sense. For, in
this case traversal of the curve in the opposite direction would lead to the
opposite sign for the integral, while e.g., the arc length, which is defined by
1
J~
dt with the positive square root, always has the same value ir-
respective of the direction of traversal. The above considerations also hold
for such positively homogeneous integrands.
198
IV. THE CALCULUS OF VARIATIONS
The value
~z.z _ _ ~zli = ~lili
iJ2 -
xiJ
x2
is customarily denoted by ~i •
From the above identities it follows that
~z
....:...
~z
=
=
+ iJ~ziJ - ~izX --:- ~z11Y ?i[~zu - ~z11 + (xy - yxm11,
X~zz
~11 - ~,; = - x[~zli - ~z11
+
~iiX - ~zliY
(Xfj - yx)~1],
so that the two equations (30) are connected by the identity
(31)
and may be replaced, for example, by the single equation
~zu
(32)
-
~z11
+ (xy
-
iJx)~1 =
0.
The situation is quite analogous when we are concerned with the
determination of several functions of one variable. Here the variational problem
1
1
F(x,
y, z, y< z') dx = min. goes over into the prob-
zo
t1
lem
lz
~(x,
to
the function
y, z, x, iJ, i) dt = min. with ~ = xF(x, y, z,
~
y/x, i/x);
is homogeneous of the first degree in the variables
iJ, and i.
x,
The advantage of the homogeneous representation is not solely
that of symmetry. Thus, for example, curves on which x does not
increase monotonically, such as closed curves, cannot be represented
in the form y = y(x); thus they cannot be readily treated by the nonhomogeneous representation.
In the case of problems in several dimensions the homogeneou~
representation is obtained as follows: If in the integral
JJF(x,
y, z,
Zz,
Zu) dx dy
the variables x and y and the function z are written as functions of
two parameters u and v in such a way that the Jacobian determinant
of x and y with respect to u and v
THE EULER EQUATIONS
199
a(x, y)
a(u, v) = XuYv - x,,yu
is different from zero, then we have
Zy
=
and the integral takes the form
ff
(33)
F(x, y, z,
Zz,
zu) dx dy
a(y, z)
a(z, x))
z - a(u, v) - a(u,v) a(x, y) dudv
a(x, y)'
a(x, y) a(u, v)
( , y, '
a(u, v)
a(u, v)
= ff F x
JJ
=
z
rr u (x'y, z'a(u,
a(y, z) a(z, x) a(x, y)) du dv.
v)' a(u, v), a(u, v)
JJ
rt;
Here the integrand lJ is homogeneous of the first degree in the last
three Jacobians. The relations derived above for the case of onedimensional integrals, in particular the identity (31) and the symmetric form (32) of the differential equation, may easily be generalized to the case of several variables. Since these generalizations
will not be needed in the present book, the reader is referred to the
literature. 1
7. Relaxing of Conditions. Theorems of du Bois-Reymond and
Haar. So far we have required that the comparison functions pos-
sess continuous derivatives up to the highest order occurring in
the Euler differential equation. This requirement appears unnaturally restrictive from the point of view of the variational problem;
for example, the variational problem with the integrand F(x, y, y')
has a meaning even if the first derivative is required to be only piecewise continuous and no assumptions at all regarding the second
derivative are made. It is a priori conceivable that if the conditions
of admissibility are broadened in this way one might obtain a new
solution which no longer satisfies the Euler differential equation.
1 0. Bolza, Vorlesungen iiber Variationsrechnung, pp. 666-671, B. G. Teubner, Leipzig and Berlin, 1909; G. Kobb, Surles maxima et minima des integrales doubles, Acta Math., Vol. 16, 1892-3, pp. 65-140.
200
IV. THE CALCULUS OF VARIATIONS
Let us first consider an actual minimum problem and assume that
y(x) is that function with continuous first and second derivatives
which yields the minimum. Then the function y(x) yields the minimum even if we enlarge the space of admissible /unctions to include functions y* which need not have second derivatives. For, according to the
Weierstrass approximation theorem, we can approximate the derivative y*' by a polynomial p' (x) and the function y* by a polynomial
p(x) as precisely as desired, where p(x) satisfies the boundary conditions p(xo) = Yo, p(x1) = Yi .1 Then J[p] also differs arbitrarily
little from J[y*]. But since p(x) is an admissible comparison function with continuous second derivatives we have J[p] ~ J[y], and
therefore J[y*] ~ J[y].
If the minimum problem, or more generally the problem of stationary values, is formulated with these broadened admissibility
conditions, we stipulate only that the solution must possess a continuous first derivative. We now pose the question: does this
function automatically possess any higher order derivatives? If so,
does it satisfy the Euler differential equation? This question is
answered in the affirmative by the following theorem of du BoisReymond: Given a variational problem with integrand F(x, y, y'), suppose F has continuous first and second derivatives with respect to all its
arguments. Let y(x) be a continuous function with prescribed values
at the boundary and with a continuous first derivative. Assume that the
first variation (15) vanishes for y(x) and for every function 71(x) with
vanishing boundary values and a continuous first derivative and that
moreover Fu'u' ~ 0, then y(x) has continuous second derivatives and
satisfies Euler's differential equation; 1:.e. the vanishing of the first variation implies that y(x) possesses a continuous second derivative and satisfies the Euler equation.
1
According to the Weierstrass theorem we may construct a polynomial
xo) from the function y*'(x) for xo ~
x ~xi , where Eis an arbitrarily fixed positive quantity. Then the polynomial
q'(x) which differs by less than (E/2) (x1 -
q(x)
=
Yo
+
z1
1
q'(t) dt
zo
assumes the prescribed initial value Yo and differs from y*(x) by at most E/2
in the interval. To obtain a polynomial which also takes on the required
final value Y1 we simply add to q(x) the linear function l(x) = [y1 - q(x1)]·
(x - xo)/(x1 - xo) and verify immediately that p(x) = q(x) + l(x) has all the·
properties specified in the text.
201
THE EULER EQUATIONS
We preface the proof by a simple lemma: If cp(x) is a function piece1
wise continuous in the interval of integration, and if the equation
.z1
1
cp(x)7J(x) dx
=
0
.zo
holds for arbitrary continuous functions 11(x) satisfying the condit-ion
.z1
1:ro 71(x) dx
= 0
then cp(x) is a constant. To prove this we first note that the above
relation is certainly satisfied for constant cp. We now determine a
constant c such that
lx
J:r
1
(cp - c) dx = 0 for our given cp; we then have
.zo
1
(cp - c)11 dx = 0 because
J:r
1
cp71 dx = 0.
But now we may set
zo
zo
11 = cp -- c, obtaining the equation
j
•Xl
2
('P - c) dx = 0, from which
.zo
our assertion follows immediately.
The following more general theorem is proved in exactly the same
way: If cp(x) is a piecewise continuous function which satisfies the con-
dition
ri cp71 dx
=
0 for all continuous functions 71(x) satisfying the
J.zo
conditions
.z1
.z1
1
11 dx = 0,
zo
1 X71
zo
dx = 0,
then cp is a polynomial of then-th degree:
cp =
Co
+
C1X
+ ·· · +
CnXn.
To prove du Bois-Reymond's theorem we note that the equation
holds for any continuously differentiable function r(x) for which
1
See Ch. II, p. 48.
202
IV. THE CALCULUS OF VARIATIONS
We introduce the abbreviations FY = A', Fy' = B,
lz
F 11 dx = A
zo
and obtain, integrating by parts,
z1 (A't + Bt') dx 1z1 ((B 1
=
zo
A) dx = 0.
zo
t' =
We choose an arbitrary function
tions that it must be continuous, that
11,
subject only to the condi-
1·zi
11 dx
= t(x 1 )
zo
-
t(x0)
holds, and that the boundary conditions on t must be satisfied.
plying the preceding simple lemma we obtain
(34)
B - A = F'V' -
lz F
11
zo
where c is independent of x.
equation.
But since
lz
dx
= O
Ap-
= c,
This equation takes the place of Euler's
Fydx is differentiable with respect to the
zo
upper limit x and since c also is, it follows that FY' is differentiable;
therefore Euler's equation
(34a)
d
dx F11 ,
-
F11
=
0
holds. Now if F is twice continuously differentiable with respect to
its arguments, and if furthermore the Legendre condition Fy'y' ~ 0
is fulfilled, it follows that the piecewise continuous function y' is
actually continuous and possesses a continuous derivative. For, in
the first place, since F y'y' ~ 0, y' may be expressed as a continuously
differentiable function cp(x, y, Fy'). Since Fy' is a continuous function
of x in virtue of (34), the same must be true of y'. Thus the arguments y and F y' of cp are continuously differentiable and the same is
therefore true of cp = y'.
Du Bois-Reymond's result may immediately be extended to an
integrand of the form F(x, y, y', · · · , y<n» by using the generalized
version of the above lemma. The details may be left to the reader.
In the case of variational problems with several independent
variables the situation 1s a little more subtle. If in the problem with
integrand F(x, y, u, U:r:, u11 ) the space of admissible functions is extended to include all continuous functions with piece-wise continuous
derivatives, it is no longer true that the vanishing of the first variation
THE EULER EQUATIONS
203
necessarily implies that the second derivatives exist and are continuous
and that Euler's differential equation holds. However, for several
dimensions there does exist an analogue to du Bois-Reymond's theorem (theorem of Haar) : The vanishing of the first variation of the
integral of F(x, y, u, Uz , u.) for continuous u with continuous derivatives Uz and u. is equivalent to equation
(35)
Ji
Fu dx dy =
l
(Fu:i: dy - Fuy dx),
where the integral on the left is taken over any arbitrary simply connected
subregion B of G bounded by piecewise smooth curves, and the integral
on the right is extended over the boundary R of B in the positive sense.
In the particular case where F does not depend explicitly on u:, Haar's
theorem states that the integral on the right vanishes for a closed
curve R; this is equivalent to the statement that a function cl>(x, y)
exists for which the system of differential equations
holds in every simply -connected subregion of G. Therefore the
above integral relation or this system of differential equations of the
first order takes the place of the Euler second order differential
equation.
To prove Haar's theorem we need only show that the integral
relation is valid for the special case that B is a square. It is then
immediately valid for any region consisting of a finite number of
squares, and the theorem for an arbitrary region follows in the usual
manner. Let us consider the square B: Xo ~ x ~ x1 , Yo ~ y ~ Y1 •
The vanishing of the first variation for B is expressed by the equation
s
if vanishes on the boundary of the square. We now specialize the
variation s(x, y) to the form s(x, y) = v(x)w(y), where v(x) vanishes
for x = Xo , X1 and w(y) vanishes for y = y 0 , Yi . Thus we obtain
the equation
204
IV. THE CALCULUS OF VARIATIONS
from which the desired result follows by repeated application of du
Bois-Reymond's theorem: We introduce the abbreviations
lx Fu
11
1 Fuz
dy
= A(x, y),
zo
110
11
dx
= B(x, y),
11
1jx Fu
110
dx dy = C(x, y)
zo
and integrate by parts; we obtain
r dy{r (-
C,v'w
+ A,v'w -
Bv'w')
dx}
=
O
or
{ dx v'{f (- C,w + A,w - Bw') dy}
=
O.
Since v' is the derivative of an arbitrary function which vanishes on
the boundary, it follows from the above lemma that
111
1
(- C 11 w
+A
11
w - Bw') dy
= c,
110
where c is independent of x; integration by parts transforms this to
111
1
(C - A - B)w' dy = c.
110
In this equation we set x equal first to X1 and then to Xo and subtract
one of the resulting equations from the other; we obtain
111
1
[D(xi, y) - D(xo, y)]w' dy = 0,
110
where D = C - A - B.
once more we have
Applying du Bois-Reymond's theorem
D(x1, Yi) - D(xo, Yi) = D(xi , Yo) - D(xo, Yo),
which is just the equation
(35')
1
1111z1
YO
Fu dx dy
zo
that was to be proved.
205
THE EULER EQUATIONS
8. Variational Problems and Functional Equations. So far we
have been concerned with variational problems involving functionals formed by integrating a given differential expression in
the argument function. However, more general classes of functionals
are often encountered in variational problems. We sh::-.Jl show by a
few examples how the above pattern may be applied to obtain functional equations which replace the Euler differential equations.
(a) We want to make the expression
J[cp] =
ff
K(s, t)cp(s)cp(t) ds dt
+
f
2
[cp(s)] ds - 2
f
cp(s)f(s) ds
stationary, where K(s, t) is a given continuous symmetric function of
s and t, f(s) a given continuous function of s, and cp(s) the unknown
continuous argument function. All integrations are to be extended
over a given interval a ::::; s ::::; b, a ::::; t ::::; b. If we replace cp by
cp + Er and consider J[cp + Er] = cl>(E) as a function of E, we obtain,
after a simple transformation,
SJ
.. t
•~:I = 2<
=
!(!} [ { K(s, t)¥>(s) ds
+
¥>(!) - f(t)
J
dt.
Thus the requirement OJ = 0 leads to the Fredholm integral equation
{
K(s, t)cp(s) ds
+ cp(t)
- f(t) = 0
as the Euler equation of the problem.
The extremum problems treated in Chapter III for integral equations with symmetric kernels K(s, t) likewise can be easily treated
by our variational scheme.
(b) The expression
J[cp] =
L:
[p(x)(cp' (x))
2
+ 2cp(x + l)cp(x -
1) - /(x) - 2cp(x)f(x)] dx
is to be made stationary, where the argument function is continuous
and has a piecewise continuous derivative in the entire interval
- oo < x < oo • Forming the first variation we obtain, after a simple
transformation,
fJJ =
Edd J[cp
L:
E
= 2E
+ Et] I
e-0
t(x)[ - (pep')'+ cp(x
+ 2) + cp(x -
2) - cp(x) - f(x)] dx;
206
IV. THE CALCULUS OF VARIATIONS
the Euler functional equation which expresses the vanishing of the
first variation for arbitrary f' is
(w')' - 'P(X
+ 2)
- 'P(X - 2)
+ 'P(X) + f(x)
= 0.
Thus we obtain a differential-difference equation rather than a differential equation.
§4. Integration of the Euler Differential Equation
In Volume II a systematic method of integrating the Euler differential equations will be developed by means of the Hamilton-Jacobi
theory. At this point we shall consider briefly the integration of the
simple preceding examples. We confine ourselves to the problem
l
zi
F(x, y, y') dx = min.
zo
If the function F does not involve the derivative y', the Euler
equation reduces to F'II = 0; this equation determines the function
y(x) implicitly. We note that in this case the boundary values cannot be prescribed arbitrarily if the problem is to have a solution.
If the function F does not contain the dependent variable y, we
obtain imi:p.ediately dFy'/dx = 0, i.e. Fy' = const. = c; therefore
J'P(x, c) dx.
y' = 'P(x, c) and y =
The Euler equation can then be
solved by quadrature.
If the function F does not contain the indepenaent variable x, we
can again achieve the integration by quadrature. For then
(y'F11
1
F)' = y"Fy'
-
+
y'F;, - Fy y" - F'lly' = y'(F;, - Fy) = O;
1
thus it follows immediately from the Euler equation that
F(y, y') - y'F,J'(y, y') = c,
from which y' may be obtained as a function 'P(Y, c) of y and c, and
x
=
J
dy
'P(Y, c)"
Alternatively, we may obtain this result, at least formally, by reduction to the previous case, noting that the extremal curve leads to
the vanishing of the first variation if y is regarded as the independent
and x as the dependent variable. If differentiation with respect to y
207
INTEGRATION OF EULER DIFFERENTIAL EQUATION
is denoted by a dot we obtain the variational problem of extremizing
JF(y, 1/x)x dy, in which the new dependent variable does not occur.
Examples (b), (c), and (d) on page 187 can now be integrated:
.
- . 11 +
y'2
----;
(b) with 1/1 = 1/'\/y, that is, F =
1
y F11
!
Setting y =
x
1
-
-
F =
c2 (1 -
vy(1
y
1
.
+ y'
cos t), we have
1
= const. = -.
2
)
c
•
lc2-y
--
t
=
y
Jdyy' = Jtan (t/2) dtdy dt
= c2 Jsin2 (t/2) dt = c1 + ! c2(t -
cot (t/2)
=
sin t).
Thus the brachistochrones are the cycloids described by a point on
the circumference of a circle of radius c2/2 which rolls on the x-axis.
(c) F =
YV 1 + y'
I
yF11
2 ;
y
1
-
F =
= -c1 cosh (ex+
-y
vi+ y'
2
=
1
- c'
c1).
The surface of revolution of least area connecting two given circles is
thus obtained by revolving a catenary about its axis.
(d) F
= YV 1
- iJ2
iJF· - F =
;
11
y =
- y
v11="Y2
=
!c sin (cs + ei).
The other coordinate xis given by
x =
Jv
1 - iJ2 ds =
J
sin (cs
+ c1) ds
= -
~ cos (cs + c1) + ~;
thus the solution of the isoperimetric problem can only be a circle.
208
IV. THE CALCULUS OF VARIATIONS
§5. Boundary Conditions
In the previous sections we have postulated that the functions to be
determined assume prescribed values at the boundary of the region
of integration. However, in many problems there are no a priori
conditions for the boundary values, or the behavior of the function
at the boundary may be restricted by more general conditions. If
no boundary conditions are prescribed for the unknown functions in a
fixed basic region we speak of free boundary values. Problems also
arise in which the boundaries themselves are free subject to suitable
restrictions. In geometry, for example, we often wish to determine
curves with end points on a prescribed curve or surface, or surfaces
with boundaries on a given ~urface. Here it is part of the problem
to determine the region of integration of the independent variables.
Problems of these types may be treated by a simple generalization of
our previous procedure: the expression for the first variation f>J of
the integral J is adapted to our more general problems since the variations of the functions at the boundary do not necessarily have to be
assumed equal to zero.
1. Natural Boundary Conditions for Free Boundaries. We consider the variational problem of §3, 1 with the integral
z1
J
=
1
F(x, y, y') dx
zo
but no longer impose conditions on the argument function y(x) at
x = x 0 , x1 • The necessary condition for J to be stationary is that
the first variation
(equation (15), page 186) vanish. It is evident that the Euler equation [F]y = 0 must be satisfied. For, if J is stationary with respect to
variations which do not have prescribed boundary values, then it is
certainly stationary with respect to the smaller class of variations for
which f>y = 0 on the boundary which implies Euler's equation.
Therefore we need only consider that part of f>y which depends on the
boundary. Because of the arbitrariness of f>y at the boundary we
obtain as a necessary condition the "natural boundary condition"
Fu'= 0
for
x
=
Xo
and
x
=
Xi •
209
BOUNDARY CONDITIONS
In the same way, from the expressions for the first variation (pages
188-191) we obtain as necessary conditions for the stationary char-
acter of the integrals
x1
1
(36)
F(x, y, z, · · ·, y', z', · · ·) dx,
zo
(37)
JL F(x, y, u,
fl
(38)
F(x, y, u,
Uz, Uy,
Ux, Uy)
v,
dx dy,
Vz, Vy, • • ·)
dx dy,
respectively, the natural boundary conditions
Fy'
=
Fz'
=
for x = Xo and x =
0
dx
dy
F "'z ds - F "'11 ds = 0,
X1 ,
dx
dy
-=
- - F 1111
F
ds 0 ·' · ·
Vz ds
in addition to the Euler equations. The last two sets of conditions
are to be satisfied on the boundary r of G whose arc length is denoted
bys.
The concept of natural boundary conditions is significant because
it is easily applied to more general types of variational problems, including those in which boundary values occur explicitly. The following important examples illustrate this:
J =
(39)
lx
1
F(x, y, y') dx - cp(yo)
zo
+ 1/t(y1),
(not prescribed)
and
(40)
J = JJF(x, y, u,
Uz, Uy)
dx dy
+
l
<l>(s, u, u.) ds
( u.
=
~;).
The variations are given by
( )
41
aJ =
J.:'
-
+ [il-' (y + F•' (x y(x
['P'(yo) + F,Axo, y(xo), y'(xo))] oyo
[F], 8y dx
1)
1,
1),
y' (xJ)] ay1
210
IV. THE CALCULUS OF VARIATIONS
and
(42)
fJJ =
fl
[F]uOU dx dy
l
+ (Fu,.~; -
Fuv
~= + [<P]
11)
OU ds,
respectively, with
(43)
d
ds <Pu.•
[<P]u = <Pu -
The corresponding natural boundary conditions are:
+ <ie'(y)]lzo
[Fii'
=
0,
[Fii'
dy
dx
Fu,. ds - Fu 11 ds +<Pu -
+ 1J.''(y)Jlz1 = 0;
d
ds <Pu~ =
1
0.
In the particular case
(44)
J
=
Ji ( +
u!
u!) dx dy
+
£
2
<TU ds
with a continuous boundary function u(s) the expression for the
variation is
(45)
8J = -2
JJ. (u.. + u,,.)
&u dx dy
+2
£(:: +
au) &u ds,
where a/an denotes differentiation in the direction of the outward
normal. For the more general integral
J =
(44')
Ji
[p( u!
+ u!)
2
- qu ] dx dy
+
£
2
p<Tu ds,
in which p(x, y) and its first,derivatives are continuous in G, q(x, y)
is continuous in G, and u(s) is continuous on r, we obtain similarly
(45')
OJ
=
-2
fl [(puz)z + (pu
1) 11
+ qu] ou dx dy
+2
£ (:~ +
p
<TU) OU ds.
If in (39) and (41) we set
'f(Y) = l(y - a)
The notation
at x == xo.
1
2
,
lzo means that the expression at the left is to be evaluated
211
BOUNDARY CONDITIONS
the natural boundary conditions become
}z F
11'
l:i:o
+ Yo -
~l F
a = 0,
11
1
lzi
+ Y1 -
b = 0.
The passage to the limit l ~ oo yields the condition for tlxed boundary
values
Y1 = b,
Yo= a,
so that the si,mplest variational problem with fixed end points of the
extremals appears as a limiting case of a problem with free boundaries.
1
In general we can, by adding boundary terms or boundary integrals,
essentially modify the natural boundary conditions without altering the
Euler equations.
2. Geometrical Problems. Transversality. In problems where the
end points of the required curve are free on given curves or surfaces, 2
or, more generally, where the boundary of the region of integration is
not fixed, parametric representation is useful. We shall derive the
boundary conditions for an extremum if the unknown plane curve
y(x) begins on a fixed curve T(x, y) = 0, while the end point at x1 is
fixed. We introduce a parameter t varying between the fixed limits
1
t0
J
=:::;;
t =:::;; t1 and transform J = [ : F(x, y, y') dx into the integral
=
1'
0
1
ty(x, y, x, y) dt, with ty
to
~ xF(x, y, y/x).
The initial con-
dition is T(x(to), y(to)) = O; the values x(t1) and y(t1) are fixed. We
have thus eliminated the bothersome variability of the interval of
integration. We now introduce two functions ~(t), 71(t) which vanish
at t = t1 but are otherwise arbitrary, and two parameters E1, E2 satisfying the condition
'1'(E1, E2) = T[x(to)
+ E1Wo), y(to) + E271(to)]
= 0.
1 Instead of adding such boundary integrals one might add divergence expressions to the integrand over the entire region of integration (see pages 195
and 251, footnote).
2 The "free boundaries" just considered are, of course, a special case of
such problems. Thus, for example, the problem
l
s._
F(x, y, y') dx =min.,
zo
where the values y(x 0), y(x1) may be arbitrary, may also be formulated as
follows: Find the curve with end points on the vertical straight lines x = xo ,
x = x1 which minimizes the integral.
212
IV. THE CALCULUS OF VARIATIONS
If our curve is extremal, the function
is stationary for e1 = 0, e2 = 0 when E1 and e2 are subjected to the condition 'l'(e1 , e2) = 0. According to the theory of ordinary extrema
there exist two constants Ao , A, not both equal to zero, such that
!la (A'11
vE1
+ Ao<I>) I
= 0,
!la (A'11
vE2
E1=E2=0
+ Xo<I>) I
= 0.
E1=E11=0
We assume that aT/ax and aT/ay do not both vanish at t = to.
Then we may take Ao = 1. Since the functions x(t), y(t) must satisfy
the Euler equations, we find from dur expressions for the first variations that the equations
HATx - ~z) =
hold at t = to.
o,
Eliminating A we obtain the transversality condition
(46)
If the end point is also variable on a given curve a corresponding
condition must, of course, be satisfied there.
The transversality condition is a relation between the direction
of the extremal curve and that of the given boundary curve. It is
linear in Tx and Ty ; therefore, if the direction of the extremal is
given, that of the boundary curve is certainly determined uniquely.
(The converse is not necessarily true.) For every given boundary
curve it is possible to construct a one-parameter family of transverse
extremals by drawing through each point of the boundary curve a curve
which is a solution of the Euler equation and starts in the transverse
direction.
Returning to the inhomogeneous representation y = f (x) of the
curve we obtain the transversality condition in the form
(47)
(F - y'F,,.)Ty - F,,•T:e
=
0
since
(48)
~z
= F - iiJ Fy' = F -
y 'F,,•;
~ti=
F,,,;
213
BOUNDARY CONDITIONS
if the boundary curve is given in the form y = g(x), the transversality
condition becomes
F
+
(g' - y')F11
1
= 0.
Note that the latter formulation fails whenever the tangent of the
boundary curve is parallel to the y-axis at the point under consideration. In this case we see from (47) that we return to the natural
boundary condition F 11 • = 0.
The situation is very similar in the problem of determining a space
curve y = y(x), z = z(x) which begins on a given surface T(x, y, z) = 0,
passes through a given point (x1 , y 1 , z1), and renders the integral J =
lx
1
F(x, y, z, y', z') dx stationary.
Just as before we
zo
introduce a parametric representation with lj(x, y, z, ±, iJ, z) =
±F(x, y, z, iJ/x, z/x) and obtain the transversality conditions
(49)
or, in inhomogeneous notation, the conditions
(50)
Tx:T 11 :Tz = (F - y'F11 ,
-
z'Fz•):F11 ,:Fz'.
Once again these conditions assign one (or more) directions to each
point of the boundary surface T = 0; therefore for each boundary
surface we have a two-parameter family of extremals. To every
direction of the extremal there corresponds exactly one transverse
direction of the surface.
It goes without saying that the same transversality conditions
apply to the end point of the curve if this point is variable on a
surface.
In the case of the geodesic curves on a surface or of the shortest
curves in space, transversality coincides with orthogonality. Thus
for F = V 1 + y' 2 + z' 2 the transversality condition is T x: T 11 : Tz
l:y':z'. For F = Ve+ 2fy' + gy' 2 we obtain
Tx: T 11 = (e
+ fy') :(f + gy'),
which is the condition for an orthogonal intersection.
Therefore, if on a surface we draw the pencil of geodesics from a point
P, this pencil intersects its orthogonal trajectories transversely. Let Q
be any point near P. As Q moves along an orthogonal trajectory, the
length of the geodesic from P to Q is statiQnary. Hence this length is
214
IV. THE CALCULUS OF VARIATlONS
constant and the trajectories, the so-called geodesic circles, are closed
curves.
In the second volume we shall treat in more detail the connection
between transversals and extremals. Here we merely point out that
in the case of the propagation of light the transversals are simply the
wave fronts of the light waves and the extremals are the light rays. By
a transversal we mean a curve or surface which is everywhere transverse to a family of extremals.
§6.
The Second Variation and the Legendre Condition
The Euler differential equation is a necessary condition for an
extremum. It turns out that a particular extremal satisfying given
boundary conditions can furnish an actual extremum only if it satisfies certain additional necessary conditions, which take the form of
inequalities. The formulation of such inequalities, together with
their refinement into sufficient conditions, is an important part of the
classical calculus of variations. This will be discussed in the second
volume; at present we merely state Legendre's necessary criterion:
If the extremal 'P = u(x) makes the integral J[<P]
i=
lx
zo
1
F(x, "'' <P') dx
a minimum with respect to continuous comparison functions <P(x)
with piecewise continuous first derivatives, then the condition
F ,,,,,,(x, u, u')
~
0
is satisfied everywhere along the extremal.
To prove this we expand the expression
Zl
J[<P]
=
1 F(x,
zo
by Taylor's theorem:
Here
'P,
<P') dx
215
SECOND VARIATION AND LEGENDRE CONDITION
where the bar means that in the expressi~ns F ,,,, , F ,,,,, , F ,,, ,,, the
1
1
p11',
p11, ip = 'P
arguments 'P, <p 1 are to be replaced by ip = 'P
for
stationary
is
J
Since
E.
and
0
between
p being a number
'P = u, Ji[u, 11] vanishes and a necessary condition for a minimum is
evidently J2[i/J, 11] ~ 0 for arbitrarily chosen 11·
If in J2[i/J, 11] we let the parameter E tend to zero, J2 goes over into
the integral
+
+
and we obtain the necessary condition for the extremal u
J2[u, 11]
~
0.
2
If we define the "second variation" fJ J of J by
2
2
fJ J =
~ J2[u, 11]
the condition becomes
tiJ ~ 0.
From this integral condition we obtain the above differential condition of Legendre by making use of the arbitrariness of 11· We
choose for 11 a special piecewise linear function which is different from
zero only in the vicinity of the point x = a, i.e. we choose
x- a)
_;- (l+-CT-
for
a -
X- a)
-1-(1--CT-
for
a :=;; x :=;; a
11=vCT
11=vCT
11 =
0
CT
:=;; x :=;; a,
+ CT,
everywhere else.
The integral J2[u, 11] then reduces to an integral over the interval
2
CT, and in this interval 11' = 1/CT. If we now let
a - CT :=;; x :=;; a
CT tend to zero the first two terms of the integral tend to zero, while
the limit of the third term is the value of 2F ,,,,,,at x = a. Thus this
value must be non-negative, and the necessity of the Legendre condition is established.
In the case of several unknown functions <p, t/I, · · · the correspond-
+
216
IV. THE CALCULUS OF VARIATIONS
ing Legendre condition is: the quadratic form whose coefficient matrix
IS
must not assume negative values; i.e., it must be positive definite.
Instead of this form of the Legendre condition with the sign ~ the
more stringent Legendre condition
F,,,,,,,,
>
0
is often considered. If this condition is not only fulfilled for the
extremal cp = u, but also holds for arbitrary values of x and u in a
given region and for completely arbitrary u', we speak of the strong
Legendre condition.
If in addition to this condition the more restrictive inequality
F,,,,,,,,F,,,,,, - F~,,,, ~ 0
holds for all cp and x in a given region and for arbitrary 'P', then the
quadratic form in the integrand of J 2 is positive definite, and therefore an extremal in the given region certainly furnishes a minimum.
This simple but very rough sufficient criterion will be refined in Volume II.
§7.
Variational Problems with Subsidiary Conditions
In the problems studied so far the argument functions could be
chosen arbitrarily, subject only to boundary conditions, and the
solution of the variational problem was determined from the Euler
equations with given or natural boundary conditions. · We shall now
consider problems in which additional conditions are imposed on the
argument functions; these conditions refer to the entire course of the
argument functions and lead to essential modifications of the Euler
differential equations themselves.
1. Isoperimetric Problems. A simple example is the generalized
isoperimetric problem (see §1, 3(d)): Find the function y which makes
the integral
J =
l
:q
:i:o
F(x, y, y') dx
VARIATIONAL PROBLEMS WITH SUBSIDIARY CONDITIONS
217
stationary, has given boundary values y(xo) = Yo , y(x1) = Y1, and is
subject to the subsidiary condition
:z;i
(51)
K
=
J
G(x, y, y') dx
=
c
:r:o
with constant c.
Let us suppose that y = y(x) is the desired extremal. We consider the family of neighboring curves y + oy = y(x) + E117(x) + E2r(x),
Where E1 and E2 are parameters and 17(X) and r(x) are arbitrary functions satisfying the conditions 17(xo) = 17(x1) = r(xo) = t(x1) = 0.
Then the function
must be stationary at E1 = E2 = 0 with respect to all sufficiently smal1
values of Et and E2 for which
:r:1
'11(E1, E2) =
1
G(x, y
+
E111
+
E2r, y'
+
E111'
+
E2r') dx = c.
:r:o
According to the theorems on ordinary maxima and minima (see §1)
there exist two constants Ao and A, not both equal to zero, such that
We thus have
:r:1
1
{Xo[F]11
+ X[G]
11
}11 dx = 0,
.zo
1 {XolFlu + X[Glulr
.zi
.zo
dx = 0•
We may conclude from the first of these equations that the ratio of
Ao to A does not depend on r. Since r is arbitrary, it follows from the
second equation that Ao[F} 11 + A[G] 11 = 0. If Ao ¢ 0, i.e. if equation 1
(52)
1 It is easily seen that we have the exceptional case (52) whenever there
exists only one function which satisfies the given subsidiary condition.
218
IV. THE CALCULUS OF VARIATIONS
does not hold, we may set Ao = 1, and we have
!!:___
(53)
a(F
+ 'AG)
- a(F
ay'
dx
+ 'AG)
=
ay
o.
Thus we have the result: Except in the exceptional case where
equation (52) holds, the Euler equation of our variational problem is
obtained by forming the Euler eqootion of the variational problem with
integrand F* = F +AG for a suitable value of the parameter A, disregarding the subsidiary condition.
The general integral of the differential equation (53) contains the
parameter A in addition to the two constants of integration. These
three quantities must be determined from the boundary conditions
and the equation K = c.
A simple example is given by the ordinary isoperimetric problem
in which F = VI
y' 2 and G = y. We obtain immediately
+
or
d
y'
V 1 + y' 2 =
dx
'A,
from which we find that the extremals are circles.
Another example concerns the equilibrium position of a homogeneous string suspended at its end points. Here F = yyl + y' 2
and G = v'l
y' 2 ; we obtain (using the Euler equation for Fx = 0,
page 206)
+
(y
-
+ 'A) ( v 1 + y'2 y
+ 'A
v 1y'2+ y'2) = v 1 +"A
+ y'2 = c,
y
= c cosh ( ~
thus the required curve is a catenary.
+ C1) ;
The exceptional case may be il-
lustrated by the subsidiary condition [
yl + y' 2 dx
=
1, with
y(O) = y(I) = 0. Evidently y = 0 is the only admissible comparison
function, and it does in fact satisfy equation (52). Whatever F may
be, the solution cannot be anything but y = 0. 1
1 For a study of the exceptional
case see C. Caratheodory, Uber die diskontinuierlichen LOsungen in der Variationsrechnu ng, GOttingen, 1904, pp. 45 ff.
VARIATIO NAL PROBLEM S WITH SUBSIDIA RY CONDITIO NS
219
2. Finite Subsidiar y Conditions. We shall now consider another type
Jz F(x,
1
of variation al problem: Make the integral J =
y, z, y', z') dx
zo
stationar y in comparison to functions y(x), z(x) which satisfy,
in addition to the boundary conditions y(xo) = Yo , y(x1) = Yi ,
z(xo) = zo, z(x1) = z1, a subsidiar y condition of the form
G(x, y, z)
(54)
0.
=
a given surGeometrically speaking, a space curve y(x), z(x) lying on
1
ent.
requirem
extremum
the
by
ed
determin
be
to
is
face
A natural way to obtain necessary conditions for the functions y(x)
and z(x) is to solve the equation G(x, y, z) = 0 for one of the
functions, say z(x), thus reducing the problem to that of determin ing
one independ ent function y(x). According to elementa ry theorems
of analysis this solution z = g(x, y) can certainly be obtained if
aG /oz ¢ 0 on the extremal in question. We may then regard z' as a
function of x, y, y' and thus eliminate it from F(x, y, z, y', z') by noting the relation Gz + y'Gy + z'Gz = 0 or z' = y'og/oy + og/ox.
Thus we have
F(x, y, z, y', z')
=
F ( x, y, g(x, y), y', :: + y' ::)
and y must satisfy the Euler equation
d (
dx FY'
+ Fz'
ag)
ay
-
[ FY +. F. ag
ay +
2
g)J
,a
( a2g
F•. ax ay + y iJy2
= 0,
which is easily transform ed to the form
I
- Fu )
(Fu'
+ (F••
f
- F. ) -iJg
ay
=
0.
But, since
G,,
ag
+ G. oy = O,
the proportio n
(F~, - F 11 ):(F~, - F.) = Gu:G.
must hold.
Therefor e either G 11
= Gz =
9 identically along the ex-
not all curves on the
1 Note that the coordinate x is singled out, so that
surface G = 0 are necessaril y admiseible ,
220
IV. THE CALCULUS OF VARIATIONS
tremal (which contradicts our hypothesis) or there exists a factor of
proportionality X = X(x) for which
(55)
If we set F* = F + XG the result may be written in the form of the
Euler equations for F*:
-[F*]~ =
F:: - F:
= 0.
These equations are necessary conditions for an extremum unless the
two equations
G.
=0
hold simultaneously on the extremal; in the latter case the third
equation Gz = 0 holds because of the relation Gz + y'Gy + z'G. = 0.
The factors X which occur here and in the preceding example are
known as Euler or Lagrange multipliers; analogous terminology is
used in differential calculus. A formal similarity exists between the
problems discussed in subsections 1 and 2: the expression F* = F + XG
is formed and the Euler equation for F* is set up in both cases.
However, in the first case X is a constant while in the second case it is a
function of x. The Euler equations, together with the subsidiary condition and the boundary conditions, supply the correct number of conditions for determining the extremal.
A special case of the problem just discussed is to determine the
geodesics on a given surface G(x, y, z) = 0. Here F =VI+ y' 2 + z' 2 ,
and we obtain for the geodesic curves, represented parametrically
by x = x(t), y = y(t), z = z(t),
d
dt
x
ii
d
i
d
v x2+ iJ2 + z2 dt v x2 + y2 + z2
=
=
dt
-v--x-2_+_y_2-+~z2
= Gz:Gu:G.
or
d
dt
x
vx2 + iJ2 + z2. d
ii
dt v x2 + iJ2 + z2
-
d
dt
i
"'J...G
z
'AG
11
= 0
,
= 0
'
v x2 + iJ2 + z2 -XG• =O·
VARIATIONAL PROBLEMS WITH SUBSIDIARY CONDITIONS
221
These three equations, together with the fourth equation G = 0, determine the geodesic curves and the multiplier X(x). This representation brings out. the most important geometrical properties of
geodesics; for example, it shows that the osculating plane goes through
the normal to the surface. The proof is left to the reader.
3. Differential Equations as Subsidiary Conditions. Up to now the
multiplier 'A has been used merely as an elegant artifice. But multipliers are indispensable if the subsidiary condition takes the
general form
G(x, y, z, y', z')
(56)
=
0,
where the expression G(x, y, z, y', z') cannot be obtained by differentiating an expression H(x, y, z) with respect to x, i.e. where G is a
nonintegrable differential expression. Such subsidiary conditions are
also called nonholonomi c conditions. A simple example of such a
condition is y' - z = 0. If this condition were holonomic, i.e. equivalent to a finite condition H(x, y, z) = const., the values of x, y, and
z could not be chosen independent ly everywhere, but for every set of
values x, y, z it is evidently possible to choose y' so that it satisfies
the condition y' . . ;. . z = 0. Nonholonomic conditions occur in mechanics when the equations of constraint contain direction as well as
position coordinates, such as in the motion of a ship, a skate, or a
rolling sphere.
The problems with subsidiary conditions already treated may be
regarded as special cases of this general problem. This is obvious
for the problem of subsection 2. But the isoperimetric problem may
also be brought into this category. In this case z and z' do not occur at all in F, while the subsidiary condition takes the form
z' - G(x, y, y') = 0. The boundary conditions are
z(xo) = 0,
y(xo) = Yo,
The ordinary minimum problem with higher derivatives under the
integral sign is also a special case of this problem. For example, the
x1
1
problem of the extremum of
Xl
of the extremum of
J
F(x, y, y', y") dx is equivalent to that
zo
F(x, y, y', z') dx with the subsidiary condition
xo
z - y' = 0.
In all these special cases the necessary conditions can obviously be
expressed as follows: If the solution does not satisfy the Euler equations
222
IV. THE CALCULUS OF VARIATIONS
corresponding to the expression G, then there exists a multiplier A(x) such
that the Euler equations corresponding to the expression F* = F
AG
are satisfied.
This multiplier rule also holds for the general problem formulated
above. We shall omit the proof, referring the reader to the liter-
+
ature.1
In conclusion, it should be emphasized that our method remains
valid if the number of unknown functions and subsidiary conditions
is increased. For functions of several independent variables, our
results are presumably valid, although a general proof has not yet
been given in the case where the subsidiary conditions are partial
differential equations.
§8.
Invariant Character of the Euler Equations
1. The Euler Expression as a Gradient in Function Space. Invariance of the Euler Expression. The stationary character of a function
f (x1 , X2 , • • • , Xn) at a particular point is equivalent to the equation
gradf
=
O,
where grad f denotes the gradient of the function, i.e. the vector in
n-dimensional space whose components are fx 1 , fx 2 , • • • , fxn . This
gradient vector has the following characteristic property: if the n
variables x1 , x 2 , • • • , Xn are all differentiable functions of a parameter
t, then f(x1, x2, · · · , Xn) goes over into a function oft, and we have
(57)
j(t) =
t
Xi !:l:i = v· grad
i-1
f,
where the dot denotes differentiation with respect to t and v is the
xi ; thus the rate of change
of the function is the inner product of the translation vector of the
argument point and the gradient vector.
The Euler differential expression, which vanishes if and only if the
functional is stationary, may be regarded analogously as the gradient
"translation vector" with the components
of a functional in function space.
For example, in the case of a functional
Xl
J[cp) =
1
xo
F(x, cp, cp') dx
1 See D. Hilbert,
Zur Variationsrechnu ng, Math. Ann., Vol. 62, 1906, pp.
351-370. A detailed exposition is also found in the texts by Bolza and Hadamard referred to in the bibliography.
223
INVARIANT CHARACTER OF EULER EQUATIONS
we may suppose that the argument function cp depends on a parameter
t as well as on the independent variable x. Then J[cp] = J(t) is a
function oft, and from the expression for the first variation we obtain
J(t)
=
:r1
1
4'(x)[FJ,, dx
:ro
if, independent oft, cp remains fixed at the boundary of the interval.
Here the dot again denotes differentiation with respect to t. This
formula is completely analogous to formula (57) for the function
f(x1, X2, • • • , Xn). We express this analogy by calling the expression [F],, the gradient of J[cp] in function space.
In general one may define the gradient of a functional J[cp] as an
expression G[cp] ·such that, if cp is replaced by a family of functions depending on a parameter t and the independent variables, the relation
d
dt J[cp] =
1:r1
:ro
4' G[cp] dx
hol.ds.
For example, if K(x, y) = K(y, x), the expression 2
f
K(x, y)cp(y) dy
is the gradient of the functional { { K(x, y)cp(x)cp(y) dx dy.
The gradient of a function has some well-known invariance properties under transformation s of the independent variables. The Euler
differential expression has analogous invariant (or rather covariant)
properties when the functfons in the integrand are transformed to
new independent variables.
In the simplest case let us transform x to the new variable ~(x)
and let
F(x, y, y') =
1
so that {
F dx
1
:r1
ZO
(58)
=
1h
Eo
q
F(x(~), y, ~~~~~) = ~ (~, y, ~~),
<I> ddx
£
d~.
a
[F] 1171 dx = -
aE
Then
1
:r1
zo
F(x, y
1h (
= 1h
a
= !I
uE
~o
Eo
<J>
+ Efl, y' + E71') dx I
1E=0
~' y
a,.,) dx
dt.., + E dt..:; dt., d~
+ E7J, dy
[<I> dx] 71
d~
Y
d~;
I
:
.E=
224
IV. THE CALCULUS OF VARIATIONS
therefore, since 71 is arbitrary (except that it must vanish at the
boundary), we have
[F] 11 =
(59)
d~ [<I>
dx
dx]
d~
11.
In the case of two independent variables we obtain in the same way
F(x, y, u, Ux, u11 )
= F(x(~, 71), y(~, 71), u, u~~x + u,,71x, u~~11 + u,,7111)
=
Ji
Ji
(60)
<I>(~, 71,
u,
F dx dy =
(F]. ! dx dy =
[F]u =
u~,
u,,);
Ji <I> :~;: ~~ d~
Ji [ :1;: !?l ! d~ d~,
d71,
'P
a(~, 71) [<I> a(x, y)] .
c!(x, y)
a(~, 71) u
Analogous transformation properties hold when there are more than
two independent variables.
The invariance property, elegantly expressed by these formulas, is
of great practical advantage. It simplifies computations involved in
transforming Euler differential expressions to new independent variables, since the transformation of the second derivatives need not be
carried out explicitly.
2. Transformation of 4.u. Spherical coordinates. One important
example is the integrand u! + u; + u! . Suppose that the transformation x = x(g1, g2, ga), y = y(g1, g2, ga), z = z(g1, g2, ga)
changes the square of the line element dx 2 + dy 2 + dz 2 into
Li,k Y1·;. dgi dgk , where
the determinant a = I Yik I of these quantities is the square of the
Jacobian determinant of x, y, z with respect to ~t, ~2, ~a. It is
easily seen that
225
INVARIANT CHARACTER OF EULER EQUATIONS
lk are defined by
g'k ~ o~i o~k + oti o~k + o~i o~k
where the quantities
ax ax
ay ay
az az
and satisfy the equations
Okz being the Kronecker delta, okz = 0 for k ~ l, Okk = 1.
We therefore obtain the following general equation for the transformation of au to curvilinear coordinates ~1, ~2, ~a :
(61)
In particular, if g12 = g1a = g2a = 0, i.e. if the new coordinate system
also is orthogonal (the coordinate surfaces ~1 = const., ~2 = const.,
f,a = const. intersect orthogonally), the transformation formula
becomes
(62)
~
au = of,1
(ui 'V. /
Y22 Yaa)
gu
+ ~ (u2
0~2
• / Us3 Yn) +~
g22
0~3
vunu22g33
'V
For example, for spherical coordinates r, fJ,
x
=
r sin fJ cos <p,
y
di = dr
2
+
(u 'V
3 • /Un Y22)
g33 .
<p,
=
r sin fJ sin <p,
r 2 dfJ 2
r 2 sin2 (J
+
z
= r cos fJ,
a;/,
we obtain after a brief calculation
(63) tiu = r' s!n 8 {:, (r'u, sin 8)
+ : 8 (u, sin 8) + :,,, (si';; 8)}.
In the case of only two independent
ing formulas apply. Thus if
2
ds = e df,
2
variables~' 11,
+ 2f df, d11 + g d11
the correspond-
2
we obtain the invariant form )f the differential expression
(64 ) a
1
u = ,1eg -
r
{()
of.
(gut - f u., 2)
v eo -
!
() (eu., - f ue )}
+ ()11
veg -
!2
•
226
IV. THE CALCULUS OF VARIATIONS
In particular, in the case of polar coordinates,
{65)
2
ds = dr
2
+ r dr/,
2
Au =
! {~
or (rur) + ~
Otp
T
(u"')}.
T
3. Ellipsoidal Coordinates.1 The transformation to ellipsoidal coordinates is also important. These coordinates are defined as the
three roots p, u, T of the cubic equation ins
x2
y2
2
2
--+--+--=
s - e1
s - e2
s - ea
(66)
1,
where e1 , e2 , ea are given real numbers. These roots are real for
real x, y, z and, if e1 > e2 > e3, they can be so labelled as to satisfy
the inequalities
p ~
e1 ~ u ~ e2 ~
T
~
e3 .
The surfaces p = const., u = const., T = const. are ellipsoids, hyperboloids of one sheet, and hyperboloids of two sheets, respectively.
Cartesian coordinates are expressed in terms of ellipsoidal coordinates
by
x
2
=
(p - e1)(u - ei)(T - ei)
(e1 - e2)(e1 - ea)
(p - e2)(u - e2)(T - e2)
(e2 - e3)(e2 - e1)
(67)
z2 = (p - ea)(u - e3)(r - ea)
(ea - e1)(ea - e2)
and the line element is
4
(68)
dl =
(p - U) (p - T)
dp 2
(p ·- e1) (p - e2) (p - ea)
+
+
(u - r)(u - p)
du2
(u - e1)(u - e2)(a - e2)
(r - u)(r - p)
(r - e1)(r - e2)(r - e3)
d
2
T •
1 See C. G. J. Jacobi, Vorlesungen Uber Dynamik (given at Konigsberg
1842-3, published by A. Clebsch, Berlin, 1866, reprinted as a supplement to
Jacobi's Collected Works, Berlin, 1884), Lecture 26, where the details of the
calculation may be found. It is to be emphasized that the following considerations may be immediately generalized to more than three dimensions.
227
INVARIANT CHARACTER OF EULER EQUATIONS
This suggests the introduction of the new variables
dX
J vJC>..)'
P
+ e2 + e3 = 0, which can be achieved by the substitution
s = s' + i(e1 + e2 + e3), and if the lower limit of the integrals is
taken to be oo , we obtain simply
If e1
where p is Weierstrass's p-function ;1 furthermore
ds 2 = (p -
<T) (p
- . T) dti
+ (<T -
T) ( <T
~ p) dt~
+(
T -
p) ( T
-
<T)
dti .
For a function T of the coordinates ti we have, in accordance with
(62),
+
1
(r - p)(r -
oT
<T) ati .
2
One advantage of the introduction of the integrals ti is that the Cartesian coordinates are single-valued functions of the ti since the expressions in the numerators of
X=
VP(t1) - e1 VP(t2) - e1 VP(t3) -
e1
Ve1 - C2 Ve1 - C3
(70)
y
VP(t1) - e2 VP(t2) - e2 VP(t3) Ve2 -
z =
C3
v e2 -
e2
e1
vPCt1) - e3 vp(t2) - dvp(t3) - e3
y
e3 -
e1
y
e3 -
e2
See, for example, Hurwitz and Courant, Vorlesungen iiber allgemeine
Funktionentheorie und elliptische Funktionen, pp. 161-171, 3rd ed., Springer,
Berlin, 1929; Interscience, New York, 1944.
1
228
IV. THE CALCULUS OF VARIATIONS
are single-valued functions of ti, t2, t3 om~e the signs of the square
roots are fixed. As the point (x, y, z) ranges over an octant, each of
the quantities p, u, r traverses its appropriate interval, and when one
of these assumes one of its end values, the point (x, y, z) lies on a
boundary plane of the octant as indicated in Figure 1. In this dia-
Figure 1. Confocal surfaces of the second order.
gram, the planes are cut off at their intersections with an ellipsoid
p = Pi > e1 ; the inner boundary curves are portions of the
"focal ellipse"
( x = 0,
2
2
)
_Y_+_z_=l
e1 - e2
ei - e3
and of the
"jocal hyperbola" ( y = 0,
x2
z2
)
--+--=1.
e1
e2 - e3
e2 -
Now if w and w' are, respectively, the real and pure imaginary
periods of the integrals ti , i.e. if
1
00
w =
2
Ill
d>.
vJC>-),
w' =
2
l
ea
-oo
d'A
vJ ("A)'
+
we may let ti vary from 0 to w/2, t2 from w/2 to !(w
w'), and t3
from !(w + w') to !w', obtaining all the points of the octant. If the
interval for each ti is doubled the point ranges over the whole space.
If a single-valued function of the ti is to be single-valued in space, it
must remain unchanged under all substitutions of the ti which leave
x, y, and z unchanged, for example it is single-valued if ti and t2 are
replaced by w - ti and w - t2 , respectively.
If we write t1 = u, t2 = w/2 +iv, t3 = w' /2 + w, p(t1) = f(u), p(t2) =
229
INVARIANT CHARACTER OF EULER EQUATIONS
g(v), p(ta) = h(w), we may take u, v, and w to be real.
ds2
=
[f(u) - g(v)][f(u) - h(w)] du
+ [f(u)
(71)
+
Then
2
- g(v)][g(v) - h(w)] dv 2
[j(u) - h(w)][g(v) - h(w)] dw2
and for real u, v, w all the coefficients are non-negative, since
f(u) ~ e1 ~ g(v) ~ e2 ~ h(w) ~ e8 • The fact that dt2 is pure imiginary in the symmetric form in t1, t2, ta is important; for, the positive definite character of ds2 is assured by the negative value of the
coefficient of dt~.
Among the degenerate forms of ellipsoidal coordinates we may mention (aside from spherical coordinates, which may also be regarded
as a degenerate case) the spheroidal and paraboloidal coordinates. If
two of the ei , say ei and e2 , coincide we obtain
(72)
The two roots s = A1 , s = A2 of this equation, together with the
angle 'P defined by
x = r cos
'(J,
form the new coordinates.
y = r sin '(J,
We have here
(>-1 - ea) (X2 - ea)
e1 - ea
(73)
(74)
r2 d'P2
+
>-1 - >-2
dXi
4(>-1 - e1)(X1 - ea)
= r 2d'P2
+
(>-1 - >.2)(dti - dt~)
ds2 =
+
A2 - A1
dX~
4(>.2 - e1)(A2 - ea)
with
(75)
Hence
(76)
If we now let one end of the ellipsoids tend to infinity we obtain;
230
IV. THE CALCULUS OF VARIATIONS
after passage to the limit,1 the paraboloidal coordinates as the roots
Ai , A2 of equation
x2 + y2
·___ - 2z
s - ei
(77)
+
s - ei = 0,
where r and z are given by the following expressions:
(78)
r 2 = -(Ai - ei)(A2 - ei),
2z = 2ei - Ai - A2.
Here the coordinates of a point in space are Ai, A2, and cp.
element is (7 4)
ds2
+
The line
A2 - ~ dA~
4(A2 - e1) -
= r2 d<p2
+
Ai - A2 dAi
4(Ai - ei)
= r 2 dcp2
+
(Ai - A2) (dt.i - dtD
with
(79)
and the differential expression
2
ll.T = !_2 a T2 +
r a<p .
r(Ai
ll.T takes the form (76)
1
[~(r
- Az) ilti
aT)- ~(r aT)].
ati
atz at2
If in the above expressions the terms containing <p are omitted, one
is immediately led to theformulas for elliptic and parabolic coordinates
in the r, z-plane. In both cases we obtain, from formula (64),
ll. T
in which Ai and
1
= A1 - A2
(a2T
at~
a T)
2
- at~ '
ti are connected by
and
ti= VA: - e1,
respectively.
1
These coordinates may, of course, also be defined without reference to the
foregoing treatment by starting out from the system (77) of confocal paraboloids of revolution.
TRANSFORMATION OF VARIATIONAL PROBLEMS
231
§9. Trans/ormation of Variational Problems to Canonical
and lnvolutory Form
The Lagrange multiplier method leads to several transformations
which are important both theoretically and practically.
By means of these transformations new problems equivalent to a
given problem can be so formulated that stationary conditions occur
simultaneously in equivalent problems. In this way we are led to
transformations of the variational problems which are important because of their symmetric character. Moreover, for a given minimum problem with minimum d, we shall often be able to find an
equivalent maximum problem with the same value d as maximum;
this is a useful tool for the practical problem of bounding d from
above and below. 1
1. Transformation of an Ordinary Minimum Problem with Subsidiary
Conditions. Before discussing these transformations, we briefly con-
sider ordinary minimum problems with a finite number of variables.
Our discussion is based on the following self-evident principle:
If a function f(x1, X2, • • • , Xn), subject to certain subsidiary conditions,
has a stationary value at the point Xi = fi (i = 1, 2, · · · , n) and if
the quantities fi satisfy any relation r(f1, f2, · · · , fn) = 0, then
f remains stationary at Xi = f, provided the additional condition
r(x1 , X2 , • • • , Xn) = 0 is appended to the subsidiary conditions.
We begin by considering the problem
I: f (x, y) is to be made stationary under the subsidiary condition
g(x, y) = 0; the ordinary continuity and diff~entialYility requirements
g! ~ 0 at the stationary
are to be satisfied, and we suppose that g!
point. By the multiplier rule, problem I may be replaced by the
equivalent problem
II: F(x, y; X) = f(x, y) + >..g(x, y) is to be made stationary as a
function of the three arguments x, y, X.
The condition dF = 0 is then equivalent to the three equations
fz
>..g:z: = 0", f 11
Xg 11 = 0, g = 0. Had we started with problem
II, then by adding expl~citly the condition g = 0, which is automatically fulfilled by the solution of problem II, we would immediately have arrived at problem I by our general principle.
+
+
+
1
See E. Trefftz, Ein Gegenstuck zum Ritzschen Verfahren, Verh. d. 2. Int.
Kongr. fUr Technische Mechanik, Zurich, 1927, p. 131, where such an approximation procedure was first given. See also Trefftz, Konvergenz und Fehlerschatzung beim Ritzschen Verfahren, Math. Ann., Vol. 100, 1928, pp. 503-521.
232
IV. THE CALCULUS OF VARIATIONS
But we may also obtain another problem equivalent to problem
II (by equivalence we mean that stationary behavior occurs at the
same point) by appending as subsidiary conditions, not g = 0, but
the other two equations which the solution of II satisfies. We thus
arrive at problem
III: F (x, y; ).. ) = j + >..g is to be made stationary under the subsidiary conditions f:r: + >..gx = 0, jy + >..gy = 0.
If we assume that the latter two equations can be solved, in the
vicinity of the stationary point, for x and y as functions of >.., then
F(x, y; >..) goes over into a function 1/;(>..) of).. and we obtain problem
IV, which once again is equivalent to the other three problems:
IV:!/;()..) is to be made stationary.
We shall now investigate the stationary points; are they or are
they not maxima or minima? Let us suppose in problem I, henceforth denoted by I', that f possesses an actual minimum f(x, fj) = d
at the point x, y. We then consider problem
II': F(x, y; >..) = f + >..g =min. with fixed>... Let us assume that,
if ).. is chosen arbitrarily in a certain neighborhood of the value X
defined by the Lagrange multiplier rule, there exists an actual minimum, which we denote by d>.. = !/;()..), and which is characterized by
the equations f :r: + >..g:r: = 0, f y + >..gy = 0. Then we certainly have
d>..
~
d.
Indeed, problem I' with the minimum d is obtained from problem II'
with minimum d>.. by appending the condition g = 0, which restricts
the domain of comparison values. If we further assume that for
every ).. in the neighborhood of X the equations f :r: + >..g:r: = 0,
f Y + >..g11 = 0 uniquely determine x and y as functions of >.., then
d>. = d, and
d
=
max. (d,.).
Thus dis the maximum of the minimum!/;(>..) of F = f + >..g where
the minimum is to be taken for fixed ).. and then the maximum taken
with respect to>... Under these conditions ·we may also characterize
d by the problem
·
III': F(x, y; >..) = f + )..g = max. = d subject to the subsidiary
conditions f :r: + >..gx = 0, f y + AY11 = 0.
The problem f = (x + 1) 2 + y 2 = min. subject to the condition
g = 2x = 0 may serve as an illustration of our maximum-minimum
TRANSFORM ATION OF VARIATION AL PROBLEMS
233
considerations. Geometrically the problem is to find the lowest point
or vertex of the vertical parabola which is formed by the intersectio n
2
2
of the paraboloid z = (x + 1) + y with the plane x = 0. We obtain immediate ly the value d = 1 for the desired least value of
2
2
X the paraboloid
z = (x + 1) + y • We now note that for fixed
2
2
2
contains the
always
>..
2X
y
+
1)
+
X
+
(x
=
Xg
z ~ f +
above parabola and that the vertex of the paraboloid lies lower than
that of the parabola. By varying X we vary the vertex of the paraboloid which will at most move up to the vertex of the parabola. but
never any higher. Thus the vertex of the parabola is the highest
point of the set of lowest points of our paraboloids.
2. Involutory Transform ation of the Simplest Variational Problems.
Analogous transforma tions of variationa l problems are based upon
the following general principle: If a functional J[u, v, ···]is stationary
for a certain admissible system of functions u, v, · · · , which may be
required to fulfill certain subsidiary conditions, {hen J remains stationary
for this system of functions when the set of subsidiary conditions is enlarged to include any further relations alread,y .satisfied by the functions
u, v, ....
We shall call conditions that are necessary for the vanishing of the
variation (such as the Euler equations and the natural boundary
conditions) natural conditions; subsidiary and boundary conditions
imposed a priori will be known as constraints. Then, from our principle: If a variational problem for a given functional is changed by the
explicit addition of one or more natural conditions to the set of constraints, the stationary character of the functional is not affected.
We shall now turn to some problems of the simplest type:
1
F(x, u, u') dx is to have a stationary value subject to the
I: J =
lx
zo
usual continuity conditions, the boundary conditions
(80)
u(xo) - Uo = 0,
and the subsidiary condition
(81)
du - u'
dx
=
·
0.
That is, we regard the variationa l problem as one involving two unknown functions u and u' subject to the differential equation of constraint (81). In accordanc e with the multiplier rule the solutions of
234
IV. THE CALCULUS OF VARIATIONS
I are simultaneously solutions of the following problem
H[u, u', X; µo, µ.1]
II: =
J.:'[F + :>.('.!:- u') ]dx -
µo[u(x.) - Uo)
+ µ,[u(x,) -
u,J
is to be made stationary, where u(x), u'(x), A(x) and the parameters µo
and µi- are to be determined, and there are no subsidiary or boundary
conditions, i.e. the problem is free. The variational equations, i.e~ the
Euler equatio~s and the natural boundary conditions of the .problem,
are
(82)
Fu, - X = O,
(83)
dA
Fu - -dx = 0,
(84)
du - u'
dx
= 0
in the interior of the interval, and
+
(85)
A(Xo)
+ µo
=
0,
A(X1)
(86)
u(xo)
~ Uo =
0,
u(x1) - u1 = 0
P.l
= 0,
at the end· points, as is found immediately when the first variation is
set equal to zero. If we eliminate A, µo, µ. 1 , we obtain the Euler
equation.
If we apply our general principle to problem II by appending to
the set of constraints the conditions du/ dx - u' = 0, u(xo) - Uo = 0,
u(x1) - u1 = 0, we are led back to problem I. If, on the other
hand, we adjoin equations (82), (83), (85), which correspond to the
natural conditions of problem I, we obtain a transformation which is
of importance in applications; we shall call it the trans!ormation into
1
the reciprocal form of the variational problem. In this way we obtain
problem III, which may be expressed as a problem of type I if we
remove the derivative du/ dx from the integral H by partial integration and then introduce new argument functions p, p' and a new
integrand 'lt(x, p, p') by means of equations
(87)
Fu'= p,
Fu = p',
pu'
+ p'u -
F
= '11.
1 The significance of this transformation was first discovered by K. 0.
Friedrichs, Ein Verfahren der Variationsrechnung, ... , N achr. der Ges. d.
Wiss., Gottingen, 1929, pp. 13-20.
TRANSFORMATION OF VARIATIONAL PROBLEMS
235
In order that this "Legendre transformation" be meaningful we
must require that u and u' can be obtained from the first two equations as functions of p, p', and x; these functions are then to be substituted in the left side of the third equation. This procedure certainly can be carried out, if the requirement
(88)
is· fulfilled for all sets of values x, u, u' of the fundamental domain.
1
We thus obtain the following "reciprocal" problem, equivalent to I,
1
IV:
i'(x, p, p') dx
- {
zo
+
p(x1)u1 - p(xo)uo
is to be made stationary under the subsidiary condition
dp
dx - p'
= O·,
no boundary conditions are imposed.
The natural conditions for problem IV are
in the interior and
Vp
1
lzo -
Uo
= 0,
Vp'
lz 1
-
U1 = 0
on the boundary. It is apparent from their derivation that these
constraints are identical with those of problem I; this may be independently verified with the help of the inversion
i'p' = u,
'11,, =
u',
up'
+
u' p - '11 = F
of the Legendre transformation (87).
By use of the same formulas we see that the reciprocal transformation applied to the free problem IV leads back to the original problem I. Thus this transformation is of involutory character, and the
natural conditions of the one problem go over into the constraints of the
other.
The degener_ate case in which the integrand F of the variational
problem does not depend on u explicitly or contains u only linerarly
requires special treatment. We have here
F(x, u, u') = g(x, u')
1
+
uf(x).
This may be regarded as the analogue of problem IV of subsection 1.
236
IV. THE CALCULUS OF VARIATIONS
In this case the Legendre transformation considered previously is not
reversible for arbitrary p and p'. But by applying our transformation principle directly we find that the following variational problem
is equivalent to the original problem I:
x1
1
-
zo
tf>(p) dx
+ p(x1)u1 -
p(xo)uo = stationary,
with the subsidiary condition dp/dx = f(x). Here p and tf>(p) are
related to the expressions in the original problem by the transformation
-tf>(p) = g(x, u') - u'p,
P = gu',
and it is assumed that the equation gu' = p can be solved for u' in
terms of p. The new problem is essentially simpler than the original
one in the following respect: the required function p(x) can be obtained, aside from an additive parameter, by quadrature from the
subsidiary condition. Thus in this degenerate case the variational
problem goes over into an ordinary extremum problem of determining
a parameter.
As in subsection 1 we now examine the effect of these transformations upon the maximum or minimum character of our expressions.
By repeating the arguments of subsection 1, we shall arrive at the
following result: If the original problem I (hence!orth to be denoted by
I') pertains to a minimum d, then the same value d will occur as a
maximum in the corresponding reciprocal problem IV (to be denoted by
IV').
Again, this statement is true only if certain restrictions are imposed. We require, specifically, that for any arbitrary A(x) with a
piecewise continuous derivative, for which A(x1) + µ1 = 0,
1
A(xo)
µo = 0, the expression H of problem II, page 234, possess a
minimum d>.. depending on A. Then, removing the derivative of u
by partial integration, we obtain problem
+
II':
r
= jzo
1
[
J
F - d>dx u - >..u' dx - >..(xo)u0
+
>..(x 1)u1
1 Without this condition,
which is automatically satisfied by the solution,
it would certainly be impossible to obtain a minimum with arbitrary >.., µ.
TRANSFORMATION OF VARIATIONAL PROBLEMS
237
is to be made a minimum, A(x) being a fixed function. The functions u(x), u'(x) that solve this problem then satisfy equations
(89)
flu, -
A. = 0,
d'A
Fu - dx
=
•
O.
We now assume, in analogy with the procedure of subsection 1,
that these equations determine the functions u and u' uniquely for
arbitrary values of A and dA/ dx.
Since problem I' arises from problem II' when we impose the
additional subsidiary conditions du/ dx - u' = 0, u(xo) - Uo = 0,
u(x1) - u1 = 0, we certainly have d 2::: dx .
On the other hand equations (89) are satisfied by the solution of
problem I' with A = X = Fu' and, because of the assumption of
uniqueness, we have d>.. = d.
It follows that
d =max dx.
But the problem of maximizing dx is just the problem IV', and
thus our assertion is proved.
A sufficient criterion for the validity of our assumptions is that
the inequalities
(90)
Fu'u'
>0
hold for all u and x in the domain under discussion and for arbitrary
u'. As we have already seen (page 216), if these inequalities hold a
solution of the Euler equations furnishes a minimum for problem I'.
The existence of a minimum dx in problem II' follows from these
inequalities similarly; for, equations (89) and inequalities (90) together express the fact that at each value of x the integrand of H
possesses a minimum for the appropriate values of u and u'. Certainly, then, H itself is a minimum.
In conclusion we may point out that the transition from a minimum problem to a maximum problem, as accomplished by the reciprocal transformation, may be exhibited directly under assumption
(90). In the following reasoning, the reciprocal transformation will
be obtained once more. From Taylor's expansion, in view of the
inequalities (90), we obtain at once the inequality
F(u, u') - F(v, v') -
(u - v)Fv -
(u' - v')Fv' 2::: 0,
238
IV. THE CALCULUS OF VARIATIONS
in which the equality holds if and only if u = v, u' = v'.
the above expression in the form
If we write
F(u, u') - .[F(v, v') - vFv - v'Fv'] - uFv - u'Fr:'
and replace v and v' by the quantities p and p' by means of the
Legendre transformation
p' = F",
p = F"''
'lt(x, p, p') = vp'
+ v'p
- F,
the inequality becomes
F(x, u, u')
+ 'lt(x, p, p')
- up' - u'p ~ 0
for arbitrary u, u', p, p'; equality occurs if and only if p and p' correspond to the functions v = u, v' = u'. We now integrate this inequality between the limits xo and X1 , regarding u, u', p, p' as functions of x subject to the constraints
du - u' = 0
dx
'
dp - p'
dx
=
0
u(xo) - Uo = 0
'
u(x1) -
u1
=
0.
The left side can certainly not be negative; it vanishes if and only if
u is the solution of problem I' and p the solution of problem IV'.
Thus the problem
z1
1
[F
zo
+ 'It -
up' - u'p] dx
with the given constraints possesses this solution, and the minimum
value in this problem is zero. This statement is equivalent to the
above assertion concerning the relation of the two problems I' and
IV'.
3. Transformation of Variational Problems to Canonical Form.
The general principle formulated in subsection 2 leads to another
well-known transformation, the transformation into canonical form.
Here the Euler second order differential equation is replaced by a
system of differential equations of first order. This transformation,
which has no exact counterpart in subsection 1, is obtained by imposing equations (82) and (86) as constraints in problem II. Thus
one first obtains the problem
Ila:
(
[F(x, u, u') + F•. ( : - u')] dx
TRANSFORMATION OF VARIATIONAL PROBLEMS
239
is to be made stationary with the boundary conditions u(xo) = Uo ,
u(x1) = u1 , where u and u' are to be regarded as two independent
argument functions.
If we introduce a new argument function 1
p =Fu'
in place of u' and a new integrand
4>(x, u, p) = pu' - F(x, u, u')
in place of F(x, u, u')-we expressly assume
Fu'u' "F- 0,
so that u' can be determined as a function of p, u, and x from the
relation p = Fu' -we obtain the equivalent problem
Ilb:
J.:' [p: -
il>(x, u, p)
Jdx = stationary
with the boundary conditions u(xo) = Uo, u(x1) = u 1 • It is easily
seen that the quantities occurring in the equivalent problems I and
Ilb ·are connected by the Legendre transformation
Fu'= p,
pu' - F = 4>,
whose inverse is expressed by the equations
tl>p =
u',
pu' - 4>
= F.
This form of the variational problem is known as the canonical
farm. The formulation of the variational equations for p and u
yields the canonical dijferential equations of the variational problem:
dp
dx
+
tPu = 0,
du
dx -
ti>p = 0.
In an analogous manner we may transform a variational problem
· · · , un(x) of the independent
variable x into canonical form.
Summarizing the above discussion, we state without explicitly repeating qualifying sufficient conditions: Suppose, in problem I, d is a
minimum; then, in the canonical problem, d becomes a maximumminimum if for fixed p we find a minimum by varying u and then
fol n unknown functions u1(x), u2(x),
1
Thus pis equal to the multiplier occurring in II.
240
IV. THE CALCULUS OF VARIATIONS
determine the maximum of this minimum (a function of p) by
varying p.
4. Generalizations. Our transformation theory is easily extended
to problems involving several unknown functions, higher derivatives,
or several independent variables. We confine ourselves here to the
treatment of a particularly simple example which corresponds to the
degenerate case of subsection 2, namely the transformation of Dirichlet's classical variational problem
~ fJ. [(!~)' + (:~)'] dx dy =
I:
min.
in which u is a function of x, y with piecewise continuous derivatives
in the region G and prescribed boundary values u = f (s). The
boundary r of G is assumed to be a curve with continuously turning
tangent (except possibly at a finite number of points) and with arc
lengths.
If in problem I we replace the two partial derivatives by the functions p and q and append the subsidiary conditions au/ax = p,
au/ay = q, the multiplier rule immediately leads to the equivalent
problem
II:
ff. [w +
>-(!> P) + "(!~ - q) Jax ay
+
q')
-£
p(s)[u - f(s)] ds = stat.
Here ~(x, y), µ(x, y), p(s) are multipliers. Transforming the double
integral by partial integration we obtain II in the form
fJ. [HP'+
q') -
+
u(:: +~)-hp
£[
U(
+ ii :
X:
- µq] dxdy
- p(s))
+ p(s)f(s)
J
ds = stat.
Here a/ an denotes diffei:entiation with respect to the outer normal,
and boundary values are indicated by a bar (u, X, µ). We now impose as explicit subsidiary conditions the Euler differential equations
and certain natural boundary conditions:
p -
X = 0,
ax+ aµ=
ax
ay
0
'
q -
- ax
ay
an
an
µ = 0,
X-+µ--p=O
'
241
TRANSFORMATION OF VARIATIONAL PROBLEMS
thus obtaining the equivalent problem
-!
III:
Jl
(p
2
+q
2
)
dx dy
+
£
p(s)f(s) ds
= stat.
with the subsidiary conditions
_ay
( ) - p_ax
an - q an = 0
p 8
on the boundary and
ap
ax
+ aq = 0
ay
in the interior. The reciprocal transformation is completed.
We can simplify the result if we satisfy the latter subsidiary conditions by introducing a function v(x, y) such that
av
P=
ay'
q =
av
-ax·
Then
_ax+_ ay _av
q an - as'
pan
where the differentiation on the right is to be performed in the direction of the positive tangent of r; the problem goes over into problem
IV:
-
i
~ JJ. [(::)' + (:;)'] dx dy + ~ f(s} ds =
stat.
The boundary integral on the right, incidentally, may be transformed
by partial integration into the form
-£
vf'(s) ds.
In this new problem, the integrand is of the same form as in problem
I. The solution of problem I defines a function satisfying the potential equation AU = a2u/ax2 + a2u/ay2 = 0 with the boundary values
f (s) ; problem IV also defines a potential function v which, in virtue
of the natural boundary conditions, turns out to be the potential
function conjugate to u.
A minimum in problem I corresponds to a maximum having the
242
IV. THE CALCULUS OF VARIATIONS
same value in problem IV. This becomes clear if we subtract the expression IV from I; a simple transformation leads to the expression
!!. [(:= - :~)' + (:~ +:~)']
dx dy.
Problems I and IV together are equivalent to the problem of minimizing this last integral subject to the single boundary condition
u = f(s). The minimum has the value zero and is attained when u
is the solution of the appropriate boundary value problem of potential theory and v is its conjugate function satisfying the differential
equations
au
ay
-
av
ax·
For a more direct and general treatment of reciprocal quadratic
variational problems see §11.
§IO. Variational Calculus and the Dijferential Equations of
Mathematica l Physics
1. General Remarks. The calculus of variations is a reliable guide
both in formulating and in treating differential equations of mathematical physics. Problems of (stable) equilibrium are governed by
the variational principle of the minimum potential energy; the laws
of motion are most simply formulated in terms of Hamilton's variational principle. By these two principles we shall derive some of the
fundamental differential equations of mathematical physics.
Let us first conside~ a system with a finite number, say n, of degrees of freedom. Let the position of the system be characterized
by the values of n parameters q1 , q2 , · · · , qn ; the object is to determine these as functions of the time t. We suppose that the mechanical properties of the system are determined by two quantities, the
kinetic and potential energy. We assume that the kinetic energy
T(q1, (]2, · · · , iJn, q1, q2, · · · , qn, t) is a known function of the n
velocities qi 1 then coordinates qi, and the time t, specifically that
it is a quadratic form in the velocities:
n
T =
L
i.k-l
Pi1c(q1 , q2, · · • , qn, t)qi qk •
The potential energy U(q1, q2, · · · , qn, t) is assumed to be a known
iunction of t and the coordinates qi. Then Hamilton's principle
VARIATIO NAL CALCULU S AND MATHEMA TICAL PHYSICS
243
states: Between two instants of time to and t1 the motion proceeds in
s-uch a way that for the functions qi(t) the integral
J =
!
ti
(T - U) dt
to
is stationary in comparison with the neighboring functions qi(t) for
which Mto) = qi(lo) and f/i(t1) = qi(t1). In other words: The actual
motion makes the value of the integral J stationary with respect to all
neighboring virtual motions which lead from the initial to the final position of the system in the same interval of time.
In accordance with §3 Hamilton 's principle leads at once to Lagrange's general equations of motion
~ ~'!
(91)
dt iJqi
-
j__ (T - U)
iJqi
=
0
(i = 1, 2, · · · , n).
The equilibrium conditions can be obtained from the equations of
motion, under the assumpti on that T and U do not depend explicitly
on t, "by setting the derivatives with respect to time equal to zero in
(91). The resulting conditions are
au= o.
(92)
aq..
Thus: A mechanical system with the potential energy
U(q1, q2, · · · , qn) is in equilibri am at a particula r set of values
of the coordinates q1, q2 , • • • , qn if and only if the potential energy
is stationar y for this set of values.
In order that the equilibrium be stable, it is, moreover, necessary
and sufficient that the stationar y value of U in the equilibrium position be a minimum.
This fact follows easily from the principle of conservation of
U = const., an immedia te consequence of equation s
energy, T
(91). We shall henceforth accept the principle of minimum potential
energy as the condition of stable equilibrium.
A motion takes on a particula rly simple characte r if confined to
the vicinity of a stable equilibrium position of a system. We may
assume without loss of generality that at the equilibrium in question
all the coordinates qi vanish. If we now consider those motions
close to the equilibrium position for which higher powers of the
coordinates and of their time derivatives are negligible compared
with lower powers, and if we assume that T and U do not contain
+
244
IV. THE CALCULUS OF VARIATIONS
.t explicitly, then we may regard T as a positive definite quadratic
form in
q, with constant coefficients aik
:
n
T =
L
a,kqiqk •
i,k-1
Under these hypotheses, U also is a positive definite quadratic form
in qi with the constant coefficients bik :
n
u = 2:
i,k=l
bikqiqk •
Thus the equations of motion go over into the linear second order
differential equations with constant coefficients
n
n
2: aikiik + k-1
2: bikqk
k-1
=
o
which govern "small vibrations" about a stable equilibrium positiow
and which will be discussed in detail in the next chapter.
In the case of problems of continuum mechanics, in which the
position of the system no longer can be described by a finite number
of variables, we may also proceed from Hamilton's principle or from
the principle of least potential energy. Here the kinetic and potential energies are not functions of a finite number of variables, but
are functionals represented by integrals over appropriate space
regions, surfaces, or lines.
2. The Vibrating String and the Vibrating Rod. The simplest example of continuum mechanics is given by the homogeneous vibrating
string, subjected to a constant tensionµ, which executes small transverse vibrations about the position of stable equilibrium, the interval
0 S x S l of the x-axis. Our problem is to determine the perpendicular deflection u(x, t) of a point on the string from its equilibrium
position. We assume that the motion is "small" in the sense, specifically, that higher powers of the function u and of its derivatives
may be neglected compared with lower powers. First, let us consider the case in which the string is fixed at its end points, i.e.
u(O, t) = u(l, t) = 0. The kinetic energy of the string is given by
the integral T ;;.
! [ z pu~ dx, where
p
denotes the linear density of
the string. The potential energy U is proportional- to the increase
in length compared with the length at rest, the factor of propor-
VARIATION AL CALCULUS AND MATHEMAT ICAL PHYSICS
Since, apart from quantities
tionality being equal to the tensionµ.
lz ~ dx -
of higher order, the change in length
1u!
245
l approxi-
z
mately equals !
dx, the potential energy takes the form
U
=!
1zµu!
dx.
Hamilton' s principle, therefore, l~ads to the problem of finding an
admissible function u(x, t) for which the double integral
t1 (T -
1
U) dt =
rt1il (pu~ -
! },
to
to
o
µu!) dx dt
is stationary compared with all the continuous functions u(x, t) which
have piecewise continuous first derivatives, vanish at x = 0 and at
x = l, and coincide with the functions u(x, to) and u(x, t1) corresponding to the actual motion at t = to and t = ti, respectively. For
constant p andµ, we then obtain from the general laws of the variational calculus the partial differential equation of the vibrating string
(93)
·putt -
P,Uxx
= 0.
If an external force f (x, t) acts on the string the term
fo 1 f (x, t)u dx
must be added to the potential energy, and we now obtain the differential equation
(94)
pUtt -
µUxx
+ J(x, t)
=
0.
The stable equilibrium of a string under the influence of an external force is given, in accordance with the minimum principle, by
the minimum of the integral fo 1 (!µu! +ju) dx, where the external
force f (x) is, of course, assumed to be time-indep endent.
to the Euler equation
P,Uxx -
This leads
f(x) = 0
as a special case of the equation of motion (94).
We now formulate the corresponding equations for the states of a
laterally movable rod. The rod is defined as a one-dimensional continuum, lying in a straight line when at rest, whose potential energy
246
IV. THE CALCULUS OF VARIATIONS
at deformation is proportional to the integral of the square of the
curvature, extended over its length. If we again assume that higher
powers of the deformation function u(x, t) and of its derivatives are
negligible compared with lower powers, we obtain an expression of
the form !µ
Li u!x dx for the potential energy of deformation.
The
kinetic energy has the same form as that of the string. With an
external force f (x, t) acting, the equation of motion, derived from
Hamilton's principl~, becomes
PUtt
+ µUxxxx + J(x, t)
=
0,
and the condition for equilibrium under an external force f(x) is
µUxxxx
+ J(x)
= 0.
The boundary conditions or other constraints to be imposed are of
great importance in the solution of our variational problems. We
may, for example, fix the boundary by conditions such as
u(O) = u(l) = 0 for the string or u(O) = ux(O) = u(l) = Ux(l) = 0
for the rod, or we may instead leave the boundaries free. For free
boundaries, the methods of §5 lead to the natural boundary conditions
(95)
Ux(O, t) = Ux(l, t) = 0
for the string and
(96)
Uxx(O, t) = Uxx(l, t) = 0,
Uxxx(O, t) = Uxxx(l, l) = 0
for the rod.
If the end points of the string are neither fixed nor free but are
held by elastic forces, b·oundary terms !h1µu 2(0, t) and !h'?µu 2(l, t)
must be added to the potential energy. These terms do not alter
the equation of motion (94), but lead to the natural boundary conditions1
ux(O, t) = h1u(O, t),
· Ux(l, t) = -h2u(l, t).
3. Membrane and Plate. For the plane membrane and plate the
situation is similar to that for the string and rod. A membrane
is a portion of a surface, plane at rest, with potential energy
proportional to change in area; the proportional ity factor is known
1
For further remarks
con~rning
rods, see §12, 12-13.
VARIATION AL CALCULUS AND MATHEMAT ICAL PHYSICS
247
as the tension. Suppose that the membrane at rest covers a region
G of the x, y-plane; let the deformatio n normal to the equilibrium
plane be denoted by u(x, y), and suppose this deformation to be small in the sense that higher powers of u, Ux , Uy
are negligible compared with lower ones. Then the expression
Jl
Jl
(1
[1
+ u! + u;)i
+ !(u! + u;)]
dx dy for the area may be replaced by
dx dy, and the required potential energy,
apart from a constant factor, is given by the double integral
(97)
!
Ji (u! + u!)
dx dy.
We consider first the equilibrium problem for the membrane. If we
suppose that the displacement u(x, y) of the membrane possesses
prescribed values u = u(s) on the boundary r of G-where s denotes
the arc length of r -and that no external forces act on the membrane, then the equilibrium position is characterized by the following
variationa l problem: The displacem ent u(x, y) in the equilibrium
position is that function for whfoh the integral
Jl (
u!
+ u!)
dx dy
attains the least possible value if all those functions u are admitted
for competitio n which are continuous in the closed domain G, take
continuous first
on the prescribed boundary values u(s), and possess
1
interior. The
the
in
s
and piecewise continuou s second derivative
Euler equation is
AU
=
Uxx
+
Uyy
= 0.
Thus the problem of finding the equilibrium position is equivalent to
the boundary value p,,:oblem: find in Ga solution of the above partial
differential equation (the potential equation) which assumes prescribed
values on the boundary r of G.
We now make the more general assumptio ns that the interior of
the membrane is subject to an external force of surface density
f(x, y), the boundary of the membrane (supposed to be freely
movable above r) to an external force oflinear density p(s), and, finally,
1 Later, for the complete treatment of the problem, it will be important that
the continuity requiremen ts for the second derivatives may be relaxed without
altering the solution.
248
IV. THE CALCULUS OF VARIATIONS
that the boundary is tied to its rest position by elastic forces which
are characterize d by a modulus of elasticity of linear density u(s).
Then the potential energy of a membrane with the displacemen t
u(x, y) is given by the expression
Ji
[!µ(u!
+ u!) + Ju]
dx dy
+
£
[p(s)u
+ !u(s)u
2
]
ds.
Once more we obtain the equilibrium position by finding a function
u(x, y), subject to no boundary conditions but only to the above
continuity conditions, for which this integral is a minimum. The
Euler equation (which expresses the equilibrium condition in the
interior) is, if we take µ = 1,
Au
= f(x,
y),
and the natural boundary condition is
an + uu + p( s) =
au
O.
These two requirement s again represent a boundary value problem
for a partial differential equation.
From this general case, we again arrive at the problem of solving
the differential equation Au = f with the boundary values u = 0, if
we set p equal to zero and let u increase beyond all bounds.
If <T = 0 our equilibrium problem does not in general possess a
solution. Physically, this is plausible; for, a freely movable membrane, subjected to arbitrary external forces, cannot have a stable
equilibrium position unless the external forces happen to be balanced.
This fact is easily proved from the variational principle: In order
that our expression for the energy possess a lower bound in the case
u = 0, equation
(98)
Ji .f
dx dy
+
£
p ds = O
must be satisfied. Indeed, for a constant displacemen t u(x, y) = c,
the energy is c times the left side of (98) and can therefore have
arbitrarily large negative values, unless the left side vanishes. But
if we impose condition (98), the solution of the variational or equilibrium problem is not determined uniquely, since an arbitrary constant can be added to u without altering the value of the energy; its
VARIATIONAL CALCULUS AND MATHEMATICAL PHYSICS
249
minimum is therefore also unchanged. We must impose a further
condition in order to fix the solution uniquely; It is customary to
use the condition
flu
dx dy = 0,
which means that the center of mass of the membrane is fixed in the
rest position.
We arrive at the equations of motion of a membrane by Hamilton's
principle, noting that the kinetic energy is given by
(99)
T =
!P
fl u~
dx dy.
If external forces act with surface density f(x, y, t) in the interior of
the membrane and line density p(s, t) on the boundary, and if the
boundary is again subjected to elastic forces with the elastic modulus
u(s), Hamilton's principle requires that the expression
1
{
fl [! pu~
-
!
µ( u!
+ ~) -
f (x,
y, t)u] dx dy dt
- Jt £(! uu
1
to
be made stationary.
2
r
+
pu) ds dt
The Euler equation for this problem is
µ.6.u - pu,, - f(x, y, t) = O;
the natural boundary conditions are
(100)
au
-an + CTU + p(s, t) =
0.
If the membrane has a fixed boundary, i.e. if the values of u on the
boundary are given as a function of the arc length, then the prescription of these boundary values takes the place of condition (100).
In the equilibrium case the minimum principle not only gives the
pertinent formulation but also proves a powerful tool for solving and
analyzing the problem. However, in dynamic problems the chief
use of Hamilton's principle is to derive formally the differential
equation; for the further analysis of these equations Hamilton's
principle is often not suitable since it requires that the functions ad-
250
IV. THE CALCULUS OF VARIATIONS
mitted for comparison be prescribed at two fixed instants of time to
and t1 , conditions not generally encountered in actual problems.
The information usually given in dynamical problems comprises
boundary conditions and, in addition, initial conditions, i.e. the
values of the functions u(x, y, t) and Ut(x, y, t) at one instant t = 0.
Thus dynamic problems lead to mixed boundary and initial value
problems.
The situ~tion is analogous in the case of the plate. A plate is an
elastic two-dimensional body, plane when in equilibrium, whose
potential energy under deformation is given by an integral of a quaddratic form in the principal curvatures of the plate. If the principal
radii of curvature of the deformed plate are denoted by p1, P2 the
pot~ntial energy density is given by an expression of the form
where A and B are constants determined by the material of the plate.
Assuming u, U:i:, u 11 , ···to be small, we may write
-Pi2 + P2-2 =
dU
.
=
U:i::i:
+U
1
1111 ,
= U:i::i;Uy11
PlP2
.
-
2
U:i:11
•
Thus the desired potential energy of deformation is given by an expression of the form
·
(101)
U1
=
fl
[(Llu)
2
-
2(1 - µ)(u:i::cU1111
-
u!11 )] dx dy,
apart from a constant factor (depending on the material) which we
may set equal to unity.
To this must be added th~ energy due to external surface forces,
boundary forces, and bending moments on the boundary if they are
prescribed:
U2 =
fl
ju dx dy
+
l
p(s)u ds
+
l
m(s)
:~ ds;
heref(x, y), p(s), m(s) represent, respectively, the densities of applied
surface forces, boundary forces, and bending moments normal to the
boundary curve.
Equilibrium is once more characterized by the requirement that
VARIATIONAL CALCULUS AND MATHEMATIC AL PHYSICS
251
U1 + U2 be a minimum for a suitable admissible function u(x, y).
(Here the admissible functions have continuous derivatives up to the
fourth order, but these conditions may actually be greatly relaxed
without affecting the solution of the problem.) To find the Euler
equations and the natural boundary conditions for our minimum
problem we must form the variation oU = 0U1 + 0U2, as outlined
m §5, and equate it to zero. We obtain
0U1 =
Ji
(.1.1u ou) dx dy -
£
_M 0 :~ ds -
i
p OU ds,
with
+ 2UxyXs Ys + Uy y!),
~ (ux:i:XnX• + UxiXnYs + XsYn) + UyyYnYs)·
dS
M(u) = -.1u
+ (1
- µ)(Ux:i:X!
P(u) = _!___ .1u
+ (1
- µ)
an
11
Here Xn, Yn and Xs, Ys are the direction cosines of the outward normal
and the tangent vector, respectively. From the condition oU = 0
we obtain as the equilibrium conditions, the Euler differential equation
.1.1u
+ f = 0,
and, since no conditions have been imposed a priori on the boundary,
1
the natural boundary conditions
P(u) - p = 0,
M(u) - m = 0.
If the plate is clamped at the edge, i.e. if u and au/ iln have the
prescribed value zero at the boundary, these natural boundary conditions are replaced by the conditions u = 0, au/an = 0. If the
boundary of the plate is supported, i.e. if the boundary is fixed at
zero while the tangent plane at the boundary is not constrained, the
boundary conditions are
(102)
u
= 0,
To formulate the differential equation for the motion of the plate
(the differential equation of the vibrating plate) we again utilize
1 It is worth noting that in the variational problem for the plate the expression uuu1111 - ui11 , because it is a divergence expression, does not affect the
Euler differential equation; yet it is decisive for the form of the natural boundary conditions.
252
IV. THE CALCULUS OF VARIATIONS
Hamilton's principle, employing the expression (99) for the kinetic
energy. We obtain the equations
µ!l!lu
+
pUtt =
0
or' more generally'
µ!l!lu
+ pUtt + f(x, y, t)
=
0
with appropriate boundary conditions just as above in the equilibrium
problems. For the characterization of an actual physical problem
these boundary conditions have to be supplemented by initial conditions which describe the initial state, i.e. the functions u(x, y, 0)
and Ut(X, y, 0).
§11. Reciprocal Quadratic Variational Problems 1
Linear functional equations of mathematical physics result from
quadratic variational problems. It is illuminating, especially in
regard to the reciprocity phenomenon discussed in §9, to consider
such quadratic problems from a somewhat general and abstract viewpoint (compare Ch. I, §5, 3) and at the same time to use suggestive
geometrical language. 2
We consider, in a linear vector space A of vectors p, q, · · · , a quadratic form Q(p, p) which we may assume to be positive definite.
The vectors may be functions or systems of functions; for example
1)
Q(p, p)
1
=
fl
p2 dx dy
This section has been added for the present edition.
Recently several authors have rediscovered and advanced the theories indicated in §9. See J. L. Synge, The method of the hypercircle in functionspace for boundary value problems, Proc. Roy. Soc. London, Ser. A, Vol. 191,
1941, pp. 447-467; W. Prager and J. L. Synge, Approximations in elasticity
based on the concept of function space, Q. App. Math., Vol. 5, 1947, pp. 241269; J. L. Synge, The method of the hypercircle in elasticity whep. body forces
are present, Q. App. Math., Vol. 6, 1948, pp. 15-19 and the literature quoted
there. Synge's papers have called attention to the advantage of a geometrical
interpretation from which the distance of the approximations to the exact solutions can be deduced. See also J. B. Diaz and H. J. Greenberg, Upper and
lower bounds for the solution of the first boundary value problem of elasticity,
Q. App. Math., Vol. 6, 1948, pp. 326-331; H.J. Greenberg, The determination
of upper and lower bounds for the solution of the Dirichlet problem, J. Math.
Phys., Vol. 27, 1948, pp. 161-182; J. B. Diaz and H. J. Greenberg, Upper and
lower bounds for the solution of the first biharmonic boundary value problem,
J. Math. Phys., Vol. 27, 1948, pp. 193-201; J.B. Diaz, Upper and lower bounds
for quadratic functionals, Proc. of the Symposium on Spectral Theory and
Differential Problems, Stillwater, 1951.
2
RECIPROCA L QUADRATIC VARIATION AL PROBLEMS
where p
grad rp(x, y) in a region G, or
Q(p, p) =
2)
where p
Ji
p2 dx dy
!:irp(x, y), or a form such as
1l
1
Q(p, p) =
3)
253
1
K(x, y)p(x)p(y) dx dy.
0
The definition of the corresponding polar forms Q(p, q) is obvious.
We interpret Q(p, p) as the square of the length of p, and call
vectors p, q orthogonal if
Q(p, q) = 0.
For orthogonal vectors p, q we have
Q(p
+ q, p + q)
=
Q(p - q, p - q) = Q(p, p)
+ Q(q, q)
("Pythago rean Theorem" ).
If n is a linear subspace of the vector space A, we can define another
linear subspace ~ orthogonal ton as the space of all vectors d orthogonal to all vectors <.> of n :
Q((J), d) = 0.
In all instances considered here the following facts are true: if ~
is the linear subspace orthogonal to n, then n is the linear subspace
orthogonal to~- Every element pin A can be uniquely decomposed
into the sum of "projectio ns" (J) ·and d:
((J) inn, din
~).
If we restrict our vectors by imposing suitable continuity and
differentiability conditions, then the theorems which guarantee the
existence of a solution for our variationa l problems also insure the
validity of our statements . Accordingly, although we then operate
in an incomplete Hilbert space, we assume that the projection theorem
holds for the. respective incomplete spaces.
We now consider two variationa l problems.
I: Given two vectors Po and qo in A, minimize
Q(p, p) - 2Q(p, qo)
po is in a preadmitting for competitio n vectors p for which p
two admissible
of
difference
the
that
Note
n.
subspace
scribed linear
254
IV. THE CALCULUS OF VARIATIONS
vectors belongs to n, whereas p itself is restricted to a linear set no
of vectors "parallel" to the linear subspace n.
As a first example, let the vector space A consist of vectors p
which are pairs of functions defined in a region G; the norm is given
by
Q(p, p) =
Ji
2
p dx dy
Take Po = grad 'Po where 'Po is a given function, and take as n the subspace of vectors w = grad 'P with 'P = 0 on the boundary r of G.
The subspace ~' the orthogonal complement of n, consists of those
vectors d for which div d = 0. If we choose q0 = 0, then for this
example problem I is simply Dirichlet's problem: minimize
Ji
('P!
+ 'P;)
dx dy subject to the condition 'P - 'Po = 0 on
r.
Next, let the vector space A consist of functions p defined in a region
G; the norm is given by
Q(p, p) =
Ji
2
p dx dy.
Take Po = A<po where <po is a given function, and take as n the subspace of functions w = A<p with 'P and its normal derivative vanishing
on r. The subspace~ then consists of those functions u for which
Au = 0. If we choose qo = 0, then we have for 'P the variational
problem associated with the clamped plate.
Suppose that u minimizes the functional of problem I. Since p =
u
a.> (for any number E and any element win Q) is also admissible,
the vanishing of the first variation amounts to the condition
+
Q(u - qo , w) = 0
for all w in
n.
Hence the equivalent of Euler's equation and the accompanying
natural boundary conditions for problem I is that u - qo is an element of }; , the orthogonal complement of n.
The uniqueness of the minimizing vector u is now apparent. For,
if u' also furnishes a minimum, then, as. we have just seen, u' - qo
is in }; ; consequently (u - qo) - (u' - q0 ) = u - u' is a vector
of };. But, in view of the admissibility conditions, u and u' are
in no so that their difference u - u' lies in n. Since the subspaces n
and·.~ contain only the vector zero in their intersection, we see
that u' = u.
RECIPROC AL QUADRAT IC VARIATIO NAL PROBLEM S
255
The reciprocal problem to problem I will now be formulat ed.
II: Minimize the expression
Q(q, q) - 2Q(q, Po)
admittin g for competit ion all vectors q for which q - q0 is in ~.
If the minimum is attained for q = v, then by the same reasoning as
above we can conclude that v - Po is in n.
Since v - qo is in~ and u - qo is in ~' the difference u - v
is in ~; in the same way, from problem I we see that v - u is inn.
But since the two orthogon al subspace s have only the vector zero
in common , it follows that u = v.
Thus problems I and II have the same solutions u = v.
By adding to the variation al expressions in problems I and II the
constant terms Q(qo , qo) and Q(po , po), respectively, we can state
the problems in the following form:
I: Find the shortest distance from a fixed vector qo to the linear set
no ' i.e. minimize
d(p) = Q(p - qo , P - qo)
over all p in no .
II: Find the shortest distance from a fixed vector Po to the linear
set ~o , i.e. minimize
d(q)
=
Q(q - Po, q - Po)
over all q in ~o .
Both minima d1 and d2 are attained by the same solution p =
q
= u.
The reciprocal character of the two problems is expressed by the
fact that the admissib ility condition s of the one are the Euler conditions of the other.
Geometri cally, the functions p and q are represen ted in Figure 2.
A glance at this figure and the theorems of Pythagor as or Thales
suggest, moreover, the relations
and
Q(u - p, u - p)
+ Q(u -
q, u - q)
256
IV. THE CALCULUS OF VARIATIONS
where p, q are any two vectors admissible respectively in the two
reciprocal problems. These relations follow analytically from the
fact that u - p is in n, u - q in ~; hence u - p and u - q are orthogonal and
4Q(u-p~q' u-p~q)
+u-
q, u - p
= Q(u -
p, u - p)
+ Q(u -
= Q(p -
q, p - q).
=
Q( u - p
+u-
q)
q, u - q)
0
Figure 2
These considerations imply a remarkable fact: an upper bound for
one of the two variational problems furnishes at the same time a
lower bound for the reciprocal problem, since the sum of the minima
is a priori known. This remark will prove relevant in connection
with direct methods of the variational calculus.
Furthermore, the second relation, 1 corresponding to the theorem of
Thales, states: The solution u common to both problems deviates
from the arithmetical mean of two arbitrary functions p, q admissible,
respectively, in the two reciprocal problems by an amount equal to
half the distance between the two functions p and q. Hence, once
1 In the present connection
this relation was pointed out by Synge and
Prager.
257
SUPPLEMENTARY REMARKS AND EXERCISES
p and q are chosen, the sphere of radius I (p - q)/2 I about the point
(p + q) /2 is a geometric locus for u.
Special cases· (see also §12, 12) are easily fitted into the general
scheme. For example, as we have seen, Dirichlet's problem for the
harmonic differential equation in a region G with boundary r (see
Ji
§8) corresponds to Q(p, p) =
2
p dx dy.
The space
no
of p is
defined by p = grad <P(X, y), <P - <Po = 0 on r, Po = grad <Po(x, y),
where <Po is a prescribed function. The space 2:0 = 2: of q is defined
by div q = 0, qo = 0. Incidentally, it should be noted that in problem II the expression Q(q, po) can be transformed into the boundary
integral
£
<Poqn ds
where
qn
is the normal component of q.
Thus the reciprocal problem II can be formulated if we merely
know the prescribed boundary values of <P; no explicit knowledge of a
function <Po(x, y) is necessary.
A similar fact is true of other examples, e.g. the problem of the
clamped plate, where
Ji
Q(p, p) =
2
p dx dy,
ll.q = 0,
Po = ll.<Po,
qo = 0,
and where <P - <Po and its normal derivative are supposed to vanish
on r. Since
)= 1(
Q( q, Po
r
aq - -a<Po q ) ds
<Po --
an
an
'
where a/ an denotes the normal derivative, problem II actually refers
only to the given boundary values of <P and a'P/an.
§12. Supplementary Remarks and Exercises
1. Variational Problem for a Given Differential Equation. For a
given ordinary second order differential equation y" = f(x, y, y')
one can always find a function F(x, y, y') such that the equation
[F]u = 0, when solved for y", is identical with the differential equation.1
1 Cf. 0. Bolza, Vorlesungen uber Variationsrechnung, Teubner, Leipzig
and Berlin, 1909, pp. 37-39.
258
IV. THE CALCULUS. OF VARIATIONS
2. Reciprocity for Isoperimetric Problems.
The extremals of the
problem
Jxxo
J =
1
F(x, y, y') dx
= stationary
under the condition
K
=
Jxxo
1
G(x, y, y') dx
= constant
are identical with those of the problem K = stat., J = const., except
for the singular case of §7, I.
3. Circular Light Rays. The following statement is a consequence
of this chapter. Suppose light travels in the x, y-plane in such a
way that its speed is proportional toy; then the light rays emitted
from any point are circles with their centers on the x-axis.
4. The Problem of Dido. "Dido's problem", to enclose the largest
possible area of land within a fence of given length, may be generalized by introducing a weight function p(x, y) where p(x, y) denotes,
say, the fertility of the land. The problem is to find a closed curve of
given length such that the integral
Ji
p dx dy, taken over the en-
closed region, assumes its largest possible value. Formulate the
differential equation of the extremals.
5. Examples of Problems in Space. The surface with smallest area
which encloses a given volume is the sphere. 1
Consider the surface of least area bounded by a given curve which,
together with a given surface bounded by the same curve, encloses a
prescribed volume; then the extremals are the surfaces of constant
mean curvature. If the subsidiary condition concerning the volume
is omitted, we arrive at the differential equation of minimal surfaces
(cf. §3, 4), which states that the mean curvature is zero.
6. The Indicatrix and Applications.2 In finding the extremum of
the integral
f
ti
ty(x,
y, x,
y) dt
to
of first order in
±,
mis assumed positive homogeneous
y) we consider the curve
ty(x, y, ~' 11) = 1
in
the~'
11-plane for fixed x, y.
This curve is known as the indicatrix;
Cf. references to the literature in W. Blaschke, Kreis und Kugel, Leipzig,
1916.
2 Cf.
C. Caratheodory, Uber die starken Maxima und Minima bei enfachen
lntegralen, Math. Ann., Vol. 62, 1906, pp. 449-503.
1
SUPPLEMEN TARY REMARKS AND EXERCISES
259
it enables one to interpret many important relations geometrically.
The indicatrix for a three-dimensional problem is the surface with
the equation mx, y, z, ~' 7/, s) = 1 in ~' 7/, r-space.
The direction (ox, oy) is transverse to (i, if) if lJiOX + 'ijyoy = 0.
But the equation of the tangent to the indicatrix at the point
x/t'J, iJIO: is
(~ - ~)11• + (~ -
&)11. = 0
or
mz + 7Ji5u =
1.
Thus the transverse direction is the direction of the tangent to the
indicatrix at the point where the indicatrix intersects the ray joining
the origin with the point (x, fJ). Clearly, transversa lity is equivalent
to orthogona lity if and only if the indicatrix intersects all straight
lines through the origin at right angles, i.e. if and only if the indicatrix is a circle about the origin. In this case, since O: is homogeneous,
we have t'J(x, y, x, fJ) = <P(x, y)y x2 + y2. If the directions of the
extremals and the transverse curves are symmetrically related, a
line through the origin 0 parallel to the tangent to the indicatrix at
the point P intersects the indicatrix in a point where the tangent to
the indicatrix is parallel to OP.
The indicatrix is particularl y useful in the study of broken extremals, i.e. extremals which have a corner (discontinuity of slope) at
a point xo, Yo. We wish to know under what circumstances a curve
which leads from (x1, Y1) to (x 0 , y 0 ), arrives there in the direction
(xi), fJo), continues in another direction (xt, fJt), and ends at (x2, Y2)
can provide an extremum . In the intervals where the curve has a
continuously turning tangent it must satisfy the Euler equations. To
investigate the state of affairs at the corner we suppose that the extremal is a member of a family of curves,
y(t)
+
E7J(t),
where W), 71(t) are continuously differentiable functions vanishing
at the end points. We form the first variation, i.e. we differentiate
with respect to E and set E = 0. If we do this for the two segments
separately all terms except the boundary terms corresponding to the
corner vanish; those at the ends because the er..d points are kept fixed,
and the integrals because of the extremal character of the segments.
260
IV. THE CALCULUS OF VARIATIONS
Thus we have
WoHMxo, Yo, x"O, flo)
+ 77(tom1ho, Yo, x"O, fl"O)
- Woma:(x9, Yo, xt, flt) - 77(tomuCxo, Yo, xt, flt) = 0,
and, since Hto) and 77(to) are arbitrary, the "Weierstrass-Erdmann
vertex condition"
t"h(xo, Yo, io, flo) = i5a:(xo, Yo, xt, flt),
i'5u(Xo, Yo, io, flo) = i'5u(Xo, Yo, xt, flt)
holds. Therefore, the tangents to the indicatrix at the points at which
the indicatrix intersects the vectors (io, flo) and (xt, flt) coincide.
The two directions of the curve at the vertex are those of the vectors
from the origin to the two tangent points of a double tangent of the
indicatrix.
7. Variable Domains. Consider an integral
.:q
J =
1
xo
F(x, u, u') dx
where not only the function u(x) but also the limits Xo and X1 are
variable and may depend on the paramter e; then the first variation
of the integral contains, in addition to the usual terms, a term due
to the variation of the domain. To be specific, the variation is
(103)
where we have set
ou = e(au(x;
ae
e)) ,
~
~
_ e (ax1(e))
--
vX1 -
E=O
ae
_ e (axo(e))
--
vXo E=O '
ae
E=O'
and [F]u is the Euler functional derivative of F.
A similar formula is valid in two (or more) dimensions where the
domain of integration may vary with the parameter e. To obtain
the variation of the integral
J
=
Ji
F(x, y, u,
Ux, Uy)
dx dy
we suppose that the domain G* (coordinates denoted by x* and y*)
261
SUPPLEMENTAR Y REMARKS AND EXERCISES
which depends on Eis mapped onto the original domain G by a transformation
x* = X(x, y; E)
(104)
y*
=
Y(x, y; E).
We assume that this transformation is one-to-one continuously differentiable and reduces to the identity transformation for E = 0.
We now assign to the point (x*, y*) of G* a new functional value
u* = u*(x*, y*; E), which in the old coordinates becomes
u* = u*(X(x, y; E), Y(x, y; E)) = U(x, y; E).
(104a)
Thus our original function u(x, y) represents the surface for E = 0
of the family of surfaces u*(x*, y*; E)-the separate members of
which, for fixed E, are given parametrically by the. equations (104)
and (104a), with x and y as parameters.
We now form the integral
J(E) =
rr
JJG(E) F[x*,
y*, u*(x*, y*; E), u:.(x*, y*; E)U:.(x*, y*; E)] dx* dy*
and, introducing x, y as independent variables, transform it into the
integral
J(E) =
JL
F[X, Y, u*(X, Y; E), u:.cx, Y; E), u;.cx, Y; E)]
a~~:~) dx dy
extended over the fixed domain G. The variation is formed by differentiation with respect to e. The following notation is introduced for
convenience:
ox=
E(ax)
aE ,
oy = e(aY)
aE
E-0
OU = E(aU(x, y;
ae
~ -_
uU;r;
E
e))
e))
= E(au*(X,
Y;
&E
E...()
~ -_
uUy
(au:.(x, y;
aE
,
E-0
E-0,
e)) ,
E-=0
(aU:.(x, y;
ae
E
E))
•
E-=0
Then
oJ =
Ji
[F;r;Ox
+
Fyoy
+
FuoU
+
Fu/>U;r;
+ FuyOUy
+ F(ox);r; +
F(oy) 11] dx dy.
262
IV. THE CALCULUS OF VARIATIONS
We can write this integral in a different form if, instead of the variations above, we use the variations at a fixed argument point:
5u =
u*(x, y; E)) .
E(~
t=O
aE
Here we have written (x, y) instead of the independent variables
(x*, y*), and the variation 5u (for which the argument point is fixed)
is related to the variation· ou (for which the argument point varies
with E) by the identity
ou = 5u
(105)
+ ux5x + u,l;y.
In the same way we have
OU:z; = (5u) z
ouy = (5u) 11
+ UxxOX + Uzyoy
+ u xox + Uyyoy.
11
Introducing these expressions into oJ we then obtain
oJ
=Ji {[F]u5u + (FuJu)x + (Fuu5u)
or
oJ = (( [F]u5u dx dy
lla
+
11
+ (FOx)x + (Foy)
11 }
dx dy
1(Fuz axan + Fu anay) 5u ds
+ £F(ox :~ + oy ;~) ds,
r
11
where n is the outward normal and s the arc length of the boundary
r of G. Note that in this formula the variation of J is separated, as
one might have conjectured, into a part that arises for a fixed domain
and a part that can be ascribed to the variation of the domain; the
variation of the domain is represented by the normal components of
translation vectors (proportional to E) for the boundary points.
8. E. Noether's Theorem on Invariant Variational Problems. Integrals in Particle Mechanics.1 We consider a family of transformations, depending on a continuous parameter a,
x* = X*(x, y, u; a)
(106)
y* = Y*(x, y, u; a)
u* = U*(x, y, u; a).
1 E. Xoether,
Invariante Variationsprobleme, Nachr. Ges. Gottingen
(math.-phys. Kl.), 1918, pp. 235-257.
263
SUPPLEMEN TARY REMARKS AND EXERCISES
We suppose that the transforma tion that corresponds to the value
a = 0 is the identity. To any surface u = u(x, y) this family of
transforma tions assigns a family of surfaces u* = u*(x*, y*; a), depending on a, with the parametric . representa tion
x* = X*(x, y, u(x, y); a) = X(x, y; a),
Y*(x, y, u(x, y); a) = Y(x, y; a),
y*
u* = U*(x, y, u(x, y); a) = U(x, y; a)
(the parameter s being x, y).
We now suppose that the value of the integral
J =
Jl
F(x, y, u, Ux , Uy) dx dy
remains unchanged under the transforma tions (106), i.e. that for every
region G
J* =
Ji.
F(x*, y*, u*, u:., u:.) dx* dy* =
Ji
F dx dy,
where G* is the domain over which the point (x*, y*) varies as (x, y)
varies over the domain G. Then we evidently have
= 0,
OJ= a ( aJ*)
da a=O
and from the results of the previous subsection we obtain
0
=
ff. {[F]. &u + :x (F•• 5u) + :y (F.,
3u)
(107)
+ :x (nx) + :Y (F6y)} dx dy,
Here, as before, we have put
(108)
~x
= a(aX*)
~Y
= a(aY*)
aa
da
= a(ax)
da
a=O
'
a=-0
= a(aY)
a=()
da
,
a=O
and we also have, in virtue of (105),
(109)
~u
= a(iJU)
iJa
a=O
(Ux
+
Uuux)a=o ox -
(U11
+
Uuuu)a-o oy.
264
IV. THE CALCULUS OF VARIATIONS
Since equation (107), by hypothesis, is valid for every domain G,
the integrand on the right side must vanish, i.e. we must have
- + !l(Fuzou
a
- +Fox) + -(Fu
a
- +
[]
F uou
ou
uX
oy
11
FOy )
=
0.
Analogous formulas are obtained in the case of several dependent
variables. For example, if the integral
=
J
Jl
F(x, y, u, v,
U:i:, V:i:,
u 11 , v11 ) dx dy
is invariant under the continuous transformati ons
x*
u*
=
=
X(x, y, u, v; a),
y* = Y(x, y, u, v; a),
U(x, y, u, v; a),
v* = V(x, y, u, v; a),
one obtains
[F]u 5u
+
[F]. 5v
+ :x (Fuz 5u + Fvz 5v +Fox)
a
- + F11 ov
- +Foy)
.f- -(Fu
ou
11
11
ay
=
0,
where
- = (au)
OU
a da
a=O -
(U:i:
.+
- (Uy+
(109a)
+ U,,v:i;)a-0 OX
UuUy + Uvvy)a=O oy,
UuU:i:
(Vy+ VuUy
+ Vvvy)a=O oy.
These results are easily extended to the case of one, or of more than
two, independent variables. For one independent variable, a first
integral for the extremals
Fu,ou
+ F11.ov +Fox =
a· const.
can be obtained by integration. Here the expressions ou, av, ox are
functions of x given by (108) and (109a).
For example, we may verify· these results for the problem
:i:1
1 F(u, u') dx
:i:o
= min.
Since the integrand does not contain x ex.
265
SUPPLEMENTARY REMARKS AND EXERCISES
plicitly, it remains invariant under the continuous transformation
u*
x* = x +a,
u.
=
const.Thus we obtain for the extremals the integral F - u'Fu.
a result which we already know from §4.
If the integral J is invariant under a family of transforma.tions
containing n parameters, then we obtain n linearly independent combinations of Euler expressions in the form of divergences, and hence
n linearly independent first integrals.
These facts may be illustrated by the integrals of particle mechanics.
The orbits of the particles of a system of mass points are given by the
extremals of the problem
OJ
=
f
o
ti
(T -
=0
U) dt
to
2
where T = !L m(x + i/ + i\ and where the potential energy
depends only on the relative positions of the mass points, i.e. does not
change under a translation or rotation of the system as a whole.
This integral is invariant, for example, under the continuous transformations
t*
=
t,
x* = x +a,
(i.e. tJt = oy =
z*
y* = y,
oz = o,
=
z
ox = a),
or
t* = t,
y*
x* = x cos a
= -x sin
(i.e. tJt =
oz
= 0,
a
+
+
y sin a,
z* = z
y cos a~
ox = ay,
fJy
= -ax).
Hence from the above considerations,
T;; =
yT;; - xTu =
L mx =canst.
L m(yx - xy) =
const.
These relations, together with those obtained by permuting x, y, z,
express the conservation of linear and of angular momentum.
If T and Udo not contain the time variable t explicitly, the energy
1
integral can be obtained in a similar fashion from the fact that the
1 For more
details see E. Bessel-Hagen, Uber die Erhaltungssiitze
der Elektrodynamik, Mathematische Annalen, Volume 84, 1921, pages 258-276.
266
IV. THE CALCULUS OF VARIATIONS
f
t +.a, ot
t1
integral
(T -
to
=
U) dt is invariant under the transformation t'
a.
If the integral J remains invariant under transformations which
contain an arbitrary function p of the independent variables together
with its derivatives up to order k:
x*
=
x ( x, y, u, v, p(x, y), axa p(x, y),
ak
)
... , ayk p(x, y) '
.................... '
then we obtain a linear combination of the Euler expressions and of
their total derivatives up to k-th order which vanishes identically;
i.e., the Euler equations are not mutually independent.
A simple example is the homogeneous form of the integral
J =
f
t1
~(x,
to
y, x, y)
dt.
The integral remains unchanged when t, x(t), y(t), x(t), y(t) are replaced by t(T), x(t(T)), y(t(T)), dx(t(T))/dT, dy(t(T))/dT. Accordingly the Euler expressions fmx, [my are connected by the relation
xmJx
+ 'filmy =
0.
[Cf. formula (31), page 198.]1
9. Transversality for Multiple Integrals.
Ji
Suppose the integral
F(x, y, z, Xu, Yu, Zu, Xv, Yv, Zv) du dv
is to be minimized under the condition that the boundary of the surface [x(u, v), y(u, v), z(u, v)] lie on a given surface cp(x, y, z) = O; a
formal extension of the procedure applied for curves would lead to the
boundary condition
Fz,,
F:r.,
'Px
Fu,.
Fu,,
'Pu
Fz,;
Fz.,
'Pz
I
=
0,
1
1 For a more detailed discussion
and for generalizations and applications
to mechanics, electrodynamics, and relativity, see the paper by E. Noether
referred to above and the references given there.
267
SUPPLEM ENTARY REMARKS AND EXERCISE S
if we knew that the derivativ es concerned existed and were sufficiently
regular at the boundary . However, this condition has not been
rigorously analyzed or justified.
10. Euler's Differential Expressio ns on Surfaces. Suppose that the
parametr ic represent ation of a surface in p, q, r-space is p = p(~, '1/),
q = q(~, 11), r = r(~, 11), and that the line element on this surface is
2
2
2
ds = e d~ + 2f d~ d11 + g d11 • Then the expression
+ eu!
Qlu,u] = gu~ - eg2fueu"
- f2
is independ ent of the choice of paramete rs. The Euler differential
expression corresponding to the surface integral
Ji
Q{u, u]
veg - j2 d~ d11
is then
Au= ~[gue - fu,,2
a~ veg - f
+ eu,,J
J+ 811i[-fue
veg - f
2
'
and
AU
veg+ f 2
independ ent of the choice of paramete rs.
11. Thomson 's Principle in Electrosta tics. Let u, v, w be the components of the electric field intensity in a condenser, i.e. in the region
G bounded by two closed surfaces r 1 , r 2 in space. Suppose that the
field is free of sources, i.e.
ts
(110)
U:r:
+ V + Wz
11
=
0,
and suppose that the charges on the surfaces r 1 and r 2 are given as
+Q and -Q, respectively:
(111)
f£ (uxn + VYn + WZn) ds =
2
(see page 251 for notation) .
-Q
268
IV. THE CALCULUS OF VARIATIONS
Then for electrostatic equilibrium the energy of the field, given by
~
ffl (u + v +
2
2
w
2
)
dx dy dz
(apart from a constant), must be as small as possible. Applying the
method of Lagrange multipliers and letting the multipliers of (110)
and (111) be X(x, y, z) and µ 1 , µ2, respectively, we obtain the Euler
equations
(112)
u =
v = Ay,
X:r:,
and the natural boundary conditions
X = µi =
(113)
const~
on f1
X = µ2 = const.
on r 2
•
The field vector with components u, v, w is therefore the gradient of a
potential X, which is constant on each surface and satisfies the potential equation .1X = 0. This result may also be obtained without
using the method of multipliers by representing the vector (u, v, w)
as the curl of another vector, thus eliminating the subsidiary condition (110).
12. Equilibrium Problems for Elastic Bodies. Castigliano's Principle. A further example for the theory of §11 is the equiva-
lence of Castigliano's principle and the principle of the minimum of
potential energy for the equilibrium of an isotropic elastic body.
Before formulating the equilibrium conditions for the case of a
three-dimensional elastic body, we recall some fundamentals of
the mathematical theory of elasticity.
Suppose that at rest the body in question occupies in x, y, z-space
a region G with piecewise smooth boundary surfaces r. Suppose
that some forces deform the body from this rest position into a new
equilibrium position, each point (x, y, z) of the body undergoing a
small displacement (u, v, w). We define the strains
EU
(114)
E21
E31
= U:r:,
E12
= Hvx + Uy),
= !(w:r: + Uz),
E22
Ea2
= !(uy + Vx),
= Vy,
= !(wy + Vz),
and the dilatation
E
=
Eu
+
E22
+
E33 •
E23
= !(uz
= Hvz
E33
=
E13
Wz
+
+
W:r:),
Wy),
269
SUPPLEMENTAR Y REMARKS AND EXERCISES
The elastic forces produced by the deformation are given by the
nine components of the stress tensor
@:>
=
Su S12 S13)
S22 S23
(S21
831 · S32 833
which are likewise functions of position satisfying the symmetry
relations S12 = S21 , S23 = S32 , Su = S13 .
The stresses and strains are linearly related by Hooke's law
+ bE,
(115)
S12 = aE12,
S22 = aE22
S32 = aE32,
Su = aEu
S21 = aE21 ,
S31 = aE31,
+ bE,
S13 = aE13,
S23 = aE23,
S33 = aEas
+ bE,
where a and b are positive constants of the material.
Suppose a force P whose components have the densities P1, P2, Pa
acts at each point (x, y, z) of the elastic body; suppose, moreover, a
surface force p whose components have the surface densities p1,
P2 , pa acts at each point of the surface r of the body. Then the
equilibrium conditions are
aSu
+ aS21 + 0Sa1 + p 1 = O,
aS12
+ aS22 + aS32 + p = O,
ax
(116)
ay
az
2
az
ay
ax
asl3 + aS23 + aSaa + p 3 =
az
ay
ax
O
or, in vector notation,
div@)= -P
in the interior; on the boundary they are
(117)
SuXn
S12Xn
S1sXri
+ S21Yn + Sa1Zn + S22yn + Sa2Zn + S'l3Yn + SasZn -
P1 = 0,
P2 = 0,
Pa = 0
or, in vector notation, ·
Sn= p
(the subscript n denotes the outer normal on r).
270
IV. THE CALCULUS OF VARIATIONS
The problem now is to determine the stresses and displacements
at each point if the following quantities are given: the forces P 1 , P 2 ,
P3 in G, the surface forces Pi = th, P2 = fJ2, p3 = p3 on a portion
r1 of the boundary, and the displacements
(118)
u =
u,
v =
v,
w=w
1
on another part r 2 of the boundary .
The equilibrium state is again characterized by the principle of the
minimum of the potential energy
I:
U[u, v, w]
!
!Ji
- JJi
=
[EnSn
+ 2E12812 + E22S22 + 2E23823 + E33833 + 2E31831] dx dy dz
(P1u
+ P2v + P3w) dx dy dz
- J[
(p1u
+ p2v + p3w) ds.
1
For the variation of this integral the argument functions are the
displacements u, v, w, which attain the prescribed values on r 2 • The
stresses are to be expressed in terms of the strains E in accordance
with (115); the strains, in turn, are given in terms of the displacements by relations (114).
The equilibrium conditions (116) in G and (117) on r 1 are obtained
without difficulty as the Euler equations and natural boundary conditions for U.
If we apply the theory of reciprocal variational problems developed
previously to the minimum potential energy principle, we are led to
another variational principle, known as "Castigliano's principle." It
reqqires that the work of deformation be a minimum.
To formulate these two reciprocal variational problems in accordance with the theory of §11, we shall employ vector notation and
introduce the following definitions: First, we consider two arbitrary
symmetric tensors ~ and E connected as above by Hooke's equations
(115); if in addition the tensor E is derived from a vector field s with
components u, v, w by the relations (114), i.e. if E is a strain tensor
1 The normal displacement and tangential force, or tangential displacement
and normal force, could also be given on the boundary. It goes without saying
that either r1 or r 2may constitute the entire boundary surface.
271
SUPPLEMEN TARY REMARKS AND EXERCISES
of a deformati ons, we say that@? satisfies the compatibility conditions,
or@? = {s}. These conditions, which need not be written out explicitly,
characterize~ as the stress tensor of a deformati ons. By using (115)
the first integral in our expression for U can be written as
JJl [a f,; ~~k +
b(En
or, in terms of ~' as U = Q(@?,
quadratic integral, explicitly
(119) Q(0, 0) =
+ E22 + Eaa) 2]
dx dy dz
where Q is a positive definite
~),
JJH['t;, si.- a~
3b (Su+ S,,+
s,,)'}n dy dz.
For this quadratic form the following Green identity holds: If 'B, ~
are two symmetric tensors of which ~ = { c} is a stress tensor for the
deformatio n c, then
(120)
Q('B, <£)
=
-
JJl
c div 'B dx dy dz
+
Jl
c 'Bn ds.
Now we consider two reciprocal linear subspaces n and ~ in the space
of tensors 'B, ~' @?, ~' etc. n consists of symmetric tensors 1) for
which
and
~
consists of tensors
.8
=
on
in G,
div 1) = 0
.8 for
which
s = 0
in G,
{s}
ri
The reciprocity .of these two spaces follows from (120).
0
0
Now we consider two tensors ~ , ~ subject only to the following
conditions :
rs
(a)
0
=
{s0 } in G,
0
0
where s is an arbitrary strain satisfying s = s on
(b)
0
div ~ = -P
~~ =
in G,
r 2 , and
p
Furthermo re, we call tensors rs and ~ admissible if
0
and ~ - · ~ is in n. Then the problems
rs -
I*:
Q(rs -
i:0,
11*:
Q(~ -
rs0, ~
-
0
~ ) = min.
rs0 ) =
min.
on
r1.
rs - rs0 is in
~
272
IV. THE CALCULUS OF VARIATIONS
are reciprocal in the sense of §11. By using (120) and by disregarding
terms which are a priori known from the data (including e 0, ~ 0 ),
we obtain the following form of the two reciprocai problems:
I:
Q(e, e) -2
!Ji
sP dx dy dz -2
fl
sp ds
=
min.,
1
which is the original principle of minimum potential energy, and
II:
Q(~, ~)
- 2
Jl
r2
S~n ds =
min.,
which is Castigliano's principle of minimum work of deformation.
The normal component ~n of the tensor ~is to be expressed by (117);
the variational conditions in Gare the compatibility conditions, not
written down explicitly.
In these two reciprocal problems one observes again that the
variational conditions for one are the admissibility conditions for the
other.
It is furthermore remarkable that in the formulations of problems
I and II any reference to e 0 and ~ 0 has disappeared as far as the
interior of G is concerned. These problems can therefore be formulated without constructing tensors e 0 and ~· (by solving underdetermined partial differential equations).
By Castigliano's principle one can treat certain problems of
mechanics more simply than by the principle of minimum potential
energy. For example, in the theory of elastic beams the analogue of
Castigliano's principle reduces the problem to one for an ordinary
minimum. This can be seen in the following way: we may explicitly
satisfy the homogeneous differential equation of the beam by functions containing only a finite number of parameters; then the splitting
of the functiorr,-space into orthogonal subspaces leads to spaces ~
which depend only on a finite number of parameters such that problem II reduces to an ordinary minimum problem.
13. The Variational Problem of Buckling. If a rod is compressed by
a longitudinal force P acting at botli ends, it may be in stable or in
unstable equilibrium; i.e., after a slight lateral bending it will either
return to the equilibrium position or "buckle," depending on whether
the magnitude P is less than or greater than a certain critical value
Po, the "buckling force." In the first case the straight rod is in a
position of minimum potential energy with respect to small deformations; in the second case, it is not.
273
SUPPLEMENTARY REMARKS AND EXERCISES
If the rod at equilibrium is of length l and if its lateral displacement is denoted by u(x) (O ~ x ~ l), then the potential energy isapart from material constants-
1
z
U =
1
l
2
(u") dx - P
2
(u') dx.
The first term is the energy of bending, the second term the potential
energy of elongation (as in the case of the string).
For sufficiently small values of P the minimum of U with the
boundary condition u(O) = u(l) = 0 has the value zero. 1 On the
other hand, for sufficiently large P, U can be negative; for any admissible function u we need only choose
{
P
>
(u
11 2
)
dx
oz
•
lo
2
(u') dx
The buckling force Po, i.e. the largest value of P for which' .the minimum of U is equal to zero, may evidently be expressed as the minimum of
1
l
(u
11 2
)
dx
•t
Jo
(u')2 dx
with the boundary condition u(O) = u(l) = 0.
minimum of
1
Equivalently, Po is the
l
(u
11 2
)
dx
subject to the condition
1
l
1
2
(u') dx = 1
For example, this is true for F < l/Z 2 • For, since
point xo at which u'(xo) = 0.
u'(x) =
l:z;zo u"(x) dx,
( u') 2
folu' dx = 0 there is a
Then
~l
i(
l
u") 2 dx,
j
.z
0
z
(u') 2 dx ~ l 2
1
0
(u")2 dx.
274
IV. THE CALCULUS OF VARIATIONS
and the boundary condition u(O) = u(l) = 0. This is expressed by
saying that Po = Xis the first eigenvalue of the differential equation
u"" +Xu"= 0
with the boundary conditions u(O) = u(l) = 0, u" (O) = u" (l) ·= 0.
Such eigenvalue problems and their treatment by means of variational methods wiU be discussed in the next two chapters.
References
For detailed references see the excellent bibliography by Le cat:
Lec~t,
M. Bibliographie du calcul des variations 1850:-1913. Gand, Paris,
1913.
Lecat, M. Bibliographie du calcul des variations depuis les origines jusqu'a
1850 comprenant la liste des travaux, qui ont prepare ce calcul. Gand,
Paris, 1916.
Textbooks
Here we mention only the most important textbooks:
Bliss, G. A. Calculus of Variations. Open Court, Chicago, 1924.
Bolza, 0. Vorlesungen iiber Variationsrechnung. B. G. Teubner, Leipzig
and Berlin, 1909.
Hadamard, J. Le~ons sur le calcul des variations, Vol. I. A. Hermann et
fils, Paris, 1910.
Kneser, A. Lehrbuch der Variationsrechnung. F. Vieweg und Sohn, Braunschweig, 1925.
Moigno, M., and LindelOf, L. L. Calcul des variations. Mallet-Bachelier,
Paris, 1861.
Tonelli, L. Fondamenti di calcolo delle variazioni, Vols. I and II. N.
Zanichelli, Bologna, 1921and1923.
Vivanti, G. Elementi del calcolo delle variazioni. Giuseppe Principato,
Messina, 1923.
Monographs and Articles
The following articles from the Enzyklopadie der math. Wiss., 1904-1916:
Hellinger, E. Die allgemeinen Ansatze der Mechanik der Kontinua. Vol.
4D, pp. 601-694.
Kneser, A. Variationsrechnung, Vol. 2A, pp. 571-625.
Zermelo, E., and Hahn, H. Weiterentwicklung der Variationsrechnung in
den letzten Jahren, Vol. 2A, pp. 626-641.
French edition of the encyclopedia:
Lecat, M. Calcul des variations, Tome II, Vol. 6, pp. 1-288.
CHAPT ER V
Vibrat ion and Eigenv alue Proble ms
In Chapters V and VI importan t aspects of the linear differential
equations of mathema tical physics will be discussed, especially in
relation to vibration phenomena. The method of eigenfunctions
will occupy a central position througho ut.
§I. Prelimin ary Remark s about Linear Differen tial Equatio ns
1. Principle of Superposition. A linear homogeneous differential
expression (or differential operator) for a function u is generally
written in the form
L[u] =Au+ Buz
+ · · · + Cuzz + · · ·
where the coefficients are given functions of the independ ent variables. L[u] is a linear homogeneous combination of the function u
and its derivatives up to a given order, called the order of the differential expression. The linear differential operator
L
=
A
+ B(iJ/iJx) + · · · + C(iJ /iJx + ...
2
2
)
satisfies the fundame ntal relation
(1)
where c1 , c2 are any constants. The general linear differential equation has the form
L[u]
= f(x,
y, · · · ),
=
0
where f is a given function of the independ ent variables. If f
nonotherwise
and
eous,
the differential equation is called homogen
homogeneous.
In this chapter we shall be almost exclusively concerned with
linear homogeneous differential operators; these operators are a special
275
276
V. VIBRATION AND EIGENVALUE PROBLEMS
instance of linear homogeneous functional operators. Other examples
are given by the integral expression
Jl
K (x, y;
~' 11)u(~, 11) d~ d11,
by the operator
e[u] =
2 12r
h27r
0
{u(x
+ h cos e, y + h sine)
- u(x, y)} de,
or by the difference operator
1
h2 { u(x
+ h, y) + u(x -
h, y)
+ u(x, y + h) + u(x, y -
h) - 4u(x, y)}.
If we suppose that u has continuous derivatives up to the second order,
the last two expressions tend to the differential operator ..1u for
h ~ 0; this can be easily verified. The condition that a linear combination of such linear operators should equal a known function
is expressed by a linear functional equation; integral, difference, and
differential equations are of this type. Equation (1) expresses the
linear homogeneous character of an operator L[u].
The solutions of a linear homogeneous differential equation, and
in general of any linear homogeneous functional equation, have the
following superposition property: If u1 , U2 are two solutions of the
equation and c1 , c2 are arbitrary constants, then c1u1
c2u2 is also
a solution. More generally, we can combine any number of known
particular solutions u 1 , u2 , · · · with constants c1 , c2 , • • • and obtain a new solution C1U1 + C2U2 + · · · . A convergent series L':=1CnUn
formed from an infinite sequence of solutions u1 , u 2 , · · · represents
a solution if the differential operator L[u] may be applied to the series term by term.
If a solution u(x, y, · · · ; a) of the functional equation L[u} = 0,
depending on a parameter a, is known, new solutions of the form
+
v =
J
w(a)u(x, y, · · · ; a) da
can be constructed; here w(a) is an arbitrary function and the domain
of integration can be chosen at will. The only restrictions needed are:
the integral must exist and the operation L must be applicable under
the integral sign. These conditions are, in particular, satisfied for
PRELIMINARY REMARKS
277
differential operators if w(a) is piecewise continuous and the domain
of integration is finite.
If the homogeneous equation is completely solved one solution of
the inhomogeneous equation yields all its solutions; for, all solutions
of the inhomogeneous equation are obtained by adding one such
solution to each solution of the homogeneous equation.
2. Homogeneous and Nonhomogeneou s Problems. Boundary Conditions. We consider problems in which both a linear differential
equation and boundary or initial conditions must be satisfied (cf. Ch.
IV, §10). If both the boundary conditions and the differential
equation are homogeneous, the problem is called homogeneous. In
this case, cu is a solution if u is a solution and c is constant. Homogeneous boundary conditions usually consist of relations between
the values assumed by the desired function u and its derivatives
Ux ' • . . on the boundary r of the domain G in question.
A simple
condition of this kind is u = o or au/ an = o, where aI an denotes
differentiation in the direction of the outer normal.
Given linear nonhomogeneous boundary conditions-for instance,
boundary values u = f (not vanishing everywhere)-w e can obtain
an equivalent problem with homogeneous boundary conditions: Assume that L[u] = 0 is a linear homogeneous equation, and the
boundary values .f can be extended continuously into the interior of
G in such a way that L[j] = g is a continuous function in G. Then,
for the new unknown function v = f - u, we immediately have the
differential equation L[v] = g with the homogeneous boundary condition v = 0. Conversely, given a nonhomogeneous linear equation
with homogeneous boundary conditions, a special solution of which
is known, then problem with a homogeneous equation and nonhomogeneous boundary conditions can be obtained by subtraction.
In general: Homogeneous dijj'erential equations with nonhomogeneous
boundary conditions are essentially equivalent to nonhomogeneous differential equations with homogeneous boundary conditions. 1
a
3. Formal Relations. Adjoint Differential Expressions. Green's Formulas. We shall briefly discuss certain formal relations. In par-
ticular we shall consider differential expressions which arise from a
1 However,
one should note that the first step, transforming a problem with
nonhomogeneous boundary conditions into one with homogeneous conditions,
involves assumptions on continuity and differentiability not necessarily made
for the original problem.
278
V. VIBRATION AND EIGENVALUE PROBLEM8
variational problem with a homogeneous quadratic integrand; such
expressions are called self-adjoint differential expressions.
(a) One Independent Variable. We consider the quadratic expression
Q[u, u] = au' 2
+ 2bu'u + du
2
,
where a, b, d are given functions of x, and u(x) is the argument
function. We integrate the symmetric bilinear expression
+ b(u'v + v'u) + duv
Q[u, v] = au'v'
over an interval (x0 , x1). Integrating by parts and eliminating the
derivatives of v, we obtain "Green's formula''
f
(2)
Q[u, v] dx
~
-
f
vL[u] dx
+ (au'+
bu)v
1::,
where the differential expression
L[u] = (au')'
+ (b'
- d)u
agrees, except for the factor - 2, with the Euler differential expression associated with the integrand Q[u, u]. Similarly, since Q[u, v] is
symmetric, we have
Zl
1zo Q[u, v] dx =
(2a)
-
1Zl uL[v] dx + (av' + bv)u IZl ;
zo
zo
from (2) and (2a) we obtain the symmetric Green's formula
1
(2b)
[
(vL[u] - uL[v]) dx
.ro
= a(u'v - v'u) !'xi .
.ro
If we start with an arbitrary bilinear expression
B[u, v] = au'v'
+ bu'v + cuv' + duv
instead of a symmetric one and integrate by parts, we obtain formulas of the form
1Zl
zo
(3)
B[u, v] dx
= -
1Zl vL[u] dx + (au' + cu)v IZl
.ro
= -
1
.r1
zo
(4)
1
z1
XO
(vL[u] - uM[v]) dx
.ro
uM[v] dx
+ (av' + bv)u IZl ,
.ro
= [a(u'v - v'u)
+
(c - b)uv]
l.r1
XO
.
279
PRELIMINA RY REMARKS
The differential expression
+ (bv)'
M[v] = (av')'
- cv' - dv
is associated uniquely with the differential expression
L[u] = (au')' - bu'
+ (cu)'
- du
by the requireme nt that the integral on the left in (4) can be expressed
in terms of the values of the function and its derivative s on the boundary. These two expressions are said to be adjoint to each other. If
L[u] = M[u] holds identically, the differential expression L[u] is
called self-adjoin t and may be derived, as above, from a quadratic
expression Q[u, u].
qu is
ru'
The adjoint of the differential expression pu"
(pv)" - (rv)' + qv. Hence
+
+
p' = r
is a necessary and sufficient condition for a differential expression to
be self-adjoint.
By means of the relations a = p, b' - d = q, a quadratic expression Q[u, u] that corre~ponds to (pu')' + qu can be constructe d in
various ways.
can be
An arbitrary linear differential expression pu" + ru' +
transforme d into one which is self-adjoint: we may multiply by a
suitable nonvanish ing factor
qu
p(x) =
efl<r-p
1
)/p]dz,
we may introduce a new independe nt variable
x' =
Je-f
[(r-p')/p]dz
dx.
in place of x, or we may introduce a new dependent variable
v = ue f
[(r-p 1 )/p]dr
in place of u.
(b) Several Independen t Variables. Relations analogous to for
mula (2) hold for linear second order partial differential equations
An important example is given by the quadratic integrand
Q[u, u] = p(u!
+
u;)
+
qil
280
V. VIBRATION AND EIGENVALUE PROBLEMS
with the polar form
+ UyVy) +
Q[u, v] = p(uxV:i:
quv.
We integrate QJu, v] over a domain G with a piecewise smooth
boundary r. Integration by parts then yields Green's formula .
(5)
Ji
Q[u, v] dx dy = -
Ji
vL[u] dx dy
+
£
pv
:~ ds
where
Here we assume that the function v is continuous with piecewise
continuous first derivatives in the closed domain G and that u is
continuous with continuous first and piecewise continuous second
derivatives. In equation (5), s denotes arc length and a/an differentiation in the direction of the outer normal.
If v satisfies the same conditions as u, we may interchange u and v
in the formula and, subtracting the second expression from (5), obtain the symmetric Green's Formula
(5a)
({ (vL[u] - uL[v]) dx dy =
JJG
1
r
p(v au - u av) ds.
on
an
. For p = 1, q = 0 our-self-adj oint-differe ntial expression L[u]
is the potential expression ilu and formulas (5) and (5a) are the
familiar Green's formulas of potential theory.
As in the case of ordinary linear differential equations, an adjoint
expression M[v] can be associated with any partial differential expression L[u] if we require that vL[u] - uM[v] can be expressed as a
divergence.
4. Linear Functional Equations as Limiting Cases and Analogues
of Systems of Linear Equations. Any differential equation can be con-
sidered as the limiting case of a difference equation: We replace
each differential quotient by a corresponding difference quotient for
whi~h the increment of the independent variable, the so-called mesh
width, is denoted by h; we consider the functional values of u only
at the points x, · y, · · · of a lattice whose coordinates are integral
multiples of h. Thus the differential equation is replaced by a system
of linear equations for the values of the function u at the lattice
points. In a similar way integral equations and other functional
SYSTEMS WITH FINITE NUMBER OF DEGREES OF FREEDOM
281
equations can be replaced by systems of linear equations. In Volume
II we shall develop this idea systematically, in particular for numerical .solutions of· differential equations. Here, the analogy between
differential equations and difference equations is used merely as a
heuristic principle. It leads us to expect that problems in linear
differential equations are closely analogous to corresponding problems
in the linear algebraic equations from which they originate by limiting
processes. This conjecture will be confirmed under very general
assumptions.
In particular, the following alternatives hold: If a homogeneous
problem corresponding to a homogeneous differential expression has
the unique solution u = 0, then the nonhomogeneous problem always
has one and only one solution. If, however, the homogeneous problem has a nontrivial solution, then the nonhomogeneous problem has
solutions only under restrictive linear conditions, and these solutions
are not unique. As in Chapter I, the case in which a parameter X
appears linearly in the homogeneous differential expression will play
a special role. We are interested in just those values of X, the eigenvalues for the problem, for which the homogeneous problem has a
nontrivial solution, an eigenj?J,nction.
To replace linear differential equations in the physics of continua
by difference equations corresponds to replacing the continuous medium by a system with a finite number of degrees of freedom.
§2. Systems of a Finite Number of Degrees of Freedom
As in Chapter IV, §10, we consider a system with n degrees of
freedom in the generalized coordinates q1, q2, · · · , qn ; the kinetic
and potential energies of the system are given by the quadratic forms
n
n
T
2: ahkqhqk
= h,k=l
and
2: bhkqhqk '
u = h,k=l
respectively, with constant coefficients ahk , b,,,k •
In view of its physical meaning the form T is positive definite.
We shall assume that U is also positive definite. We then know
that stable equilibrium occurs for q1 = q2 = · · · = qn = 0.
If we assign nonzero values to some of the coordinates qh, or if we
constrain the qh by other nonhomogeneous conditions, we obtain a
new state of equilibrium different from the original rest state q,,, = 0.
In the case of a finite number of degrees of freedom the problem of
282
V. VIBRATION AND EIGENVALUE PROBLEMS
finding such positions of equilibrium under constraints does not introduce a specifically new element into the mathematical formulation;
however, in the limit for n ~ oo, it leads to the typical boundary
value problems of partial differential equations.
1. Normal Modes of Vibration. Normal Coordinates. General Theory of Motion. The motion of the system under consideration is
governed by the differential equations
n
L (ahkqk +
k=l
(G)
(ahk = Ukh ,
bhkqk) = Ph(t)
(h = 1, 2, · · · , n)
bhk = bu)
where the functions Ph(t) represent the components of the given
external force acting on the system. We seek a solution qn(t) of
this system of differential equations for which the initial position
qh(O) (h = 1, 2, · · · , n) and the initial velocity <MO) (h = 1, 2, · · · , n)
are prescribed. If the external forces Pn(t) all vanish, we speak of
free motion or free vibration of the system.
The problem of motion is easily solved by the theory of quadratic
forms of Chapter I. The positive definite quadratic forms
n
G =
L ahkXhXk ,
h,k=l
n
F
=
L bhkXhXk
h,k=l
may be brought into the form
n
G
= L:~L
h=l
by means of a suitable linear transformation
n
(7)
Xh =
L Thk~k,
k-l
~h
n
=
L ThkXk·
k=l
Since U and T are definite, the values of 'X1 , 'X2 , · · · , An are positive.
It follows that in terms of new so-called normal coordinates defined
by
n
(7a)
qh
=
L
k=l
n
Thk'f/k '
1/h =
L:
fhkqk,
k=l
the kinetic and potential energies take the form
n
T = L:~L
11=1
n
u = 1.-1
L: Ak'f/L
SYSTEMS WITH FINITE NUMBER OF DEGREE S OF FREEDOM
283
the equation s of motion take the form
ijh + }..h'Ylh = N 11(t)
where
n
N1.(t) =
L P zUhu,
l=l
are normal coordinates for the external force. In these differential
equation s the unknow n functions 'Ylh are separate d.
It is often convenient to generalize the name "norma l coordin ate"
to include coordinates for which the energies have the form
n
n
T =
where Ah =
CL'iJL
h=l
u=
L: >-t11L
h=l
Xt / c = vi .
For free vibratio ns we have Nh(t) = 0, and we immedi ately obtain
the solution in the form
(vh = V}..h)
1Jh = Yh cos vh(t - 'Ph)
(8)
= ah cos vht
+ bh sin vht
(h = 1, 2, · · · , n).
Here the ah , bh or the Yh , 'Ph are arbitrar y constan ts of integrat ion.
There is a free motion for which all normal coordinates except the
h-th vanish, while the motion of the h-th normal coordinate is given
by the equatio n Tlh = Yh cos vh(t - 'Ph). This motion is called the
h-th normal mode of vibration or eigenvibration of the system with
amplitude Yh and phase 'Ph. Whenev er we speak of the h-th normal
mode of vibratio n we mean the function Tlh = cos vht , i.e. the h-th
normal mode with amplitu de 1 and phase 0. The number s Vi are
called the eigenfrequencies of the system, or the pitches, a term borrowed from acoustics. We obtain the h-th normal mode in terms of
the original coordinates qk if in the transfor mation formulas (7a) we
insert cos vht for Tlh and 0 for all the other Tli •
Every free motion of the system can be considered as a superposition of eigenvibrations with different phases and amplitudes. The
2n constan ts of integrat ion a1, a2, · · · , an, b1, b2 , · · · , bn provide
us with the number of arbitrar y paramet ers required to adjust the
solution to the prescribed initial state, i.e. to obtain a solution for
which the coordinates assume the prescribed initial values and initial
velocities.
284
V. VIBRATION AND EIGENVALUE PROBLEMS
In order to represent the solution of this initial value problem formally, let us consider the quantities qi, q2 , · · · , qn as components of
an n-dimensional vector q. If we denote the vector whose components are T1i, Tu, • · • , Tni (i = 1, 2, · · · , n) by ei, we have, by
(7a) and (8),
n
L
q(t) =
eiyi cos vi(t -
cpJ.
i=l
This representation for the general free motion leads immediately
to the relations
n
L
q(O)
eiyi
COS
(vicp,),
i=l
(9)
n
=
q(O)
L
eiyivi sin (vicpi),
i=l
where the vectors q(O), q(O) describe the given initial state.
If for simplicity we assume that the form Tis already the unit form
T = Lf=1q~, the "eigenvectors" ei constitute a complete orthogonal
system of vectors (cf. Ch. II, §1); multiplying (9) by eh we then obtain the relations
ehq(O) = Yh cos (vhcph),
ehq(O) = VhYh sin (vhcph),
from which the amplitudes Yh and the phases 'Ph can be determined
immediately.
We note that the eigenvibrations can be defined as motions of the
system for which the ratios of the coordinates qk are independent of
the time, in other words, those for which qk has the form qk = Vkg(t),
where Vk is independent of t. Substituting these expressions for qk
in equations (6) with Pi = 0, we have the equations
g(t)
n
L
- g(t).
k-1
aikVic
Since the right member is a constant independent of i and t, say X,
we obtain immediately the eigenvalue problem for G and F formulated in the equations
n
L
k=l
(bik - Xaik)vk = 0
(i = 1, 2, · · · , n).
SYSTEMS WITH FINITE NUMBER OF DEGREES OF FREEDOM
285
The connection with the preceding statements based on transformation (7a) is thus made clear.
We now consider the problem offorced motion for which the external
forces Ph(t) do not all vanish. To solve this problem it is sufficient
to find a single solution of the general differential equation
1/h
+ Ah7/h: =
Nh(t).
For the initial values 7/h(O) = 0 and 1}h(O) = 0 the solution is
(10)
7/h(t) = _ 11-
v
L'
Ah o
1
-
Nh(r) sin V°Ah (t - r) dr;
and the general forced motion arises from superpositio n of this special
motion on free motion.
Assume that the external force Nh(t) is periodic with frequency
w1., say of the form Nh(t) = ah cos wh(t - o). Then if w~ ~ Ah, formula (10) shows that the motion of the coordinates 7/h is the result
of superpositio n of a pure vibration of frequency Wh and an eigenvibration of frequency V°Ah. However, if w~ = Ah, i.e. if "resonance" occurs, the forced motion of 7/h no longer follows the rhythm
of the exciting force Nh(t). Instead, as we see easily from formula
(10),
and I 11" I no longer remains bounded as t increases.
2. General Properties of Vibrating Systems. We arrange the squares
X1 , A2 , · · · , An of the vibration numbers in the order of increasing
magnitude: Ai ::; A2 ::; · · · ::; An . By Ch. I, §4, we characterize
XP as the maximum value which can be assumed by the minimum of
h,k=l bhkXJ.Xk when the variables are subthe quadratic form F =
h,k=l ahkXhXk = 1, and then to
jected first to the condition G =
form
the
of
p - 1 auxiliary conditions
L
(11)
L
(j = 1, 2, ... , p - 1)
'vith arbitrary ah; . From this fact we obtain at once a number of
1 This solution can be interpreted in the following way: the continuous external force is repla~ed by. discontinuous instantaneous impulses acting at
time intervals At, and then the limit At -+ 0 is taken.
286
V. VIBRATION AND EIGENVALUE PROBLEMS
general theorems about the eigenvalues and the corresponding pitches
(the proofs have already been given in Ch. I, §4).
THEOREM I: The p-th overtone of a vibrating system is the highest of
the fundamental tones of the systems obtained from the given system by
imposing p arbitrarily chosen restrictions of the form (11).
THEOREM II: lfbyimposingr conditionsoftheform (11) ona system
S we obtain an "r-fold restricted" system S', then the frequencies v~ , v~ ,
· · · , v:_r of the restricted system are neither smaller than the corresponding frequencies v1, v2, · · · , Pn-r nor larger than the frequencies Vr+1 ,
Vr+2 , · · · , vn of the free system; that is,
vp
·s
v~ S vp+r
(p
= 1, 2, · · · , n - r).
THEOREM III: As the inertia increases, the pitch of the fundamental
tone and every overtone decreases (or remains the same).
Increase of inertia means change to a system with kinetic energy
T' such that T' - T is never negative, while the potential energy
remains unchanged.
THEOREM IV: If the system stiffens, the pitch of th~ fundamental tone
and every overtone increases (or remains the same).
Stiffening of the system means change to a system whose kinetic
energy is the same but whose potential energy is greater for the same
values of the coordinates.
We need hardly mention that the change in the fundamental tone
and overtones is, in each case, opposite to the change indicated in
Theorems II to IV if we remove restraining conditions, decrease ·the
masses, or relax the system, i.e. change to a system S' in relation to
which S appears stiffened.
§3. The Vibrating String
For a system with a finite number of degrees of freedom, the totality
of motions is known when the eigenvibrations (synchronous vibrations) are known. This is also true for continuous systems that can
vibrate. In such systems we shall consider free vibrations (standing
vibrations) for which the deflection u can be expressed as the product
of a factor g(t), depending only on the time, and a factor v(x), depending only on the position, the so-called form factor or vibration
.form. An arbitrary vibration phenomenon can be represented by
the superposition of eigenvibrations of this kind.
These concepts will be illustrated by a number of examples.
THE VIBRATI NG STRING
287
The deflection u (x, t)
ry
of a homogeneous string with the bounda conditio ns u(O, t) =
u(1r, t) = 0 (c.f. Ch. IV, §10) satisfies the differential equatio n
1. Free Motion of the Homoge neous String.
(12)
CUxx =
PUtt
or Uxx =
2
µ Uu
For simplicity, we choose the unit of time in such a way thatµ = 1.
We seek function s of the form u = v(x)g(t) which satisfy (12). For
these functions the differential equatio n (12) can be written in the
form
v" (x) _ g(t)
v(x) - g(t) •
The left side is indepen dent of t, the right side of x; hence both sides
must be equal to one and the same constan t, - X. From the boundary condition v(O)g(t) = v(7r)g(t) = 0 we have v(O) = v(7r) = 0.
Thus v(x) is to be determi ned by the differential equatio n
v" +Xv = 0
(13)
and the bounda ry conditions
v(O) = v(7r) = 0.
(13a)
Not all these requirem ents can be fulfilled for arbitrar y values of the
constan t X. In fact, we conclude from the form c1ev'4x + ~e-yC.>.x of
the general solution of differential equatio n (13)2 that the bounda ry
conditions can be fulfilled if and only if X = n is the square of an
integer n. The corresponding solutions have the form Vn = sin nx.
2
2
The number s 1, 2 , 3 , • • • and the function s sin x, sin 2x, · · · are
called the eigenvalues and the eigenfunctions, respectively, for the
eigenvalue problem defined by the differential equatio n (13) and the
bounda ry conditions (13a).
For g(t) we obtain g = a cos nt + b sin nt, where a and b are arbitrary constan ts. Thus for every positive integer n there is a solution
of equatio n (12) of the form sin nx(an cos nt + bn sin nt). The harmonic motions obtaine d in this way are called the eigenvibrations
of the string, the corresponding number s n = Jin the associated
eigenfrequencies. We can form more general solutions by taking
sums of the form
u = L sin nx (an cos nl + bn sin nt)
n
288
V. VIBRATIO N AND EIGENVA LUE PROBLEM S
where the summati on extends over either a finite or an infinite number of terms. In the latter case, it is sufficient to assume that the
series converges uniformly and that it may be twice differentiated
termwise with respect to each of the two variables.
Now we can fit the solution to an arbitrary initial state given by
the function u(x, O) = cp(x), Ut(X, O) = 1/;(x) if we choose the coefficients an, bn in a suitable way. For, according to the theory of
Fourier series, an, bn can be so determin ed that
00
cp(x) =
L
00
an sin nx,
n=l
1/;(x)
=
L
nbn sin nx.
n==l
The series for u(x, t) formed with the coefficients thus determin ed
represents the desired solution.1
We obtain quite analogous results if the string is subjected to other
boundary conditions. For example, suppose that the initial point
is fixed, i.e. u(O, t) = 0, and that the end point is connected elastically to its position at rest according to the equation U:c = - hu
(h > 0). 2 Then if we set u(x, t) = v(x)g(t) we obtain the following
eigenvalue problem for v(x): determin e constants A = v2 in such a
way that the differential equation v"
Av = 0 has a nontrivia l solution v satisfying the boundary conditions v(O) = 0, v'(7r)
hv(7r) = 0.
The first boundary condition shows that v must have the form sin vx.
The second boundary condition provides the transcen dental equation
h sin V7r = - .,, cos v7r. If h ~ 0, we can obtain the roots of this
equation graphically by finding in the z, v-plane the points of intersection of the line z = - (1/h)v with the successive branches of the
curve z = tan v7r. Thus we again get a sequence of eigenvalues
A1, A2, · · · with corresponding eigenfunctions sin v1x, sin v2x, · · · and
eigenvibrations (a cos v1t + b sin v1t) sin v1x, · · · . Moreover, for the
n-th eigenfrequency vn we obtain immedia tely the "asympto tic" relation limn-+oo (vn/n) = 1.
For the special case in which the end of the string is "free," that is,
in which h = 0 and thus U:c = 0, we have Vn = n - !, which gives
+
Vn
+
= sin (n - !)x.
These statement s are valid if the functions cp, i/;, cp', cp 11 , i/;' are piecewise
smooth. We can relax these assumptio ns, if we do not require a Fourier expansion of the functions and their derivative s but merely characterize them by
their Fourier coefficients.
2 Cf. Ch. IV,
§10, 2, where these boundary conditions were derived from the
existence of an additional boundary term in the potential energy.
1
289
THE VIBRATING STRING
Again we can construct a more general solution of (12) by forming an infinite series
u(x, t) =
L
sin VnX(an cos Vnt
n
+
bn sin Vnt).
By choosing the constants an , bn in a suitable way, we again hope
to make this solution satisfy an arbitrarily prescribed initial state.
For this purpose, we shall have to examine the possibility of expanding
an arbitrary function w(x) in the interval 0 ~ x ~ 7r in terms of the
functions sin vnx, the eigenfunctions of the differential equation (12),
under the boundary conditions
hv(7r)
v(O) = 0,
(14)
=
-v'(7r).
This will be done in §14. At this point one should note the orthogonality property displayed by the functions Vn = sin vnx, i.e. the property
(15)
This can be verified immediately by multiplying the equation
v':
v~Vn = 0 by Vm and the equation v::
v!vm = 0 by Vn, subtracting the resulting equations, and integrating. We obtain
+
+
(v~
- v!)
hr
Vn Vm dx
+Lr :X (v~Vm
v~vn)
-
dx = 0,
from which the orthogonality property follows by (14).
2. Forced Motion. The motion of a string with fixed end points
under the influence of an external force Q(x, t) is characterized by the
nonhomogeneous differential equation
(16)
Uxz
=
Utt -
Q(x, t).
To find the deflection u(x, t), we expand Q(x, t) at the time tin terms
of the eigenfunctions sin nx:
co
Q(x, t) =
L
Qn(t) sin nx,
n=l
Qn(t) = -2111" Q(x, t) sin nx dx;
7r
0
similarly, we suppose the desired solution expanded in the form
00
u(x, t) =
L
qn(t) sin nx,
n=l
qn(t) = 2111"
u(x, l) sin nx dx.
7r 0
290
V. VIBRATION AND EIGENVALUE PROBLEMS
Wenowattemptto satisfy the differential equation (16) by solving the
infinite sequence of ordinary differential equations
(17)
Their solutions are given by the functions
(l 7a)
qn(t)
= ~
1'
n o
Qn(t') sin n(t -
t') dt'
+ an cos nt + bn sin nt
where an, bn are arbitrary constants to be determined from the initial
conditions; Lnqn(t) sin nx is the desired solution of equation (16) if
the series converges and is twice termwise differentiable. Another
method of handling the nonhomogeneous equation will be developed
in §5, 2 and §14, 1.
For forced motion, the result can be obtained without using the
expansion theorem. Assume a solution u(x, t) exists; we attempt to
determine its Fourier coefficients qn(t) in terms of the Fourier coefficients Qn(t) of the function Q(x, t). First we multiply equation (16)
by sin nx and integrate over the fundamental domain. Then, transforming the left side by integration by parts, we obtain (17), and
thus we again arrive at formula (17a). The function u(x, t) is characterized uniquely by the expansion coefficients obtained in this way,
because the orthogonal system of functions sin nx is complete.
As in §2, we are particularly interested in the case of harmonic or
sinusoidal Qn(t):
Qn(t) = a cos wt
2
2
+ b sin wt.
Here, for w :;t. n , qn(t) can be expressed as a linear combination of
a sinusoidal function of frequency w and one of frequency n, while in
the case of resonance, i.e. w 2 = n 2 , qn(t) becomes unbounded (cf.
page 285).
The relations for the homogeneous vibrating string are typical of
those for the more general continuous vibrating system with which
we shall be concerned in the present chapter. In general, the essential points will be the determination of the normal modes, the question
of their completeness, and the validity of the expansion theorem.
Here, however, it will not be possible to refer to an existing theory
such as the theory of Fourier series. In order not to interrupt the
discussion, the completene~ proofs for the eigenfunctions will be postponed to §14.
291
THE VIBRATING STRING
3. The General N onhomogeneous String and the Sturm-Liouville
Eigenvalue Problem. We now consider the general equation of a
non-homogeneous string
(pux)x =
pUtt
where p(x) denotes the modulus of elasticity multiplied by the crosssectional area, and p(x) the mass per unit length. We seek solutions
of this equation which satisfy certain homogeneous boundary conditions. Again we try to find a solution of the form u = v(x)g(t) and
obtain
(pv')':vp
=
g:g,
which can be satisfied only if each side is equal to one and the same
constant, say -A. For v(x) \Ve then have the ordinary differential
equation
(18)
(pv')'
+ ApV
= 0,
and g satisfies the differential equation g + Ag = 0. If we set
A = 1?-it will soon be evident that negative values of A do not occur
-u takes the form
u
=
v(x)(a cos vt
+
b sin vt);
the function v(x) must be determined from the differential equation
(18) and from the boundary conditions. As in the special case of
the homogeneous string, we are faced with the eigenvalue problem of
determining the "eigenvalues" A of equation (18) for which a nontrivial
solution satisfying the boundary conditions exists. This solution is
called the eigenfunction belonging to the eigenvalue A; it is determined
1
except for an arbitrary constant factor. The following types of
boundary conditions for the initial and end points are often imposed:
1.
v(O)
= 0
and
v(1r/ = 0
(string with fixed ends)
2. hov(O) = v'(O) and -h1v(7r) = v'(7r) (elastically attached ends)
3.
4.
v'(O) = 0
v(O)
and
= v(7r) and
v'(7r) = 0
p(O)v'(O)
(free ends)
= p(7r)v'(7r);
1 The calculus of variations shows that these include the main types of
natural boundary conditions (cf. Ch. IV and VI).
292
V. VIBRATION AND EIGENVALUE PROBLEMS
condition 4 can be interpreted as a condition of periodicity if
p(O) = p(7r).
Note that, in accordance with the physical nature of the problem,
the functions p and p are positive for 0 :::; x :::; 7r; this we explicitly
assume. Furthermore, ho and h1 must be positive if the position of
the string at rest is to be a stable equilibrium. 1
The problem, as formulated, was first attacked by Sturm and
Liouville and is therefore called the Sturm-Liouville eigenvalue problem. It may be somewhat generalized by considering instead of (18)
the differential equation
(19)
(pv')' - qv
+ 'Apv
= 0,
where q is a given continuous function. The differential equations
(18) and (19) can be brought into simple normal form by transforming the independent and dependent variables, respectively. After
the transformation z = vyp, for example, equation (19) becomes
j_ (p*z')
(20)
dx
-
(q* - A)z = 0
where
p* =
e.p'
q* = - _1_.
i:...(p i:_
_1_) + ~.
dxyp
ypdx
p
If q = 0, equation (19) can be transformed into the. form
v"
+ 'Auv
=
0,
CT
= PP
by introducing in place of x the new variable
~
=
J
dx/p(x) and
then replacing ~ in turn by x.
Another important transformation of the differential equation (19)
is given by
(20a)
u=
.,Ypp
v,
t=
f tidx,
Then (19) becomes
(19a)
u" - ru +'Au= 0
where r denotes a continuous function. 2
1
2
Cf. Ch. IV, §10, 2.
Specifically, r = (f"/f)
+
(q/p) where
f = {!Pp.
293
THE VIBRATING STRING
To the eigenfunctions v and the (positive) eigenvalues A of the
differential equation (19) correspond eigenvibrations of the string
of frequency 11 = v~' represented by the functions
v(x)(a, cos 11t
+
b11 sin 11t).
Moreover, the.eigenfunctions of the Sturm-Liouville problem furnish systems of orthogonal functions; this general property follows
from the differential equation. In fact, if An, Am are two different
eigenvalues and Vn, Vm the corresponding eigenfunctions, we have, as
in subsection 1,
(Xn - Am)
'11"
10
PVnVm dx
+
1'11" d
dx (p[v:vm 0
VnV~]) dx
=
0.
Here the boundary conditions imply the vanishing of the second term
on the left, and it follows that the functions ypvi are orthogonal,
that is,
{' PVnVm dx = 0.
'We may and shall assume that these functions are normalized. In
§14 we shall show: The eigenvalues A of the differential equation (19)
for given boundary conditions, ordered with respect to magnitude, form
a denumerable sequence A1 , A2, Aa, · · · , and the corresponding system
of eigenfunctions is a complete orthogonal system. Moreover: Every
continuous function f(x) which has piecewise continuous first and second
derivatives and satisfies the boundary conditions of the eigenvalue problem can be expanded in an absolutely and uniformly convergent series
f
=
f
CnVn,
Cn =
n=l
in terms of the eigenfunctions.
sible to fit the solution
for pfVn dx
This expansion theorem makes it pos-
co
u(x, t) =
L
Vn(x)'(an cos lint
n=l
+ bn sin vnt)
to a prescribed initial state.
All eigenvalues of Sturm-Liouville problems (except those with
1
periodic boundary conditions ) are simple; i.e., no two iinearly in1
Here>. = n 2 (n = 1, 2, ···)is a double eigenvalue of y" + >.y = 0, with
.
the two eigenfunctions sin nx and cos nx.
294
V. VIBRATION AND EIGENVALUE PROBLEMS
dependent eigenfunctions v, v* can correspond to the same eigenvalue
X. Indeed, if there were two such eigenfunctions every solution of
\ (19) irrespective of boundary conditions could be expressed in the
form cv + c*v*; thus, every solution would satisfy the homogeneous
boundary conditions prescribed for the eigenfunctions. For boundary
conditions 1, 2, and 3, this leads to a contradiction since they contain
a homogeneous relation between v(O) and v' (O); on the other hand
values for v(O) and v' (0) can be arbitrarily prescribed for a solution
of (19).
The eigenvalues X are all positive for q ~ 0, ho ~ 0, hi ~ 0. We
have, in fact,
X = X 111° pv2 dx = =
f
(pv''
111°
+ qv') dx -
[(pv')'v - qv 2] dx
pv'v
1:
where the right-hand side is positive in the case of boundary conditions 1 to 4. An eigenvalue must be positive if the corresponding eigenfunction is to represent a periodic vibration. Whenever an eigenvalue
is negative an aperiodic motion occurs instead of the corresponding
normal mode. We shall see later that this can happen only a finite
1
number of times, even if q is negative.
The forced motion of the string could be analyzed in the same way
as the homogeneous string in subsection 2. But if the external force
Q(x, t) in the nonhomogeneous equation (pux)x = pUet - Q(x, t) is
periodic of the form Q(x, t) = cp(x)eiwt, 2 the following procedure3 is
ordinarily used: We write the solution u in the form u = v(x)eiwt and
obtain the nonhomogeneous equation for v(x), associated with (18),
(pv')'
+ Xpv
= -cp(x)
To determine the expansion coefficients
'Yn
=
fr
pVVn
dx
Cf. Ch. VI, §2.
As always, the use of complex quantities expresses in a simple way that
the real and imaginary parts in equations and solutions should be considered
·
separately.
a In this connection cf. the algebraic analogue in Ch. I, §3, 6.
i
2
295
THE VIBRATING ROD
of the solution v(x) we multiply our differential equation by vn(x),
integrate over the fundamental domain, transform the first term using
integration by parts, and consider the differential equation for Vn •
This implies immediately 'Yn(A - An) = -en ; hence
'Yn
Cn
= ----
where
A - An
This method becomes meaningless in the case of resonance, that
is, when the frequency v'x = w of the external force equals one of the
eigenfrequencies VX: = wn and the corresponding coefficient Cn is
different from zero.
The case of a general external force Q(x, t) can be reduced to the
special case just considered by spectrally decomposing the force
Q(x, t) as a function oft, with the help of a Fourier series or a Fourier
integral (cf. Ch. II, §§5 and 6).
§4. The Vibrating Rod
In the case of the differential equation
4
2
ox'
at''
ou+~=O
of the transverse vibrations of a homogeneous rod, we again consider
the eigenvibrations (we limit our discussion to the homogeneous rod
for the sake of brevity, since the nonhomogeneous rod offers no aspects
beyond those already treated in §3). As before we write u = v(x)g(t)
and obtain
·
vv"" = gg =
-x,
that is,
(21)
v"" - AV
=
0,
where the constant A must be determined in such a way that four prescribed homogeneous boundary conditions are satisfied at the ends
of the rod. Again we assume that the rest position of the rod is the
interval 0 s; x ~ 7r. We distinguish various types of boundary
conditions (cf. Ch. IV, §IO):
1. v"(x) = v'"(x) = 0
for x = 0 and x =
7r
(free ends)
v(x) = v"(x) = 0
for x = 0 and x =
7r
(supported ends)
2.
296
V. VIBRATION AND EIGENVALUE PROBLEMS
3.
v(x) =
v'(x)
= 0 for x = 0 and·x
4. v'(x) = v"'(x) = 0
5.
v(O) = v(7r),
v"(O) = v"(?r),
=
7r
(clamped ends)
for x = 0 and x = 7r
v'(O) =
v'(?r),}
(periodicity).
v'"(O) = v'"(?r)
In all these cases the eigenfunctions and eigenvalues can be given
explicitly, since we know the general solution of the first differential
1
equation (21). That is, assuming A ~ 0 and~ = 11 we have
or
v = ~1 cos .vx
+ ~2 sin vx + ~3 cosh vx + ~4 sinh vx.
For A = 0 the general solution degenerates into a polynomial of the
3
2
third degree v = ~ 1 + ~2x + ~aX + ~4X •
Any suitable set of four homogeneous boundary conditions to
which the rod is subjected yields four homogeneous equations of the
form L: aik~k = 0 (i = 1, 2, 3, 4) in the four quantities ~1, ~2, ~3,
~4 ; to obtain a nontrivial solution, we set the determinant I aik I = 0,
which means that we have a transcendental equation for the eigenvalues A. Every root of this equation furnishes one or more eigenfunctions, which can be taken as normalized. In particular, for the
rod with both ends free, the transcendental equation for v becomes
cosh vr cos
117r
=
1.
With the exception of the function ~1 + ~2x belonging to the double
eigenvalue A = 0, the associated eigenfunctions, not yet normalized,
are
v = (sin vr - sinh vr)(cos vx
+ cosh vx)
- (cos 11?r - cosh 117r)(sin vx
+ sinh vx).
The solutions for the rod clamped at both ends are obtained by twice
differentiating the above solutions for the free rod, since the resulting
functions satisfy both the differential equation and the boundary
conditions for the clamped rod. Moreover, every eigenfunction for
the clamped rod can be obtained in this way; i.e., each such eigen1
It is easily proved, as in §3, that X ~ 0.
297
THE VIBRATING MEMBRANE
function corresponds to an eigenfunction of the free rod, as we may
show by integrating twice and choosing appropriate constants of
integration. The eigenvalues are the positive solutions of the same
transcendental equation; the eigenfunctions are given by the expression
v = (sin v7r - sinh v~)(-cos vx
+ cosh vx)
- (cos V7r - cosh v7r)(-sin vx
+
sinh vx).
The problem of the rod differs from that of the vibrating string in
that multiple eigenvalues can occur. For example, in the problem of
a rod free at both rends, the two normalized linearly independent
eigenfunctions v = 1/y; and v = xy3/7r3 both correspond to
the eigenvalue zero. However, in the case of the clamped rod, we
lose these two eigenfunctions and their associated eigenvalue A = 0
by differentiating twice.
In all the cases considered, the eigenfunctions of equation (21)
form an orthogonal system. Again we let An, Am be two distinct
eigenvalues and Vn , Vm the corresponding eigenfunctions. Integrating by parts twice we obtain
where the right-hand side vanishes because of the homogeneous
boundary conditions. We shall show later (§14) that the system of
ei,genfunctions is complete, and that arbitrary functions possessing continuous first and second and piecewise continuous third and fourth
derivatives may be expanded in terms of these eigenfunctions.
The remainder of the theory of the transverse motions of a rod is
quite analogous to the theory of the string, and need not be further
developed here.
§5. The Vibrating Membrane
1. General Eigenvalue Problem for the Homogeneous Membrane.
The differential equation Au = Utt of a vibrating homogenous membrane also leads to an eigenvalue problem, but the eigenvalue problem pertains to a partial differential equation. Suppose that the
membrane at rest occupies a domain G in the x, y-plane with the
boundary r; the remaining assumptions and _notations are the same
as in Ch. IV, §10, 3. We consider first the simplest boundary condi-
v.
298
VIBRATION AND EIGENVALUE PROBLEMS
tion, u
O; i.e., we consider a membrane, stretched in a given frame
If we set u(x, y, t) = v(x, y)g(t) we obtain at once the relation
q=
Av=
v
g
-A
between the functions v(x, y) and g(t). From this it follows that A
is a constant, which we set equal to ·1l. The function v(x, y) and the
constant A are found by solving the following eigenvalue problem:
Determine the parameter A as an "eigenvalue" in such a way that
there exists a not identically vanishing continuous function v(x, y)
in G
r which has continuous first and second derivatives in G,
satisfies the differential equation
+
(22)
AV+ AV= 0,
and vanishes on the boundary; such a function v may be assumed
normalized. The eigenvalues A must be positive, a condition we have
alreudy expressed by writing A = v2 • This fact follows from the
equation
Ji
(v!
+
v;) dx dy
= -
Ji
vAv dx dy
= A
Ji
2
v dx dy,
obtained by multiplying equation (22) by v and applying Green's
formula (cf. §1). Accordingly, the general solution of the equation
g/g = -A = -v2 has the form g = a cos vt + b sin vt; it is a
periodic function of the time. The solution
u(x, y, t)
= v(x, y)(a cos vt
+
b sin vt)
of the equation of vibration then corresponds to an eigenvibration of
frequency v
=
VA.
The existence of eigenvibrations, or more precisely the existence of
a denumerably infinite sequence of eigenvalues Ai , A2 , · · · and
wrresponding eigenfunctions v1(x, y), v2(x, y), · · · , as well as the
associated theorems on completeness and on expansion~ in series, will
be demonstrated in §14. At this point, we note the following orthogonality property: Any two eigenfunctions vi , vk that correspond to
two distinct eigenvalues Ai and Ak are orthogonal; i.e.,
Ji
V(l'k
dx dy = 0.
299
THE VIBRATING MEMBRANE
The proof, following the previous pattern, is based on the formula
(Ai - Ak)
fl
ViVk dx dy
= -
fl (vk~Vi
- VillVk) dx dy
= 0,
derived from (22) with the aid of Green's formula and of the boundary
condition u = 0.
The motion of a freely vibrating stretched membrane with arbitrarily prescribed initial conditions given by u(x, y, 0) = f(x, y),
Ut(X, y, O) = g(x, y) can again be represented in terms of the
eigenfunctions as a series expansion
DO
u(x, y, t) =
(23)
L Vn(X, y)(an cos Vnt +
n=l
bn sin Vnt),
in which the coefficients an , bn are determined by the initial conditions
and are given by
an =
fl
f(x, y)v 11 (x, y) ax dy,
To derive this result we assume that the series (23) converges and
may be differentiated termwise a sufficient number of times.
The situation is similar if the prescribed boundary condition is of
the form au/ an = - uu, which describes an elastic attachment of
the membrane to its boundary. Here u denotes a positive quantity,
in general a function of position on the boundary. The eigenvalue
problem can be formulated precisely as above, and the solution of the
initial value problem is obtained in a similar way by means of the
· expansion theorem. Here, too, the eigenvalues A are positive numbers. To show this we multiply equation (22) by v and integrate
over G. Using Green's formula (§1) and keeping in mind the
avjan = 0 we obtain at once
boundary condition uv
+
A= A
Jl
2
v dx dy =
vx
Ji
(v!
+
v!) dx dy
+
£
2
uv ds.
are the frequencies of the corresponding
The numbers v =
eigenvibrations. Again, the eigenfunctions associated with different
eigenvalues Ai , Ak are orthogonal.
It is interesting to consider the limiting case, u = 0, of the "free
membrane" (which can be physically realized by a suitable mechanism). Although under all other boundary conditions every eigen-
v.
3()()
VIBRATION AND EIGENVALUE PROBLEMS
value is positive, in this case the eigenvalue X = 0 exists and has the
associated eigenfunction v(x, y) = const.
2. Forced Motion. The forced motion of a membrane, described
by the differential equation
(24)
Au =
Utt -
Q(x, y, t),
can be treated by the method of §3, 2. We can represent both the
external force Q(x, y, t) and the desired function u(x, y, t) as the
series expansions Q(x, y, t) = 2:";:=1 qn(t)vn(x, y) and u(x, y, t) =
Z::-1 Un(t)vn(x, y) in terms of the eigenfunctions vn(x, y) of the
freely vibrating membrane; we now determine the coefficients un(t)
from the differential equations
Alternatively, assuming a periodic external force, we can expand Q
and u into a Fourier series. We solve equation (24) for the case
of a simply periodic force of the form 'P(x, y)eiwt and obtain a function
v(x, y)eiwt. It is seen at once that the function v(x, y) satisfies the
differential equation
(25)
Av
+ Xv
= 'P(x, y)
which can be solved by expanding v(x, y) in an eigenfunction series
of the form v = 2:";:=1'YnVn . The coefficients are determined as on
page 294, and are given by
Cn
'Yn =A -
where
Cn
=
Ji
'()Vn
An
dx dy.
3. Nodal Lines. In the case of a string or a rod, the points at
which an eigenfunction Vn vanishes are of particular interest: these
points are called the "nodal points" of the associated eigenvibration
vneivnt. In the case of eigenvibrations of a membrane, we consider nodal lines, i.e. the curves vn(x, y) = 0. These nodal lines are the curves
along which the membrane remains at rest during eigenvibrations.
Various examples will be discussed presently (see also Ch. VI, §6).
4. ~ectangular Membrane. The eigenvalue problem for a membrane depends on the choice of a domain and leads to many special
questions. We shall discuss some of these questions with reference
THE VIBRATING MEMBRANE
301
s s
a,
x
to a rectangular membrane which covers the domain G(O
b). The eigenvalues and eigenfunctions which correspond,
y
0
respectively, to the boundary conditions u = 0 and au/an = 0 are
known explicitly. In the first case the eigenvalues are the numbers
2
2
2
m2/b 2 ) with n, m = 1, 2, · · · ; the associated (not
A = 7r (n /a
normalized) eigenfunctions are the products sin (n?rx/a) sin (m7ry/b).
2
2
2
In the second case the eigenvalues are the numbers A = 7r (n /a
2
2
m /b ) with n, m = 0, 1, 2, · · · and the corresponding eigenfunctions
are cos (n?rx/a) cos (m7ry/b). Here A = 0 is also an eigenvalue, as
has been previously emphasized. (Note that the eigenfunctions for
the stretched rectangular membrane can be obtained from those for
the free membrane by differentiating with respect to x and y.)
Thus we have obtained all the eigenfunctions for the problem.
This can be seen from the fact that sin (n?rx/a) sin (m?rY/b) form
a complete orthogonal system of functions in G. For, if there
were another eigenfunction corresponding to a new eigenvalue, it would
be orthogonal to each of the eigenfunctions already found. If, on
the other hand, it corresponded to one of the above eigenvalues, say
Ai, we could subtract a suitable linear combination of the eigenfunctions associated with Ai and obtain an eigenfunction orthogonal
to these (and therefore to all) eigenfunctions. Hence it would be
orthogonal to the complete system. of eigenfunctions given by the
product of sines; any new eigenfunction must, therefore, vanish
identically.
Theorems such as the expansion theorems can be reduced to the
properties of Fourier series in two variables discussed in Chapter II.
The present example shows that in the case of a membrane multiple
eigenvalues may very well occur. They always occur if the ratio
a:b of the edges of our rectangle is rational, since the equation
2
2
2
2
2
2
n /a + m /b = n' /a + m' 2/b 2 may have the nontrivial integral solutions m, n, m', n'. If, for example, G represents the
square a = b = 'Ir, then m' = n, n' = m is a solution of this kind and
the eigenfunctions which correspond to the boundary condition u = 0
are sin mx sin ny and sin nx sin my. The question of the multiplicity of an eigenvalue in the case of the square domain is thus reduced to the number-theoretic problem: in how many ways can a
1
2
2
m2 of two squares?
number v be represented as the sum v2 = n
s s
+
+
+
1
In this connection cf. Dirichlet-Dedekind, Vorlesungen Uber Zahlentheorie, §68, pp. 164-166, 4th ed., F. Vieweg und Sohn, Braunschweig, 1894.
302
V. VIBRATION AND EIGENVALUE PROBLEMS
The nodal lines for the eigenfrinctions sin nx sin my are simply
lines parallel to the coordinate axes. However, in the case of multiple eigenvalues many other nodal lines can occur. The zeros of the
{3 sin nx sin my for a square are one exfunction a sin mx sin ny
+
~S[SJ~
U,4+f ./fu4,
u,4
U14
+JfU41
U,4 + U41
Figure 3. Nodal lines for a square membrane.
ample. The above diagrams1 illustrate a number of characteristic
types. Here Umn denotes the function sin mx sin ny.
5. Circular Membrane. Bessel Functions. The circular membrane can also be treated explicitly; we assume its radius to be 1.
The differential equation for its eigenvalue problem, in polar coordinates, takes the form
(26)
1
Vrr
+ -1r Vr + r21 VBll + AV
= 0,
A number are taken from the book by Pockels referred to in the bibliog-
raphy.
303
THE VIBRATING MEMBRANE
derived in Ch. IV, §8, 2. If, as before, we consider the case of a
stretched membrane, we have the boundary condition v(l, 8) = 0.
If we attempt to solve equation (26) by setting v(r, 8) = f(r)h(8),
we are led at once to the relation
+ ~f'(r) +
2
r (f"(r)
h"(8)
Xf(r))
= - h(8) = const. = c.
f(r)
Since v(r, 8) and hence h(8) must be periodic functions of 8 with
period 21r-otherwise v would not be single-valued-it follows that
2
c has a value c = n , where n is any non-negative integer. We thus
obtain
h(8) = a cos n8
=
and, for f(r)
+ b sin n8
y, the differential equation
r2y''
(27)
+ ry' + (r X 2
2
n )y
=
0
follows. The problem is to find eigenvalues X for which there exists
a solution of this differential equation, continuous for r = 0, which
also satisfies the boundary condition f(l) = 0. If we make the
transformation rVX" = p (}.. ~ 0) or kr = p, setting X = k2, equation
(27) takes the form
2
ay
(28)
dp2
dy
+(1 + p1 dp
n
2
p2
)
y = 0.
The solutions of this Bessel equation, the so-called Bessel functions,
play a particularly important role in analysis and mathematical
be the object of more detailed study in Chapter VII.
physics and
At present, it should be pointed out that by writing y as a power
ampm we obtain for (28) the solution
series y(p) =
will
L::-o
y(p) = J n(p)
-
pn
[
- 2 n! l
11
1
-
p2
2(2n + 2)
}
p4
+ 2·4(2n + 2)(2n + 4)
-
.. ·
which is called the Bessel function of n-th order. The series converges for every value of p, as is shown by simple criteria; that is,
304
V. VIBRATION AND EIGENVALUE PROBLEMS
the Bessel functions Jn(P) are entire transcendental functions.
particular, for n = 0 we have the series expansion
2
Jo(p) = 1 -
~
4
In
6
+ 2:42 - 22~252 +
We note the relation
(29)
which follows immediately from the series expansion.
We can write the solutions of (27) in the form
Yn = Jn(kr)
(30)
where the constant k is to be determined by the boundary condition
Yn(I) = 0, that is, by the condition J n(k) = 0. Thus the eigenvalues
A = k 2 of (27) are the squares of the zeros of the Bessel functions.
As regards the existence of these zeros, we shall show later that each
function Jn ha.s, in fact, infinitely many real zeros, which we shall
denote by kn,m (m = 1, 2, 3, · · · ). Using this notation we can write
the eigenfunctions in the form
J n(kn,mr)(a cos n(}
+ {3 sin n8).
The constants a, {3 in this expression are still arbitrary, which indicates that, with the exception of the eigenfunctions belonging to
n = 0, all the eigenvalues are at least double since they have the associated linearly independent eigenfunctions Jn cos n(} and Jn sin n(}.
The nodal curves for these eigenfunctions are circles p = const. and
radial lines 8 = const., respectively. The eigenvibrations are represented by
u = Jn(kn,mr)(a cos nfJ
+
{3 sin nfJ)(a cos kn.mt+ b sin kn.mt).
If we assume the more general boundary condition au/or = - cru,
with constant u, almost all of the above considerations remain unchanged. However, the boundary condition from which the eigenvalues ar~ determined takes a somewhat different form, namely
kJ~(k) = -crJ11.(k).
/
The functions J n(kn,mr) are the only eigenfunctions for the membrane. This can be proved by noting that every eigenfunction v is
a periodic function of 8, with period 27r, having continuous deriva-
THE VIBHA'l'ING
tives up to the second order.
series
305
M.F~MBRANE
Hence v can be expanded in a Fourier
00
v(r, fl) =
L:
f ,.(r)e',.8•
Substituting this series in equation (26), we see at once that each
individual term f ,.(r)e'n9 satisfies the differential equation.
From the general expansion theorem it follows that a function
w(r, fl) vanishing on the boundary and continuous together with its
derivatives up to the second order in the interior of the circle, can be
expanded in an absolutely and uniformly convergent series of the
form
t
w(r, fl) =
a,.mJ,.(k,.,m r) cosn(fl -
fl,.,m).
n,m=O
For. example, if w does not depend on fl, this implies that an arbitrary
function of r, which vanishes for r = 1 and has continuous derivatives
up to the second order in the interval 0 ~ r ~ 1, can be expanded
in this interval in terms of the Bessel functions Jo(ko,mr).
The orthogonality relation
f
rJ ,.(k,.,i r)J ,.(k,.,; r) dr = 0
(i
~
j)
for the Bessel functions, or the functions (30), is obtained from the
general orthogonality relation for the eigenfunctions of the equation
of a membrane by integrating with respect to fl. This relation may
also be derived directly from equation (27) using the now familiar
method. Furthermore, it is clear that orthogonality is preserved
even under the more general boundary condition kJ~(k) = -uJ,.(k).
To normalize the functions J n(k,.,mr) we shall use relation
(31)
2
f J~(kr)r
dr = J'!(k),
which is proved as follows: We multiply the differential equation
for J ,.(kr) = y, i.e. the equation
(ry')'
+ (rk
2
-
:)y =
0,
306
V. VIBRATION AND EIGENVALUE PROBLEMS
by ry', and integrate from 0 to r.
the relation
2
2k
{
Integrating by parts this leads to
2
ry dr = (ry')
2
+ (r k2 2
from which equation (31) follows for r
Thus the functions
n2)y2,
= 1, since y(l)
=
Jn(k) = 0.
V2
J~(kn,m) J nCkn,m r)
are the normalized eigenfunctions for equation (27). For further
results in the theory of Bessel functions the reader is referred to
Chapter VII and to monographs on the subject.
6. Nonhomogeneous Membrane. The generalized differential equation of the nonhomogeneous membrane,
p!:J..u
+ P:eU:e + pyUy -
qu = p(x, y)utt,
in which p and p are positive in G, leads to an eigenvalue problem
analogous to the general Sturm-Liouville problem of §3. This is
the problem of determining the values of }.. for which the differential
equation
L[v]
+ Xpv
= p!:J..v
+ P:eV:e + pyVy -
qv
+ Xpv
=
0
possesses a normalized solution satisfying prescribed homogeneous
boundary conditions. With the aid of Green's formula
rr
JJa
(v2L[v1] -
V1L[v2D dx dy =
1p(v2 an
avi - V1 av2) ds = 0
r
an
[page 280, formula (5a)], we obtain the orthogonality relation
fl
PViV;
dx dy = 0
for the eigenfunctions Vi , v; corresponding to distinct eigenvalues
Ai, "A;. We should like to determine the eigenfunctions in such a
way that the functions yp Vi form an orthonormal system, i.e. so that
JJ. pv~ dx dy =
(}
1.
.
In §14 the existence of the eigenvalues and the completeness and expansion theorems will be treated from a general point of view. These
307
THE VIBRATING PLATE
theorems assert that a function f (x, y) which satisfies the boundary
conditions and has continuous derivatives of first and second order
can be expanded: f
=
~':-1 CnVn(x, y),
with Cn =
Jl
pfvn dx dy.
§6. The Vibrating Plate
1. General Remarks. For the differential equation
A!:iu
+ uu
= 0
of the homogeneous vibrating plate, we obtain the eigenvalue equation
AAv - Xv = 0,
(32)
writing u = v(x, y)g(t) with g(t) = ae±ivt or g(t) = a cos vt + b sin vt,
2
and X = v • As boundary conditions we consider for example
u
=
0,
(clamped plate).
dU = O·
'
<Jn
v
i.e.,
=
0,
dV
=
O
<Jn
The orthogonality of two eigenfunctions corresponding
to two distinct eigenvalues is proved by the same method as before
using Green's formula (§1). The only essential difference is that now
two homogeneous boundary conditions characterize the eigenvalue
problem; this corresponds to the fact that the partial differential
equation under consideration is of fourth order.
2. Circular Boundary. The problem for the plate is analytically
more difficult than that for the membrane. It is not possible, for
example, to treat the case of the rectangular boundary in terms of
functions known explicitly. The only boundary which has been
r, fJ
explicitly treated is the circle. Introducing polar coordinates
4
can
we
,
k
=
A
set
we
If
we are once more led to Bessel functions.
form:
symbolic
write the differential equation in
4
(AA - k )v
=
0
or
2
(A - k )(A
+ k )v = O;
2
the operator A is given by
iJ2
A =
ar2
+
1 iJ
r iJr
+
1 iJ2
T2 iJ(J2 •
308
V. VIBRATION AND EIGENVALUE PROBLEMS
If we suppose that v is expanded in a Fourier series,
V
=
co
L:
Yn(r)einl,
n=-co
then each term of the series must satisfy the differential equation;
in other words, Yn must be a solution of
_e)(~+!~_n2+e) -o
( ~+!!!__n2
dr
r dr
r
dr
r dr
r
y ·
2
2
2
2
We can find two linearly independent solutions of this differential
equation which are regular for r = 0: J n(kr) and J n(ikr), where
i = V-=1. Thus the function
v(r, 0) = Jn(kr)(a1 cos nO
is a solution of (32).
+ ,b1 sin nO)
+ J n(ikr)(~ cos nO + b2 sin nO)
To satisfy the boundary conditions v(l, 0) = 0,
Vr(l, 0) = 0, we must have
+ J n(ik)~ = 0,
J:(k)a1 + iJ:(ik)~ = 0,
J n(k)a1
+ J n(ik)b2 =
J:(k)b1 + iJ:(ik)b2 =
J n(k)b1
0,
0.
Hence, the eigenfrequency k satisfies the transcendental equation
J:(k)
iJ:(ik)
J n(k) = J n(ik)
in which the imaginary unit i no longer really occurs, as is shown by
the series expansions of the Bessel functions. For details the reader
is again referred to the literature.
§7. General Remarks on the Eigenfunction Method
Let us now examine the essential features of the method illustrated
by the foregoing examples.
1. Vibration and Equilibrium Problems. Let G be a domain of the
independent space variables x, · · · , i.e. an interval on the x-axis or a
domain either in the x, y-plane or in x, y, z-space which has a
piecewise smooth boundary r. Suppose the state of a continuum
filling G is characterized by a function u(x, · · · ; t) which vanishes
identically if the system is in stable equilibrium. Let L[u] be a selfadjoint linear differential expression in the independent variables
GENERAL REMARKS ON EIGENFUNCTION METHOD
309
x, · · · , defined in G, which arises from the variation of the system's
potential energy. Let p(x, · · · ) represent the mass density at any
point of G and let Q(x, · · · ; t) represent a given external force. We
wish to find a solution of the differential equation
(33)
L[u]
=
pUu -
Q
which satisfies prescribed homogeneous time-independe nt boundary
conditions on the boundary r of G, and which corresponds to a prescribed initial state defined by
u(x, · · · ; 0) = ~(x, · · · ),
Ut(x, · · · ; O) = lf;(x, · · · ).
All occurring derivatives are assumed to be continuous in G.
The case of equilibrium corresponds to the special assumption
that all functions considered are independent of t (and no initial
conditions are given). Instead of a mixed initial-boundar y value
problem for vibration, we then obtain a boundary value problem for
equiFbrium.
Among the free motions characterized as the solutions of the homogeneous differential equation
(33a)
L[u] = pUtt
which satisfy the prescribed homogeneous boundary conditions, we
distinguish the eigenvibrations by requiring synchronism: u =
v(x, · · · )g(t). Each such eigenvibration is associated with a constant value }.., an eigenvalue, with the property g + }..g = 0; hence
g(t) = a cos
v>: t + b sin V>:t,
and
(34)
L[v]
+ }..pv
=
0,
where v must satisfy the boundary conditions for u given above. The
eigenvalue problem is to determine values of the parameter}.. (eigenvalues) for which the homogeneous differential equation (34) has
nontrivial solutions (eigenfunctions) under the prescribed boundary
conditions. The vibration satisfying the original equation (33a)
is then represented by
u = (a cos V>:t
+
b sin y1>:t) v(x, · · · ).
In the case of a finite domain G the following statements are in
general true: The eigenvalues }.. form a denumerably infinite sequence
310
V. VIBRATION AND EIGENVALUE PROBLEMS
There exists a system of associated eigenfunctions V1 ,
v2, · · · which is complete in the sense of Ch. II, §1 and which satisfies
the orthogonality relations1
A1 , A2 , • • • •
l
(i F- k),
dT = 0
PViVk
l pV~
dT = 1.
Moreover, the expansion {heorem holds: Every function w with continuous L[w] which satisfies the prescribed homogeneous boundary
conditions may be expanded in an absolutely and uniformly convergent series in the eigenfunctions:
co
w =
E c.,,v.,,,
11=1
C.,, =
l
pWV.,,
dT.
On the basis of these properties-which must be proved for each
problem (cf. §14)-we obtain an infinite sequence of eigenvibrations
(a. cos vX°:t
b. sin vX°:t)v.(x, · · · ). From these eigenvibrations
we obtain the solutions of the initial value problem for (33a) by superposition if we choose the constants a.,, and b. suitably:
+
a"
=
l
p<pVv dT ,
b,,
=
.Jx: l
p'lfv,, dT.
For the nonhomogeneous equation (33) with homogeneous
boundary conditions-as we saw in §§I, 2, there is no loss of generality
in assuming homogeneous boundary conditions for a nonhomogeneous
equation-the solution u(x, · · · ; t) is found by determining its expansion coefficients 'Y,(t) with respect to the v,, . To this end we
multiply equation (33) by v,,, integrate over G, and transform the
left side by Green's formula (5a), §I, taking the boundary conditions
into consideration; using (34) we obtain
f,,
+ Av'Yv = Q.(t),
where Q,,(t) is the given expansion coefficient of Q(x, · · · ; t)p- i with
respect to the v,,. A special solution of this equation for 'Y11 is given by
1
'Y• = ~
1
The notation
domain G.
L
1' Qh)
0
sin~(t - T) dT.
f dr means the integral of the function f(x, · · ·) over the
GENERAL REMARKS ON EIGENFUNC TION METHOD
311
The function formed using these expansion coefficients is a particular
solution of (33), and all other solutions are obtained by adding a
solution of (33a). Thus the initial value problem under consideration is reduced to the problem of solving the homogeneous· equation
(33a).
In terms of eigenfunctions we can also solve the equilibrium problem, i.e. the boundary value problem for the differential equation
L[u] = -Q(x, · · · )
with homogeneous boundary conditions. In the same way we obtain
the equation }...,,-y.,, = Q.,, for the expansion coefficients of the desired
solution u with respect to the v.,, , i.e.
-y.,,
= :.
i
Qv.,, dr.
Hence, by the expansion theorem, the solution is given by
u =
t
P=l
v.,,
A.,,
r Qv.,, dr.
JG
If we could interchang e summatio n and integration in this expression
we would arrive at a function
K(x,·. ·; ~,· .. ) =
f, v.,,(x,-. ·)v.,,(~,-. ·)
A.,,
P=l
such that the solution of the boundary value problem could be written
in the form
u(x, · · ·) =
i Q(~,
· · · )K(x,- · ·; ~,. · ·) dr,
where the integration is to be carried out with respect to the variables
~' · · · . This function K, "Green's function" of L[u], will be characterized in §14 in a quite different manner and will form the basis for
a more detailed investigati on reaching beyond the formal structure
of the present method.
2. Heat Conduction and Eigenvalue Problems. The theory of heat
conduction also leads to eigenvalue problems. If the units of time
and length are suitably chosen, the differential equation of heat
conduction takes the form
L[u] =
Ut,
312
V. VI:aRATION AND EIGENVALUE PROBLEMS
where u denotes the temperature as a function of position (x, y, z)
and time t. The radiation of heat from a homogeneous body G with
surface r into an infinite medium of constant temperature zero is
characterized at the surface r by a boundary condition of the form
au
an+uu=O
where u is a positive physical constant. This condition states that
the rate of change of temperature in the direction of the inner normal
is proportional to the jump in temperature from the exterior to the
interior of the body. We seek a solution of the equation of heat
conduction which satisfies this boundary condition and prescribed
initial conditions at the time t = 0.
We write u in the form u = v(x, y, z)g(t) and obtain at once the
equation
L[v] = ~ = -A.
v
g
We thus have the following eigenvalue problem for v: L[v] +Xv = 0
in G and av I an
uv = 0 on the surface r; for a given eigenvalue A
and its eigenfunction v the corresponding solution of the differential
equation has the form
+
u =ave-u
By the eigenfunction expansion theorem we can again make the solution satisfy a given initial state; then u(x, y, z; O) equals an arbitrarily
prescribed function cp(x, y, z), which is continuous in G together with
its derivatives of first and second order and satisfies the boundary
condition. For, if the normalized eigenfunctions v1, v2, · · · and
their associated eigenvalues X1, X2, · · · form a complete system, the
desired solution is given by the formula
00
u(x, y, z; t)
=
L CnVn(x, y, z)e-X,,,t
n=l
where Cn '=
!Ji
cpVn dx dy dz.
Incidentally, the positive character of the eigenvalues }.. implies
that the solution u(x, y, z; t) approaches zero asymptotically as t increases as must be expected from the physical meaning of the problem.
VIBRATIONS OF THREE-DIMENSI ONAL CONTINUA
313
If, in place of the homogeneous equation of heat conduction, we
consider the inhomogeneous equation
L[u] =
Ut -
Q(x, y, z)
(in which we assume that the given function Q does not depend on
the time) and impose the same boundary conditions on u as above,
we obtain by our general method a solution u(x, y, z; t) which, as
t -+- co , goes over into the solution of the corresponding boundary
value problem for the equation
L[u] = -Q(x, y, z).
§8. Vibrations of Three-dimens ional Continua. Separation
of Variables
In the theory of vibrations of three-dimension al continua, e.g. in
acoustics, elasticity, or electrodynamic s, homogeneous boundary value
problems arise for the equation
dU
=
Utt
where Llu is the potential expression in three variables; we are led to
eigenvalue problems of the form
Au+ Xu= 0
with corresponding homogeneous boundary conditions.
It often happens that the particular shape of the fundamental domain permits a further separation of variables for the solutions of this
eigenvalue problem and thus leads to new eigenvalue problems involving fewer independent variables.
An example is given by a cylindrical domain erected over the
domain G of the x, y-plane and bounded by the planes z = 0 and
z = 7r. Let us take u = 0 as a boundary condition. By means of
the substitution u = f(z)v(x, y) this problem can be reduced immediately to the corresponding problem for the plane domain G,
and we obtain
j"
- -
f
Av
= -
v
+A=
k = const.,
f =sin v'kz
2
2
2
where k = 1 , 22, 3 , • • • • The equation for vis Av+ (X - n )v = O;
here the eigenvalues differ from those for the plane domain G only
314
V. VIBRATION AND EIGENVALUE PROBLEMS
by the terms -n2 , and the eigenfunctions are identical with those
for the plane domain G.
From the completeness of the system of eigenfunctions, it follows,
as in previous cases, that we thus obtain all the eigenfunctions of the
cylinder under the given boundary conditions.
If the cylinder is erected over a rectangular domain, we have a
rectangular parallelepiped; e.g., for the cube 0 ~ x, y, z ~ 7r, we
2
2
obtain the eigenvalues Z2
m
n (l, m, n == 1, 2, 3, · · · ) and the
eigenfunctions sin lx sin my sin nz.
As a further example, we consider the vibration equation for the
spherical domain x 2
y 2 z2 ~ 1 of radius 1. If we introduce polar
coordinates r, 8, <p, the vibration equation takes the form (cf. Ch.
IV, §8, 2)
+
+
+ +
.1u +Xu
J
= r' ~n 8 [:r (r'u, sin 8) + :,,, (~8 ) + ~ (u, sin 8) + hu = 0.
If we look for a solution of the form u = Y(8, <p)f(r), we obtain
2
(r f')'
f
+
2
>..r f = -
-[!___( !") +iao (Ye sin o)J = k '
-~
Y sm 0 iJ<p sm 0
where k is a constant. The value of k must be determined in such
a way that the differential equation
.1*Y
+ kY = sm
~[i_(
!") +!.-ao (Ye sin o)J + kY =
0 iJ<p sm 0
0
has a solution which is continuous on the entire surface of the sphere.
Therefore, this solution must be periodic in <p, of period 271"', and regular
at (} = 0 and (} = 7r (i.e., at both these points it must approach a limit
independent of <p). We shall see in Ch. VII, §5 that this requirement·
is satisfied only for the values k = n(n
1) (n = 0, 1, 2, · · · ) ; in
this case the solutions are the spherical harmonics Yn(8, 'P) (see also
§9). We have for f(r) the equation
+
+
+
(r2/')' - n(n
l)f
Xr~f = O;
the solutions of this equation which are regular at r
functions
Jn+t(Vxr)
V'r
0 are the
BOUNDARY VALUE PROBLEM OF POTENTIAL THEORY
315
(cf. §5 and §10). We now have to determine the parameter A from
the boundary condition. For example, if the boundary condition
is given by u = 0, then }.. is determined by the equation J n+l ( 0) =
O; denoting the roots of this equation by An,1, · · · , we obtain solutions of the form u = Yn(O, cp)Sn (V'An,hr) for the boundary value
problem. In Ch. VII, §5 we shall prove that these solutions constitute a complete orthogonal system, and hence represent all the
eigenfunctions and eigenvalues for our differential equation problem.
§9. Eigenfunctions and the Boundary Value Problem
of Potential Theory
The boundary value problem of potential theory is the problem of
determining a function u which satisfies the differential equation
Au = 0 in the interior of a domain G, and assumes prescribed values
on the boundary. In §§1, 2, it was pointed out that we may instead
solve the nonhomogeneous equation Au = f with the associated
boundary condition u = 0. We could investigate the latter problem
by the method of §7, expanding f and u in terms of the eigenfunctions
for the equation
Av+ Xv= 0.
However, for suitable special domains G we can proceed more simply:
by separating the variables one can reduce the number of independent
variables in the problem. This procedure will be illustrated by a
number of important examples.
1. Circle, Sphere, Spherical Shell. Consider first the case of two
independent variables x, y; let G be the circle of radius 1 about
the origin. If we transform the expression Au to polar coordinates
r, cp, we have the following boundary value problem: to solve the
equation
u.,., = 0
r(rur)r
+
with preassigned boundary values u(l, cp) = f(cp); here f(cp) is a continuous periodic function of period 211'" with a piecewise continuous
first derivative. If we attempt to find solutions of the homogeneous
equation-for the moment ignoring the boundary condition-which
can be written in the form u = v(r)w(cp), we are led in the usual
manner to an eigenvalue problem
w" 1
+ Aw= O;
316
V. VIBRATION AND EIGENVALUE PROBLEMS
the boundary conditions are the periodicity conditions w(O) = w(27r),
= w'(27r). The eigenvalues for this problem are given
= n 2 (integral n), the associated eigenfunctions are given by
w = an cos ncp + bn sin ncp. For v(r) we obtain the differential equation r(rv')' - n 2v = 0, of which v = rn and v = r-n are linearly independent solutions. Thus we obtain particular solutions of the
original equation, which are regular in the unit circle and have the
form
w'(O)
by A
(an cos ncp
+ bn sin ncp)rn,
where the const~nts an and bn are arbitrary. (These solutions can
also be characterized as solutions of the differential equation Llu = 0
which are integral rational functions of x and y and homogeneous
of the n-th degree.)
According to the theory of Fourier series we can obtain the desired
solution of the boundary value problem by the process of superposition,
u
=
t
rn(an cos ncp
n=O
+ bn sin ncp),
with the coefficients a, b taken from the series expansion of the preassigned boundary values (cf. Ch. IV, §2, p. 179).
The situation is quite similar in three dimensions, where we consider the unit sphere x 2 + y 2 + z2 :S 1 for G; then Laplace's spherical
harmonics occur instead of the trigonometric functions. In fact, if
we transform the differential equation to polar coordinates r, 0, cp
(cf. pages 225 and 314) we obtain the equation
2
(r ur)r
which for u
(35)
+ sm
. 12 0 u"'"' +
= v(r)Y(e, cp)
-J- (u11 sin 0)11 =
sm 0
0,
leads to the equation
(rV)' - "Av = 0.
The general solution of (35) has the form
where c1 and c2 are arbitrary constants and ai, a2 denote roots of the
quadratic equation
a(a
+ 1)
=A.
BOUNDARY VALUE PROBLEM OF POTENTIAL THEORY
317
For Y we obtain the same eigenvalue problem as in §8, given by the
differential equation
(36)
A*Y
+ XY
=
~[~
smO smO
Y.,,.,, +(YB sin
0)9] + XY = 0
where A must be determined in such a way that the equation has a
nontrivial solution with continuous derivatives up to the second order
over the entire sphere.
To justify our requirement that Y(O, <P) be regular even at the poles
0 = 0, () = r. of the sphere we observe that the operator A is invariant
under rotations of the coordinate system; for r = 1, A becomes A*,
hence A* is also invariant under coordinate rotations. The s~ngu
lari ties of equation (36) at the points(} = 0, 0 = 7r are a consequence
of the particular choice of the coordinate system. It is therefore
natural to require that the poles of the sphere be regular points of
the function Y. (In other words, Y as a function: of position on the
sphere should satisfy the same regularity conditions everywhere.)
The eigenvalue~ X and associated eigenfunctions Y are most easily
determined by investigating those solutions u = un of AU = 0 which
are integral and rational in x, y, z and homogeneous of degree n;
this investigation is analogous to §8. If we ·write these solutions in
polar coordinates in the form Un = rnYn(O, q;), we see that the Yn
are solutions of equation (36). The eigenvalues corresponding to
1 functions Yn(O, q;) are found to be
the 2n
+
X = n(n
+
1).
It will be shown in Ch. VII, §5 that the functions Y n defined in this
manner constitute the totality of eigenfunctions of our problem.
Furthermore , it will be possible to give a proof for the completeness
and expansion theorems which is similar to that used for the SturmLiouville functions. From the expansion theorem it follows that by
superposition of solutions: u = L:':=o anrnYn, we can find a solution
. of Au = 0 which assumes prescribed values on the surface of the
sphere.
Not only the function u = rnYn but also u = r-<n+i>yn, which
has a singularity at zero, is a solution of Au = 0. Thus, by superposition of solutions of the form rnYn and r-<n+i>yn, one can find
a solution of Au = 0 which assumes prescribed values on two concentric spheres and is regular in the intermediate shell.
318
V. VIBRATION AND EIGENVALUE PROBLEMS
If in particular we consider those spherical harmonics which depend only on() and not on cp, (i.e., if we assume Y"' = 0) the differential
equation takes the form
~()
sm
(Ye sin O)e
+ XY =
O;
performing the transformati on x = cos 0, this equation becomes the
equation for the Legendre polynomials (cf. eq. (20), Ch. II). The
Legendre polynomials P n( cos 0) are a special case of spherical harmonics.
We obtain a generalization of the spherical harmonics if we consider an arbitrary domain G on the surface of the sphere and attempt
to solve the differential equation (36)
Ll*Y
+ >S =
0
for a function Y(O, cp), regular in G, which satisfies homogeneous
boundary conditions on the boundary of G; for example, the function
may be required to vanish there. Eigenfunctio ns Y1, Y2, · · · belonging to this domain are called spherical surface harmonics. 1 It
follows from the above calculations that, if a and Xsatisfy the relation
a(a
+ I)
= X,
the function raY(O, cp) = u(x, y, z) is a solution of the differential
equation Llu = 0 which is continuous, except possibly at the origin,
in the cone with Gas base and the center of the sphere as vertex.
The equation Ll*Y
XY = 0 for spherical surface harmonics is
a special case of the general differential equation
+
Ll*Y+XY
eoY
- - foY
i_ oy
ox
(
Veg - f2 oy Veg - f 2
I
oY
+ ~ goxox
o~
-Joy
Veg - f 2
+ Ay
=
0
corresponding to an arbitrary curved surface with the line element
2
+ 2j dx dy + g dy2• In Ch. IV, §8, 2 we noted that this
equation is invariant. It can be considered as the equation of vibration of a "curved membrane" lying on the surface. In the case of
a sphere it becomes equation (36) if polar coordinates are introduced.
ds = e dx 2
1 Cf. W. Thomson and
P. G. Tait, Treatise on Natural Philosophy, Vol. I,
pp. 171-218, Cambridge University Press, Cambridge, 1886.
319.
BOUNDARY VALUE PROBLEM OF POTENTIAL THEORY
A further example of a region for which
the boundary value problem of potential theory may be solved explicitly is given by a cylinder erected over the domain G of the x, yplane which is bounded by the planes z = 0 and z = ?r. We assume
that the boundary values are identically zero on the vertical surface
of the cylinder and that_they are given on the plane bases by arbitrary
twice continuously differentiable functions which vanish on the boundaries r. We now seek solutions of ~u = 0 of the form u = f(z)v(x, y)
and obtain as above the conditions f" If = -tf&v/v = >..,i.e., differential equations for f and for v in which }.. is to be determine d in such
a way that an eigenfunction v(x, y) which vanishes on r exists. If
s
Vi, V2, • • • are all the eigenfunctions, with correspond ing eigenvalue
the
determine
can
we
theorem,
expansion
the
by
A1 , A.2 , • • • , then,
constants an, bn in such a way that the infinite series 2:~1 (ane~z
+ bne-~z)vn(x, y) takes on the given boundary values for z = 0
and for z = 7r. Hence this series is the solution of our boundary
value problem, provided that it converges uniformly together with
the series obtained from it by repeated termwise differentiation with
respect to any of the variables x, y, z.
3. The Lame Problem. Essentially, the most general case in which
separation of variables helps to reduce the boundary value problem
of potential theory to an eigenvalue problem for functions of a single
variable is the case of a confocal rectangular parallelepiped. By the
latter we mean a domain bounded by a portion of each of two ellipsoids, a portion of each of two hyperboloids of one sheet and a
portion of each of two hyperboloids of two sheets, all belonging to
the same confocal family
2. Cylindrical Domain.
x2
y2
z2
-=!
---+8 --+8 - ea
- e2
8 - e1
(cf. Ch. IV, §8, 3). Almost all the boundary value problems which are
usually treated explicitly may be considered as special or limiting cases
of this "Lame" problem. If we introduce elliptic coordinates p = f(u),
u = g(v), r = h(w) in accordance with the notation of Chapter IV,
the potential equation tf&T = 0 becomes
~T
- h(w)]Tvv + [f(u) - g(v)]Tw = O
·
[g(v) - h(w)][f(u) - h(w)][f(u) - g(v)]
= [g(v) - h(w)]Tuu
+ [f(u)
10
320
V. VIBRATION AND EIGENVALUE PROBLEMS
Evidently, solutions of the form
T
=
U(u)V(v)W(w)
can be obtained if we can find two constants X, µ which satisfy the
following three ordinary differential equations
V"
+· [Xj(u) + µ]U
- [Ag(v) + µ]V
=
W"
+
= 0,
(37)
U"
(38)
(39)
[Ah(w)
+
µ]W
= 0,
0,
where the variables u, v, w lie in the intervals
determined by the conditions
P2
S f(u) S
P1,
T2
S
h(w)
S
T1 •
Thus the parallelepiped is given by conditions of the form P1 2:::
P2
2:::
<T1
2:::
<T
2:::
<T2
2:::
Tl
2:::
T
2:::
p
2:::
T2 •
Using the coordinates p, <r, T instead of u, v, w and denoting the
independent variable by s and the dependent variable by Y, we can
write equations (37), (38), and (39) in the form
2
d
cp'(s) dY
)
cp(s) ds 2 + -2- ds + (~s + µ y = O,
Y
where we have set
4(s - e1)(s - e2)(s - e3) = cp(s).
The solutions of this equation (the Lame equation) are functions
which depend on the choice of the constants X, µ, and in general cannot be expressed in terms of the elementary transcendental functions.
They are called Lame functions and have been extensively investigated; even though relatively few means for their numerical calculation have been developed. Here we state only the associated eigenvalue problem. Clearly the boundary value problem of potential
theory can be solved for a confocal parallelepiped if it can be solved
for the special case in which the given boundary values vanish on
five of the six faces. Then the solution of the general boundary
value problem is the sum of six such special solutions. Suppose,
for example, that for T = T2 nonzero boundary values have been as-
BOUNDARY VALUE PROBLEM OF POTENTIAL THEORY
321
signed. We seek the solutions U, V, W of the Lame equations (37),
(38), (39) for which the relations U(u1) = U(u,z) = V(v1) = V(v2) =
W(w 1) = 0 hold but place no condition on W(w2). The product
T = U(u)V(v)W(w)
will then be a solution of llT = 0 which vanishes for p = p2, p = P1,
u = u2 , u = u1, r = r1. However, as we shall see, the given conditions cannot be satisfied for an arbitrary choice of A, µ. Thus, we
have a new eigenvalue problem, a so-called two-parameter eigenvalue
problem, in which we must determine pairs of associated eigenvalues
A, µ for which equations (37) and (38) have solutions that vanish for
u = u1, u = u.z and v = v1, v = v2, respectively.
The situation in this eigenvalue problem is similar to that in the
ordinary one-parameter problem: There exist infinitely many pairs
of eigenvalues Ai , µi and corresponding solutions U i , Vi of the eigenvalue problem. Every function of u, v which is continuous in the rectangle
u,z ~ u ~ u1 , v2 ~ .v ~ v1 together with its derivatives up to the second
order, and which vanishes on the boundary of the rectangle, can be expanded in an absolutely and uniformly convergent ser·ies of the form
co
~ CiUi(u)Vi(v)
i-1
where the summation is over all Lame products Ui(u)Vi(v) which correspond to the eigenvalue pairs. Furthermore, Lame products which
correspond to different eigenvalue pairs satisfy the orthogonality relation
J'Ull"l
'U2
[f(u) - g(v)] Ui(u)Vi(v) Uk(u)Vk(v) dv du = 0.
112
To solve the boundary va]ue problem, the next step is to associate
to each pair of eigenfunctions Ui, Vi a function Wi(w) which satisfies
equation (39) for A =·A; andµ = µi and vanishes for w = w1 • (The
existence of one such solution follows from general existence theorems
in the theory of differential equations.) The Wi(w) do not vanish
for w = w2, for otherwise T = UVW would be a nonvanishing solution of llT = 0 with vanishing boundary values, which contradicts
elementary facts of potential theory.
The boundary values assigned on w = w 2 can then be expanded in
a series of the form
co
Lai Wi(w2)Ui(u)Vi(v),
i-1
322
V. VIBRATION AND EIGENVALUE PROBLEMS
and the series
00
L:
ai Ui(u)Vi(v)Wi(w)
i=l
represents the desired solution of the boundary value problem of
potential theory for our parallelepiped. The above formulation of
the expansion theorem apparently implies that the prescribed
boundary values of T should vanish on all the edges of the parallelepiped. Further investigation would show, however, that_ this restriction need not actua1ly be imposed.
Our two-parameter eigenvalue problem is easily reduced to a oneparameter problem for a partial differential equation. Let us set
Z(u, v) = U(u)V(v), where U(u) is a solution of equation (37) and
V(v) a solution of equation (38). If we multiply the first of these
equatioPs by V, the second by U, and add, we obtain the partial
differential equation
(40)
Zuu
+ Zvv + X[J(u)
- g(v)]Z = 0
for the function Z(u, v). This differential equation might have been
obtained directly from !::i.T = 0 by writing T = Z(u, v)W(w). Evidently, the eigenvalue X = Ai and the associated eigenfunction Zi =
U;(u)Vi(v) solve the eigenvalue problem of this differential equation
for the rectangle G: U2 :::; u :::; u1 , v2 :::; v :::; V1 under the boundary
condition Z = 0. Since equation (40) has the form !::i.Z + A.pZ = 0,
where the function p = f(u) - g(v) is positive throughout the rectangle
F, this eigenvalue problem is of the type considered above, and we
have analogous questions of existence of eigenfunctions and validity
of an eigenfunction expansion theorem. These questions will be discussed later. 1 At present we postulate that, for the rectangle G, infinitely many eigenvalues X1, X2, · · · exist together with associated
eigenfunctions Z 1 , Z 2 , • • • which vanish at the boundary; arbitrary
functions can be expanded in terms of the Ai and Z i as described above.
Now we shall show that all the eigenfunctions Zi are Lame products
U(u)V(v), or the sum of at most a finite number of Lame products
belonging to the same eigenvalue X.
Proof: let A.1 , A.2 , • • • and Z1 , Z 2 , • • • denote the complete system
of eigenvalues and corresponding eigenfunctions for (40). Corre1
See §§14, 15.
BOUNDARY VALUE PROBLEM OF POTENTIAL THEORY
323
sponding to any eigenvalue Ah, we consider the ordinary differential
equation
with the boundary condition X = 0 for u = u1 and u = U2 • We
denote the infinitely many associated eigenvalues and the normalized
eigenfunctions by µ. 1 , µ. 2 , · · · and X1, X2, · · · , respectively. Given
any function which vanishes for u = u1 and u = ~ and is continuous
with its derivative s up to the second order in the interval~ ~ u ~ u1 ,
we can expand it in terms of these eigenfunctions. This is true in
particular for the function Z(u, v) which depends on an additional
parameter v. We write the expansion in the form
co
Z(u, v) =
L
n=l
Yn(v)Xn(u ),
where
Y (v)
11
=
ui
J
Z(u, v)X (u) du
11
u2
and differentiate Y twice with respect to v.
parts, we obtain
11
f
f "'
= f
d2Yn
=
-d
2
V
ui
Using integration by
Z1111(u, v)Xn(u) du
u2
1
(-Zuu - Ah[f(u) - g(v)]Z)Xn du
=
u2
2
11
1
X
Z ( - ddu
2
11
-
Xh[f(u) - g(v)]Xn ) du
112
=
(µ.n
+
Ahg(v)) Yn.
This means that Y is an eigenfunction for equation (38) for the
domain V2 ~ v ~ vi with the given boundary condition. In other
words, the numbers Ah, µ.n ·with the associated functions Xn(u),
Yn(v) constitute a solution of our two-param eter eigenvalue problemprovided the function Yn(v) does not vanish identically. As we have
a]ready seen, each product XnYn is an eigenfunction of (40) which
corresponds to the eigenvalue Ah. However, the eigenvalues of this
differential equation can have only a finite multiplicit y, as will be
11
324
V. VIBRATION AND EIGENVALUE PROBLEMS
shown in the next chapter. Hence only a finite number k of the
functions x n ' y n can be linearly independent. we may suppose
further that no X n or Y n vanishes identically for otherw.i§e we could
simply omit the term in question. It follows that any k
I products
XnY n satisfy a linear relation
+
k+l
_L: c,,Xn,,Yn,,
= 0.
l'=l
If we assign values to the variables v for which all Y nv are different
from zero we obtain a linear equation between the Xnp. This, however, is impossible since eigenfunctions which correspond to different
eigenvalues µ are linearly independent. Hence, the representation
Z = L: X ,Sn can have at most k terms, which is what we wanted to
show.
We can now expand an arbitrary function, subject to the indicated
restrictions, in terms of the eigenfunctions zi and obtain the following
result: Any function which is continuous, together with its derivatives up to the second order, in the rectangle u2:::; u:::; U1, V2:::; v :::; V1
and va.nishes on the boundary of the rectangle, can be expanded in
a series of Lame products.
§10. Problems of the
Boundary Points
Sturm-Liouville
Type.
Singular
The separation of variables sometimes leads to eigenvalue problems
for differential equations of the Sturm-Liouville type,
(pu')' -
qu
+ °'Apu
=
0,
for which singularities occur at end points of the fundamental domain. The coefficient p, for example, may vanish at an end point.
The nature of the problem is such that certain conditions are imposed at the singular end points; e.g. the solution must be continuous
or bounded or become infinite of an order less than that prescribed.
Conditions of this type replace the homogeneous boundary conditions
considered in §3.
1. Bessel Functions. Consider for example the Bessel equation
(cf. §5, 5)
(41)
(xu ')' -
2
n-u
x
+
Xxu
= 0,
PROBLEMS OF STURM-LIOUV ILLE TYPE
325
which occurs very frequently in mathematica l physics. For this
equation the assumption of §3, 3, that p > 0 throughout the entire
fundamental domain O :::; x :::; 1, is not valid because p(O) = 0. In
other words, the point x = 0 is a singular point for the Bessel equation and the requirement that a solution remain finite at this point
represents a boundary condition. In this case we have the problem
of finding a solution which remains finite for x = 0 and, e.g., vanishes
for x = 1. The eigenfunctio ns are Bessel functions Jn( v/}...x), where
the eigenvalves A = An,m are determined as the roots of a transcendental equation by the boundary condition at x = 1.
The associated orthogonal functions z = ··~/rJ n( v~x) are charaeterized by means of the differential equation
(42)
"
n2 -
.1
4
z --x2 - z+"Az=O
which is obtained directly from the Bessel equation. (This is an
example of the transformati on given in a general form on page 292.)
For the function t = z/x = Jn(v/}...x)/v x we obtain the differential equation
(43)
2. Legendre Functions of Arbitrary Order. A similar type of problem is presented by the Sturm-Liouv ille equation
(44)
2
[(1 - x )u']'
+ 'Au =
0
with the boundary conditions that u remain finite for x = + 1 and
for x = -1, i.e. at the two singularities of the equation. The funda1~ We know from Ch. II, §8 that the
mental domain is - 1 :::; x :::;
numbers A = n(n + 1) are eigenvalues and the Legendre functions
p n(x) eigenfunctio ns.
It is easy to show that the Legendre polynomials are the only solutions of this eigenvalue problem. For example, this can be deduced
from the fact that the Legendre functions form a complete orthogonal
system as shown in Ch. II, §8. We shall give an alternative proof,
which is independent of this fact: We note that, whenever a function u = f(x) satisfies equation (44), then the function u = f( -x)
also satisfies this equation. Evidently, the functions f(x) + f( -x)
and f(x) - f(-x) are also solutions; one of them is an even, the
other an odd function, and at least one does not vanish identically
+
326
V. VIBRATION AND EIGENVALUE PROBLEMS
since by assumption u is not identically equal to zero. Thus we
need only show that every even and every odd solution u of (44)
which is continuous for -1 :::; x :::; 1 is a Legendre polynomial, and
that A must be a number of the form n(n
1). If we write the solution (which is analytic) as a power series: u =
a,,x", equation
(44) leads at once to the recursion formula
+
(45)
a.,, =
L::-0
(11 - 1) (11 - 2) - A
11 (11 - 1)
a,,_2 •
If u is even, all the a,, for which 11 is odd are zero. If u is odd, the
same is true for the a,, for which 11 is even. If 11 - 2h > 0, it follows
immediately from (45) that
(46)
~[l -
(v - l)A(v - 2)] [ 1 - (v -
···[i - (11 -
2h
3~v -
4)]
+\)(11 -
2h)}ak,
where k = 11 - 2h. The series for u has a finite number of terms if
and only if X has the form n(n
1). In this case we see immediately
that u represents the n-th Legendre polynomial. For all other values
of X we obtain an infinite power series, which by elementary criteria
converges for Ix I < 1. Fix k so large that all the factors of the
above product are positive (ak may be assumed positive). By wellknown theorems, the product of the bracketed factors on the right
side of (46) converges to a positive limit as 11 ~ oo ; hence, a,, > c/ 11
for 11 > k, where c is a positive constant. It follows that the absolute
value of L~=k anxn will be arbitrarily large if Ix I is sufficiently close
to 1 and 11 sufficiently large. This implies that limz-+±1 I u(x) I = oo
and thus that X is not an eigenvalue. 1
We can easily derive other classes of orthogonal systems of eigenfunctions from the differential equation for the Legendre polynomials
by a general method. Name1y, if we differentiate equation (44)
with respect to x, we obtain a differential equation for the function
u'(x). As before it follows that a solution which is regular at both
end points of the interval exists only for X = n(n
1) and is given
+
+
1 The above discussion
is closely related to the Raabe or Gauss convergence
criterion; cf. A. Kneser, Zur Theorie der Legend1eschen Polynome, Tohoku
Math. J., Vol. 5, 1914, pp. 1-7.
PROBLEMS OF STURM-LIOUV ILLE TYPE
327
in this case by P:(x). The resulting equation for P:(x) is not yet
self-adjoint; we make it self-adjoint by introducing the function
P:(x)vl - x2 =Zn as the unknown; then the new equation takes the
form
[(1 ~ x2)z']' - - z 2 +AZ = 0.
1 -x
The associated eigenvalues are).= n(n + 1)
the eigenfunctions
Zn
=
Vl -
(n = 1, 2, 3, ···)with
x2 P:(x).
The functions Zn = P n,1(x) are called associated Legendre functions
of first order. (The functions Pn(x) = Pn,o(x) will occasionally be
referred to as Legendre functions of zero-th order.) The Legendre
functions P n,1 satisfy the orthogonalit y relation
l
1
n ¢ m.
Pn,1Pm,1 dx = 0 for
-1
In a similar way, by differentiating (44) h times we obtain for the
function
the differential equation
2
(47 )
z
h[( 1 - x 2) z ']' - 1 - x2
+ AZ =
0,
with eigenvalues ). = n(n + 1) (n = h, h + 1, · · ·) and associated
eigenfunctions Pn,h(x), which also are mutually orthogonal. These
functions are called associated Legendre functions of h-th order. They
can be normalized with the aid of the easily verified equation
1
1
-1
2
p
n ,h
dx
= 2n
2
(n
+ 1 (n
+ h)!
- h) ! .
We can prove that we have obtained all eigenvalues and eigenfunctions of (47) by the method used for Legendre polynomials.
3. Jacobi and Tchebycheff Polynomials. A generalization of the
Legendre polynomials is given by Jacobi's polynomials of Ch. II, §9.
328
V. VIBRATION AND EIGENVALUE PROBLEMS
The differential equation for these polynomials may be written in
the following Sturm-Liouville form:
[(1 - xy-q+ixqu']'
+ X(l
- xY-qxq-iu = 0.
The n-th Jacobi polynomial corresponds to the eigenvalue X =
+ n) with the boundary conditions that the solution remain
finite for x = 0 and x = 1. As above, there are two ways to show
that Jacobi's polynomials are the only solutions of this eigenvalue
problem.
A further example is offered by the Tchebycheff polynomials,
which correspond to the Sturm-Liouville equation
n(p
(yl - x2 u')'
+
v1 A- x u = 0
2
with the boundary conditions that the solution be regular at x = ± 1.
The eigenvalue corresponding to the Tchebycheff polynomial Tn(x)
2
is X = n and, as above, these X and T n exhaust all the eigenvalues
and eigenfunctions.
4. Hermite and Laguerre Polynomials. The Hermite polynomials
u = Hn(x) and the corresponding orthogonal functions v = Hne-x 212
are characterized as the solutions of the eigenvalue problems (cf.
Ch. II, §9, 4)
2
(e-z u')'
(48)
+ "Ae-x u
2
0
=
and
(49)
v"
+ (1
2
- x )v
+ Xv
=
0,
respectively, with eigenvalues X = 0, 2, 4, · · · . The fundamental
domain is the entire line - oo < x < + oo , and the boundary condition for (48) is: the eigenfunction u should not become infinite at
x = ± oo of an order higher than a finite power of x. To show that
the Hermite eigenvalue problem has no other solutions we write
equation (48) in the form u" - 2xu' + AU = 0 and u as a power
series u = L~=o anxn. We can assume that u is either an even or
an odd function (see the discussion of eq. (44)), and hence that only
odd or only even powers of x appear in the power series. The differential equation implies the recursion formula
an+2
an
(n
2n - A
1) (n
+
+ 2)
PROBLEMS OF STURM-LIOUV ILLE TYPE
329
for the nonvanishin g coefficients. Hence, either the series breaks
off-in case X = 2n is a non-negative even integer-and thus represents the Hermite polynomial Hn, or the series has infinitely many
nonvanishin g coefficients and converges for all values of x. As
soon as 2n - X becomes positive, all the coefficients an which occur
have the same sign. In the second case, terms anxn with n arbitrarily
oo of an order
large occur; therefore u becomes infinite at x =
greater than any finite power of x. Thus, u cannot be an eigenfunction for the problem and the Hermite polynomials are the only
solutions of the eigenvalue problem.
The Laguerre polynomials will be treated in more detail, since an
application will be given in §12, 4. Here the fundam~ntal domain
is the positive real axis 0 S x < oo, and according to Ch. II, §9 the
eigenvalue equation, satisfied by the Laguerre polynomials u = Ln(x)
for the eigenvalue X = n (n a positive integer), is of the form
+
xu"
(50)
+
(1 - x)u'
+ Xu =
0
or, in self-adjoint form, it is
(xe-xu')'
+ xe-xu
= 0,
where the boundary conditions are: u remains finite at x = 0 and,
as x ~ oo, u does not become infinite of an order higher than a positive power of x. For the associated orthogonal functions
v
=
Wn
= e-x12 Ln
we find the Sturm-Liouv ille eigenvalue equation
(xv')'
+
G- ~)v +
}.v
=
0,
where we require regularity at x = 0 as the boundary condition.
Finally, we remark that the functions
112
w = Sn = x- wn
which occur later (§12, 4) satisfy the self-adjoint eigenvalue equati on
(
2 '),
- x
xw
2
-
2x -
4
1
w
+
A
xw = 0
where the solution is required to vanish at x = 0. The corresponding eigenvalues are, throughout, the positive integers X = n.
330
V. VIBRATION AND EIGENVALUE PROBLEMS
Here, as in the case of Legendre functions in subsection 2, processes
of differentiation and multiplication by suitable factors lead to Laguerre functions of higher order which satisfy similar differential
equations. By differentiating (50) m times, we find that the functions
satisfy the differential equation
(51)
xu"
+ (m + 1 -
x)u'
+ (A -
m)u = 0,
which can be written in the following self-adjoint form:
(xm+le-:1V)'
+ xme-z(A -
m)u = 0.
The associated orthogonal functions
satisfy the Sturm-Liouville equation
(51a)
(xv')'+
(1 ~ m - ~ - ;::)v +Av = 0,
and the functions
satisfy the eigenvalue equation
2
(51b)
(x 2w')' - x
+
2 (m -
~)x +
2
m
-
1w
+ Xxw
=
0
for the corresponding eigenvalues X = n, where n is an integer greater
than or equal to m and the boundary conditions are evident.
In order to show that our differential equation has no other eigenvalues or eigenfunctions, we set u = I::.o avx" in equation (51) and
find (using recursion relations) the formula
+ ., -
ao (m - X) • · · (m - X
1)
av = - ---:------,:----:-----:-.,, 1
(m + 1) · · · (m + v)
for the coefficients. We see that for arbitrary fixed X the coefficients
of this series have the same sign from a certain .,, on, and that the
series converges for all values of x. Thus it does, in fact, represent a
ASYMPTOTIC BEHAVIOR OF SOLUTIONS
331
solution of (51) which is regular for 0 ,::; x < oo. In the case of
positive integral X = n, with n > m, the series breaks off after a
finite number of terms and thus represents a polynomial. For
every other X it is easy to obtain the estimate
c
I a,, I>-,-,
'P.'Pr
where c is a suitable constant and r a suitable positive integral exponent. But this implies that for x --+ oo the solution becomes
infinite of at least the order of ez/xr. Hence this solution cannot be
an eigenfunction for the problem, and our assertion is proved.
The Asymptotic Behavior of the Solutions of SturmLiouville Equations
Simple asymptotic evaluations of the solutions of Sturm-Liouville
differential equations can be obtained for large values of the parameter
or independent variable provided the coefficients satisfy certain general conditions.
1. Boundedness of the Solution as the Independent Variable tends
to Infinity. Let us write the differential equation in the form
u" + µ(x)u = 0 (cf. §3, eq. (19a)). We assume that for x --+ oo,
µ(x) approaches a positive limit which, without loss of generality,
can be taken equal to 1. Then, setting µ = 1 + p, we can base our
discussion on the differential equation
§11.
(52)
U
11
+ U + pU
= 0.
We shall, however, replace the assumption
gent one
p--+
0 by the more strin-
(53)
where a is a positive constant. With this assumption, we assert that
every solution of equation (52) is bounded for x--+ oo. This is to be
expected since, for large x, (52) approaches the equation u" + u = 0,
which has only bounded solutions.
To prove our assertion, we multiply (52) by u', integrate from a
positive lower limit a (to be suitably determined later) to x, and
obtain
(54)
-2
J." puu' dx =
-pu'
1: + J." p'u' dx.
332
V. VIBRATION AND EIGENVALUE PROBLEMS
From this it follows immediately that
2
12
u (x) ::; u (x)
+ u2(x)
::; C(a)
+ I p(x) I u (x) + {; Ip' I u
2
2
dx,
where C(a) denotes an expression which depends only on the lower
limit a. Let M = M(x) be the maximum of the function u(~) in
the interval a ::; ~
x, assumed at the point ~; then from this inequality and from (53) it follows that
s
M' S
C(a)
+ ~'a + M'aG -
D
and thus that
M'(1- ~) S C(a).
Now if we take a 2:: 2a we obtain 2C(a) as a bound for M 2, independent of x. This proves the assertion.
2. A Sharper Result. (Bessel Functions.) We consider the equation u" + u + pu = 0 and assume that p(x) vanishes at infinity of
an order higher than the first (this assumption is more stringent
than that made in subsection I); i.e. we assume for example 1
(55)
p(x) -
oG,).
We now have a closer agreement between the differential equation
and the equation u" + u = 0, which implies not only that the solutions are bounded but also that they approach trigonometric functions asymptotically.
We proceed by determining functions a(x) and o(x) with the derivatives a' and o', related to u and u' by
u =
a
sin (x
+ o),
u' =
a
cos (x
+ o)
(a cannot vanish at any point since u and u' would then vanish
simultaneously at some point and thus u would vanish identically in
virtue of equation (52)). We can calculate u" and u' in two ways:
1 Here we
use the customary notation, according to which 0 (f (x)) denotes a
function g(x) for which the quotient I g(x)/f(x) I remains bounded as the
argument increases.
333
ASYMPTOTIC BERAVIOR OF SOLUTIONS
we obtain
u"
=
+
a' cos (x
-(1
+
+
o) - a(o'
1) sin (x
+
o)
+ o),
p) a sin (x
I
tan (x
+
u' = a cos (x
2
tan (x
+ o)
+
o' =
(56)
a
We see that
o(x) = o({J) -
o)
+
a(o'
1) cos (x
o'W
+
o),
,,
a
ao'
o'
- o' -
p sin (x
+
p'
+ o),
p sin (x
+ o) cos (x + o).
and .o approach definite limits as x
t
a
-
=
2
tan x
a
+
o) =
-o'
(
+ o) = -
I
a(o'a- p);
=
o) = a' sin (x
tan (x
(57)
+ o)
~ oo •
In fact,
d~; if we let f3 increase beyond an bounds the
integral on the right side converges by (55) and (56), since the
integrand approaches zero like 1/x2 • Thus limp....oo o({3) = 000 exists,
and the above representation shows moreover that
O(x) =
O~ + 0(1).
Correspondingly, formula (57) for a'/a = d log a/dx leads to the
relation
a(x)
=
a~( 1 + o(D),
in which a.00 ¢ 0. Thus, for every solution u we have obtained the
asymptotic representation
u
=
a
sin (x
+
o) = aoo sin (x
+
ooo)
+
o(D·
This result may be immediately applied to the differential equation
u" + (1 -
2
m
-
x2
i)u = 0'
334
V. VIBRATION AND EIGENVALUE PROBLEMS
whose solutions, according to page 325, are connected with the solutions Ym(x) of the Bessel equation by the relations
YmVX•
U =
Thus all solutions Ym(x) of the Bessel equation satisfy asymptotic
formulas of the form
~;;cos (z + 6~) + o(;k).
Ym(x) =
The constants a 00 and 000 for the Bessel function J m(x) will be determined from other considerations (cf. Ch. VII, §6, 2). It will be
found that
000
= -
mr
r
2 - 4·
3. Boundedness as the Parameter Increases. Considerations similar
to those of subsection 1 are used to prove the following theorem:
The absolute values of the solutions of the Sturm-Liouville equation
(with continuous r)
(58)
+ Xu
u" - ru
=
0
s
in the interval 0
x ~ 1 remain less than some bound independent of
X and x provided the solutions are normalized by the condition
}~
1
2
u dx
=
1
and satisfy the boundary conditions u(O) = u(l) = 0.
It suffices to prove the theorem for large positive :A; we again multiply the equation by u' and integrate from 0 to x, obtaining
(59)
u' 2(x)
+ :Au (x)
2
-
2[
ruu 1 dx = u'2(0)
+ :Au (0).
2
To evaluate the right side we integrate this equation between the
limits 0 and 1, obtaining
(60)
2
u' (0)
+ :Au (0)
2
=
f
12
u dx
+ A-
21
1
dx
.C: ruu' dt.
Inserting this value in (59) and estimating the resulting integrals by
means of Schwarz's inequality, we have
335
ASYMPTOTI C BEHAVIOR OF SOLUTIONS
(61)
2
2
'Au ~ u' + 'Au2 ~ 'A
+
1
1
2
u' dx
+
c./£' .n/£'
u"
u' dx,
where C1 denotes a positive constant independent of x and X.
the equation
1 u' dx + 1 ru dx
From
1
1
2
2
= 'A,
obtained in the familiar manner by multiplying (58) by u and transforming the resulting expression by Green's formula, we have
1
1
2
u' dx
~ 'A + C2 fo
1
2
u dx.
Inserting this in (61) we obtain the inequality
Xu\x) ~ 2X
+ C30 + c.. ,
where C2, C3, c.. again are positive constants independe nt of x and
X. This implies
2
u (x) ~ 2
c..
+ 0 + X,
c3
and .our assertion is proved.
Finally, we remark that both our result and our method of proof
remain valid if no boundary conditions are imposed. However,
functions of more than one variable do not have the analogous
1
boundedness property .
4. Asymptotic Representa tion of the Solutions. Having shown that
the solutions are bounded, we shall now prove the following theorem:
If u is any normalized solution of u" - ru + Xu = 0 in the interval
0 ~ x ~ 1, with X > 0, then there exists a solution v of v" + Xv = 0
such that
u=v+o(~)·
A simple counter-exa mple is provided by the normalized eigenfuncti which vanish
on the boundary of the unit circle. (Cf. W. Sternberg, fiber die asymptotisc he
Integration partieller Differential gleichungen II, Math. Ann., Vol. 86, particularly pp. 292-295.)
1
( v'2/Jo(ko,mr)/J~(ko,m) of the differential equation ~u +Xu
336
V. VIBRATION AND EIGENVALUE PROBLEMS
For large values of A this formula furnishes an asymptotic representation of the solutions u in terms of the trigonometric functions v.
To prove the theorem, consider the solution of v" + Av = O for
which u(O) = v(O), u'(O) = v'(O). The function u - v = w is then
a solution of the equation
w" +AW= ru.
If we multiply this equation by 2w' and integrate from 0 to x, taking
into account the fact that w(O) = w' (O) = 0, we obtain
w' 2(x)
(62)
+
Xw2(x) = 2
lx
ruw' dx.
Let M deriote the maximum of I w(x) I and M' the maximum of
I w' (x) I in the interval 0 S x 1. Schwarz's inequality applied
to (62) yields, because of A > 0,
s
M' 2 S M'C,
M'
s
C,
where C is some positive constant independent of X and x.
from equation (62) that
and hence
It follows
c
M -< vX
- r'
which completes the proof.
5. Asymptotic Representation of Sturm-Liouville Eigenfunctions.
The problem of representing solutions of the equation (pu')' - qu +
Xu = 0 asymptotically may be formulated somewhat differently if,
instead of considering arbitrary solutions, we consider eigenfunctions,
say for the interval 0
x S 7r and boundary conditions u(O) =
u(?r) = 0. Suppose that by (20a) we transform the differential
equation into
s
(63)
z" - rz
+ Xz
= 0,
where the new independent variable t ranges over the interval
0 S t
l and r is a continuous function in this interval. We wish
to compare then-th eigenfunction Zn, associated with the eigenvalue
An, with the corresponding n-th eigenfunction of the differential
equation v" + AV = 0.
s
337
ASYMPTOTIC BEHAVIOR OF SOLUTIONS
If, in equation (10), we replace the function Ni by rz we obtain a
useful tool: The solutions of (63) which vanish at t = 0 satisfy the
"Volterra integral equation" for z
(64)
z(t) = asin0t
1
+ VX
lt
r(r)z(r) sin
0
VX(t -
r)d.r
with arbitrary constant a.
According to subsection 3 the functions z(t) satisfying (64) and the
boundary condition z(l) = 0 remain bounded for all X; it follows
1
immediately from (64) that a is also bounded. This fact, together
with equation (64) and the relation
1l z2
dt = 1, leads to the precise
estimate
for a, which in turn implies
z(t) -
Vi
sin -VXt =
o(~).
Since the n-th eigenvalue An of the differential equation becomes
infinite as n increases (cf. Ch. VI, §2, 2), we see immediately that
the n-th eigenfunction Zn(t) has the asymptotic representation
z.(t) =
Ii
sin V>;: t +
JA.
0(1).
Furthermore we have the asymptotic estimate for An (cf. Ch. VI,
§2, 3)
2
7r
+ 0( 1) •
An = n 2 1,2
Hence
VX: =
+ 0(1/n), so that
sin V>;:t = sin n j t + o(~).
n(7r/l)
Accordingly, the normalized eigenfunctions of the
z" - rz + Xz = 0 have the asymptotic representation
(65)
•• c1> =
Vi
sin"
equation
y1 + o(~).
1 The boundedness of z(t) may also be proved directly from the integral
representation (64).
338
V. VIBRATION AND EIGENVALUE PROBLEMS
Similarly, if we differentiate the integral equation (64) we obtain
the corresponding formula
')
(66)
Zn(t = n
7r./2
y V l cos n 7ry t + 0(1).
In terms of the original equation, these results are expressed by the
relations
(67)
_ sin(nyf
!~ax) + 0 (-1 ) ,
V'1JP
Un(x) -
Cn
PP
where the normalizing factor
~
= ['
Cn
Cn
n
.
is determined by
sin' ( n
T ~ dx) dx
O
fVPP
and
l=
f ~dx
holds.
Correspondingly, we have
(68)
,< )
Un X
COS
= Cn n?r
l
(n ~z lx v•/~ d:t) I!!.- + O(l)
p
o
_4./
p
VPP
•
Asymptotic expressions for the eigenfunctions and their derivatives
in the case of more general homogeneous boundary conditions are
derived in the same way. We obtain the expressions
{69)
z.(t) =
t'1
cos n
yt
+om
and
(70)
z~(t)
=
-7 .y1
sin n
yt + 0(1),
which are valid as long as the coefficient h in the boundary condition
z'(O) - hz(O) = 0 remains finite.
Incidentally, the Volterra integral equation (64) makes it possible
to obtain much more precise expressions for the eigenfunctions and
EIGENVALUE PROBLEMS WITH CONTINUOUS SPECTRUM
339
their derivatives. This may be expected because the Neumann series
for such a Volterra integral equation always converges (cf. Ch. III,
§10, 8). 1 We can obtain these expressions directly, without referring
to the general theory. In (64) we set a equal to 1, i.e. we no longer
require that the functions be normalized; under the integral sign in
the right side we then substitute for z( T) the value given by the
integral equation. If we iterate this process and set v(t) = sin0t,
we obtain formula
11' 1"1
+x
dn
0
0
dT2V(T2)r(T1)T(T2) sin 0(t - Ti)
·sin 0(n - T2)
(71)
+~
f
1
dT2
dn {
f
2 dTav( Ta)r( T1)r( n)r( Ta)
·sin y'x(t - T1) sin 0(n - T2) sin 0(T2 - Ta)
+
+~
f
dn ...
f
"-l
clTnZ(T,JT(Ti) ... r(Tn)
·sin 0(t - Ti) · · · sin 0(Tn-l - Tn).
This series can be continued indefinitely in decreasing powers of 0 ;
the result is an infinite series for z(t, X) in terms of decreasing powers
of 0 which converges for all X > 0. The error which results if we
take only the first n terms is of a smaller order of magnitude than
c110r.
Eigenvalue Problems with a Continuous Spectrum
The eigenvalues of the problems considered previously form a
denumerably infinite sequence. However, if the coefficients of the
differential equation are singular at the boundary points of the
fundamental domain or, in particular, if the fundamental interval
is infinite the spectrum, or totality of eigenvalues, may behave
§12.
1 Cf. J. Liouville, J. de math. pures et appl., Vols. 1, 2, 1836/37 (see bibliography for Ch. VI), where Volterra's integral equation and the Neumann
series occur.
340
V. VIBRATION AND EIGENVALUE PROBLEMS
quite differently. In particular, continuous spectra may occur, i.e.
spectra which contain all the values of a >.-interval; in this case a
Fourier integral theorem replaces the eigenfunction expansion theorem.
I. Trigonometric Functions. A simple problem of this kind is
given by the eigenvalue equation
u" +Xu= 0
< x < oo with the boundary condition:
for the interval - oo
u remains bounded at plus and minus infinity. Clearly, any nonnegative number ). is an eigenvalue, with the associated eigenfunctions sin 0 x, cos v'x x. For this eigenvalue probJem the special
Fourier integral theorem of Ch. II, §6 takes the place of the expansion theorem.
In general, the possibility of continuous spectra becomes plausible
through a limiting process. We start with an eigenvalue problem
for a finite interval and then pass to the infinite interval; the discrete
spectrum of the finite interval may pass into a continuous spectrum
and the Fourier expansion in eigenfunctions may pass into a Fourier
integral expansion for the infinite interval.
2. Bessel Functions. The situation is similar for the Bessel equation
(xu')'
+ (Ax
- :}
=
0
in the interval 0 :::; x < oo, with the boundary condition that u
remain finite for x = 0 and x ~ oo . All the Bessel functions u =
Jn(v">..x) with). ~ 0 are eigenfunctions, so that we have a continuous spectrum consisting of all the non-negative values of>..
Here, too, the expansion theorem for the representation of an
arbitrary function f (x) is replaced by an integral theorem, in which
the domain of integration is the spectrum, i.e. the continuum of
positive real numbers. This integral representation is
1
00
f(x) =
tJ n(tx)g(t) at,
Such a representation is possible if we assume that f (x) is piecewise
smooth for x ~ 0, that the integral
f.
00
x I J(x) I ax
EIGENVALUE PROBLEMS WITH CONTINUOUS SPECTRUM
exists, and that f(O) = 0.
given in Ch. VII, §2.
341
The proof of this integral theorem will be
3. Eigenvalue Problem of the Membrane Equation for the Infinite
Plane. The eigenvalue problem for the differential equation
.1u +Xu= 0
for the entire x, y-plane, with the boundary condition of boundedness
at infinity, may be solved in two different ways.
The set of products u = sin a(x - t) sin [j(y - 11) of trigonometric functions, where t, 11 and a, {j are arbitrary and the eigenvalues
2
2
are the numbers A = a
{j , can be considered as eigenfunctions.
An eigenvalue is then any non-negative number X, and every such
eigenvalue clearly determines a continuum of eigenfunctions. The
corresponding integral representation is simply the Fourier integral
theorem in the plane (cf. Ch. II, §6).
If we introduce polar coordinates r, <p, we obtain eigenfunctions of
the form
+
u
=
Jn(0r) sin ?Up,
where n is an arbitrary integer and X any non-negative real number.
In this case the spectrum is again the continuum of non-negative
real numbers A ~ 0; but, since n is an integer, only a countable
number of eigenfunctions is associated with any eigenvalue X > 0.
Here the representation of an arbitrary function is obtained as a
Fourier series expansion with respect to n and each coefficient is
represented as an integral with respect to r, in accordance with the
preceding subsection (see also Ch. VII, §2).
Incidentally these eigenfunctions are linear combinations of sine
2
2
products corresponding to a given value of A = a
{j • In fact,
we have
+
(cf. Ch. VII, §2.)
1
In his work on quantum
theory, Schrodinger was led to a type of eigenvalue problem with a
4. The Schrodinger Eigenvalue Problem.
2
Cf. also Ch. VI, §5.
E. Schrodinger, Abhandlungen zur Wellenmechanik, Johann Ambrosius
Barth, Leipzig, 1927.
1
2
342
V. VIBRATION AND EIGENVALUE PROBLEMS
spectrum of an entirely different structure. This spectrum consists
of a continuous and a discrete part; the discrete part does not extend
to infinity but has a finite point of accumulation. In the simplest
Schrodinger problem we consider the eigenvalue equation
(72)
l:l.u
c
+ -u
+ Xu
r
0,
=
where c is a given positive constant, and r, fJ and "' are polar coordinates; we require that the eigenfunctions u be continuous at zero
and remain finite as r ---+ oo • If we multiply the differential equation
by a spherical harmonic Yn(8, <P) and integrate over the unit sphere,
we find in the usual way that the function
v(r) =
ff
u(r, fJ, "') Yn(8, "') sin() dfJ·d<P
satisfies the differential equation
(73)
Vrr
+ ~Vr
+ (x + r~ r
n(n
d l))v
r
0.
=
From the eigenfunctions of this equation, under the same boundary
conditions as above for r = 0 and r ---+ oo, we obtain eigenfunctions
for (72) by forming the products u = vYn.
Replacing A by the new parameter
c
l = 2y-X
and r by the variable
z
=
2-v-=>: r,
we are led to the differential equation
v••
+ ~v. + ( -~ + ~ -
n(n z;:- 1) )v
=
0,
which we encountered in a somewhat different form in §10, formula
(51b). From the theory given there, we conclude: For real l, i.e.
for negative A, the condition that the. solution be continuous at zero
and remain finite as r---+ oo can be satisfied only for integral values
l > n; the solutions are given by the derivatives of the Laguerre
polynomials in the form
v = z" e-• 12 Li~'!.+ 0 (z).
343
PERTURBATION THEORY
For the original equation we obtain the values
c2
X
=
-4l2
(and only these) as the negative eigenvalues associated with the
eigenfunctions
(B )
n -cr12zLc2n+n
_
l y n '"' •
l+n
u - r e
(c r)
In this expression, for a given integer l, n can assume all integral
1 linearly
values from 0 to l - 1, and Y n denotes one of the 2n
independent spherical harmonics. The discrete spectrum found in
this way consists of infinitely many numbers with the limit point
zero.
Furthermore, we state that Schrodinger's equation (72) has all
positive real numbers A as eigenvalues, i.e. it has a continuous spectrum
+
consisting of the non-negative real numbers.
To prove this assertion, we introduce the function w = rv in place
of v in equation (73). This leads to the equation
w"
+ (x + ~r -
n(n
+ l))w =
r2
0
'
which is of the type treated in §11, 1. Accordingly, its solutions w
remain bounded for all positive A, and the solutions v = w/r of
equation (73) approach zero as r increases without bound. To demonstrate that every positive value of A is an eigenvalue, we need only
show that a solution v which is regular at the origin exists for all r.
This fact is proved in the general theory of linear differential equations. However, a solution with the required properties may also
be obtained directly, by a method which we have used before, in the
form of an everywhere-convergent power series. This is best accomplished if we introduce z = r-neiry'xv in the differential equation;
thus we obtain an equation in z for which the assumption that the
solution has a power series expansion leads to a two-term recursion
formuls..
§13.
Perturbation Theory
If the eigenvalues An and the associated orthonormal eigenfunctions
Un Otthe self-adjoint linear differential equation
(74)
344
V. VIBRATION AND EIGENVALUE PROBLEMS
are known for a prescribed domain 1 and prescribed boundary conditions, then the eigenvalues and eigenfunctions of an eigenvalue
problem corresponding to a "neighboring" or "perturbed" differential
equation
(75)
can be calculated by a method of approximations which is important
in applications-the so-called Calculus of Perturbations or Perturbation Theory. It is understood that the boundary conditions and
the domain remain unchanged. Let r denote a given function which
is continuous in the fundamental domain and E a parameter; Un and
Xn are the eigenfunctions and eigenvalues of the new problem. Without proof we assume that the new eigenvalues as well as the new eigenfunctions may be expanded in powers of the perturbation parameter E.
I. Simple Eigenvalues. Suppose, first, that the original unperturbed problem has only simple eigenvalues. Then we write
+ EVn + E Wn + ••· ,
An + Eµn +. ivn + •· • .
(76)
Un = Un
(77)
Xn =
2
Insertion in (75) leads at once to equation (74) and to the additional
equations
(78)
(79)
L(vn)
L(Wn)
+ AnVn =
+ AnWn
TUn -
µnUn ,
= TVn -
µnVn -
Jin Un •
From these, in turn, we can calculate the perturbations of various
orders, i.e. the quantities µn, vn, · · · and Vn, Wn, · · · , respectively.
In order to do this, we introduce the expansion coefficients
an;
=
J
VnU;
dg
of the function Vn with respect to the eigenfunctions 'lt; . We multiply
equation (78) by Uz , integrate over the fundamental domain, and
transform the first term on the left by Green's formula using the
boundary conditions (such as vanishing boundary values); we obtain
anzCAn -
Az)
=
dnz -
µnonz ,
Here the number of dimensions of the domain is arbitrary. Integration
is always to be carried out over the entire domain; the volume element is
denoted by dg.
1
345
PERTURBATION THEORY
where Onz
0 for n
=
¢
land and
f TUnUz dg.
dni =
\
By taking l
n we obtain
=
(80)
and, taking l
¢
n,
f
Th~ quantity ann is found from the normalization condition u! dg =
1, which leads to the relation
J
UnVn dg
= 0 and hence to
ann
= 0.
Thus we find that if Vn can be expanded in terms of the u;,
~I
(81)
Vn = LJ
1-1
dnj
U;
Xn - X;
holds, where the sign L:' denotes summation over the indicated
values of j omitting the term for which j = n.
Having determined the first approximation, we find the second
ao
Wn
=L
bn;U;
;-1
in a similar way using equation (79), from which it follows, as above,
that
ao
(82)
bn i('An -
"A z) =
L:
an;d;z
i-1
-
µnanz -
vnonz.
Setting n = l we obtain the second perturbation term for the eigenvalue, namely
ao
.,," = L:
an;d;n ,
1-1
while for l
(83)
¢
n we find
bnz = '\
l\n
~,
"l
{t
1=1
an;d;t -
µnanl} •
346
V. VIBRATION AND EIGENVALUE PROBLEMS
To determine bnn we must again employ the normalization condition
Ju~ dg
=
e2 equal to zero. It follows
1 and set the coefficient of
easily that
(84)
which completely determines the second approximation.
Further approximations may be determined successively in precisely
the same manner.
2. Multiple Eigenvalues. A supplementary discussion is necessary
in the case of multiple eigenvalues, the so-called "degenerate" case.
It will be sufficient to assume that the first eigenvalue of (74) is
a-fold, i.e. that A1 = · · · = Aa = X, while for n > a the remaining
eigenvalues An are simple. In the case of a multiple eigenvalue the
eigenfunctions are determined only up to an orthogonal transformation. Thus, if the equation is perturbed, we cannot expect the
individual eigenfunctions to vary continuously unless we have selected
the system of unperturbed eigenfunctions for the multiple eigenvalue
in a suitable way (see also Ch. III, §8, 4). Accordingly, we suppose
the a eigenfunctions associated with the eigenvalue X transformed
into another system of such eigenfunctions by means of an orthogonal
transformation
a
u! = L
'Yn;U;
;-1
which will be determined later.
Un =
(n = 1, 2, · · · , a)
We now assume
u! + EVn + lwn + . . . '
and attempt to determine both the 'Yni and the functions Vn, Wn, • • • •
For n > a we have u! = Un, and the observations of subsection 1 remain valid. Thus we can limit ourselves to the cases n = 1, 2, · · · , a.
Our assumption and equation (75) lead to the following equations
for Vn and Wn , respectively:
(85)
(86)
L(vn)
L(wn)
+
a
AnVn
+ AnWn
=
L
T"f njUj
i=l
a
-
µn
L
'Yn; U; ,
i-1
a
= TVn -
µnVn -
Jin
L
'Yni U; •
i-1
347
PERTURBATION THEORY
If we multiply (85) by UL and proceed as in subsection 1, using the
same notation, we obtain
a
(87)
anz(An -
Al) =
(djl
L
i=l
- µnOjlhnj,
and hence in particular for l = 1, 2, · · · , a we have
a
0
=
(djl
L
i-1
(l,n = 1,2, ···,a).
µnOjlhnj
-
2
From these a equations, the quantities µ1 , µ2 , • • • , µa may be
determined uniquely as roots of the characteristic equation
I diz - µnoil I = 0 (see Ch. I, §2). For simplicity, we may assume
that these roots are all different, i.e. that the form Li.z=1 diLXjXL
possesses only distinct eigenvalues. Then the orthogonal matrix
('Yni) is also determined uniquely. This means that our choice of
unperturbed eigenfunctions u! = Li=1 'Yniui is now fixed. Let us
designate these u! by Un. The matrix (dnl) in the new notation is a
diagonal matrix with the elements
From equations (87), for l
the remaining elements are zero.
we have immediately
(88)
anl =
>
a,
dnl
An - Az •
As in subsection 1, the normalization condition implies ann = O; to
determine the quantities anl (l, n = 1, 2, · · · , a; n ¢ l) we must
use equations (86) for the second approximation.
l, n = 1, 2, · · · , a, lead to
The latter, for
00
an;d;l
o = I:
;-1
-
vnonl
µnanl -
or, in view of the fact that (diL) (j, l = 1, 2, · · · , a) is a diagonal
matrix with the diagonal elements µn, to
Thus when l = n we have
00
(89)
'Jin =
L: an;d;n
i-a+l
J
348
V. VIBRATIO N AND EIGENVA LUE PROBLEM S
a relation in which the coefficients an; are already determin ed by
(88). For n ¢ l we obtain
We summarize the result: Given the a-fold degenerate multiple
eigenvalue X1 = X, select a system of orthonor mal eigenfunctions
U1 , U2 , • • • , Ua
for which the matrix dnz =
matrix with elements dnn.
eigenvalue is given by
JrunUz dg is a diagonal
Then the first order perturba tion of the
and the first order perturba tion of the eigenfunctions by
00
Vn =
L
an;u;
;-1
with
ann
and
=
0
if at least one of the indices l or n is greater than a, and by
-
anz -
1
~
dn;d;1
£..J - - dnn - du ;-a+l An - Aj '
if neither index is greater than a and l ~ n.
The perturba tion terms of second and higher orders are obtained
in a corresponding manner. The second order perturba tion of the
n-th eigenvalues, in particula r, is given by
Jin=
~
d!;
£..J - - i=a+l An - Aj
as we see from (89).
3. An Example.1 Consider the problem of a freely vibrating string,
with ends fixed at x = 0 and x = 'Ir, having the coefficient of elasticity
1
Cf. Rayleigh, The Theory of Sound, Vol. I, pp. 115-118.
349
PERTURBATION THEORY
p = 1 and a mass density p(x) which differs slightly from a constant
value Po in the interval 0 ~ x ~ 7r and thus has the form p(x) =
Po + Eu(x), where u(x) is a given function and E denotes a "perturbation parameter". By §3 the associated eigenvalue problem is of
the form
(90)
u: +
"X.npoUn = 0 which
For E = 0 we get the undisturbed problem
2
has the solution "X.n = n / Po, Un = V2/7rpo sin nx.
Since all the eigenvalues are simple, we obtain the first approxima1
tion for the disturbed problem (90) by substituting the expressions
n2
Un=
Xn = - ,
Po
V
2.
- sm nx
7rPo
and
n2
r(x) = -Xnu(x) = - - u(x)
Po
in formulas (80) and (81) of subsection 1. Thus for the first order
perturbation µn of the eigenvalues we obtain
µn = - n: ~ f u(x) sin2 nx dx
Po 7r Jo
T
and for the eigenfunctions Vn we have
00
(91)
Vn
=
an;U;
L
i=l
where
2
(92)
11-r () . . .
n -- u x sm nx sm JX
an; = 2- - : 2
2
7rJ-npoo
ann
=
,J_
(£;(;
(j
¢ n);
0.
1 In subsection 1 we assumed that the function r(x) in the perturbation
term Er(x) did not depend on E; however, the function Xnu(x) in the corresponding perturbation term of (90) depends on E. Since we shall be concerned
only with first order perturbations, we may set r(x) = -Xnu(x), where Xn no
longer depends on E.
350
V. VIBRATION AND EIGENVALUE PROBLEMS
In order to apply these results, we calculate the displacement ox
of the first nodal point, which corresponds to n = 2 and lies at the
midpoint in the case of a homogeneous string.
Since we have assumed that Un can be expanded in powers of E,
we can write ox in the form ox = tr+ l( · · · ). We find the following
equation for r:
li2(~ +ET+ •••)
0 =
=
u.(i +
=
u2 (~) + .[ru;(~) + v2(i)J+ .'( · · ·).
ET
+ "••) + Ell2(~ +
ET
If in this equation we set the coefficients of
sider (91) we obtain
+ •••) + .'( •••)
E
equal to zero and con-
T=
since u 2(x) = const. sin 2x. If, for example, the nonhomogeneity
of the string is due to the presence of a small mass PoK at the point
x = 7r/4, we obtain from (92) by means of a simple limiting process
the following expression for r:
T
=
. 7r
sm 4
(
7r 12 - 4
4. K
4K(
1
= 7rv2 12
= - ..
-
. 37r
- sm
32
-
4 - 32
4
. 57r
)
sm 4 - ...
4 . 52 - 4
1
-
+
4 - 52
1
-
1
[
T
=
-K./2.
)
+ .. .
~( 1 + ~ - ~ - ~ + ~ + ii - ... ).
The value of the series in parentheses is
and
4
0
1
1
+ x2
7r
+ x' dx = 4 v2
GREEN'S FUNCTION AND DIFFERENTIAL EQUATIONS
351
Green's Function (Influence Function) and Reduction
of Differential Equations to Integral Equations
In this section we shall consider and supplement the theory from
an essentially different point of view. Instead of beginning with a
vibration or eigenvalue problem, we shall start with a boundary
value problem and develop the method of Green's function (or
influence function) to represent its solutions. In this way, the
eigenvalue differential equations will be reduced to symmetric integral
equations; the existence of the eigenfunctions and validity of the
completeness and expansion theorems will thereby be automatically
proved.
1. Green's Function and Boundary Value Problem for Ordinary
Differential Equations. First we consider a linear homogeneous selfadj oint differential expression of second order
§14.
L[u] = pu"
+ p'u'
- qu
for the function u(x) in the fundamental domain G: Xo ~ x ~ x1 ,
where p, p' and q are continuous functions of x and p > 0. The
associated nonhomogeneous differential equation is of the form
(93)
L[u] = -cp(x),
where cp(x) denotes a piecewise continuous function defined in G.
We are concerned with the boundary value problem: find a solution
u = f(x) of equation (93) which satisfies given homogeneous boundary
1
conditions at the boundary of G, e.g., the boundary condition u = 0.
It is natural to start with the following_heuristic considerations: We
again consider equation (93) as the condition for equilibrium of a
string under the influence of a time-independent force distributed
continuously with density cp(x) over the string. We visualize a limiting process from the continuously distributed force to a "point force,"
i.e. to a force acting at a single point x = ~ with a given intensity.
The string is subjected to boundary conditions. Let K(x, ~) denote
the deflection of the string at a point x as a result of the action of a
point force of unit intensity applied at the point ~; then the effect
at x of the continuously distributed force cp can be considered as the
superposition of the effects of continuously distributed point forces
1 It will be recalled that the boundary value problem of the homogeneous
differential equation with nonhomogeneous boundary conditions can always
be reduced to this problem. (Cf. §1, 2).
352
V. VIBRATION AND EIGENVALUE PROBLEMS
whose linear density at each point t is equal to cp(t).
expect the desired solution to be of the form
Thus we can
z1
(94)
u(x) =
1
K(x, t)cp(t) dt.
zo
The function K(x, t), called the influence function or Green's function
for the differential expression L[u], satisfies the prescribed conditions
at x = Xo and x = x1. It follows that the function u(x), which is
represented in equation (94) by an integral in terms of the kernel
K(x, t) and the density cp(x), also satisfies these boundary conditions.
The influence function K(x, t) satisfies the differential equation
L[K] = 0
everywhere except at the point x = ~' since K corresponds to a force
zero when x ¢ t· At the point x = ~ the function K(x, ~) must
have a singularity, which can be found heuristically in the following way: We consider the point force as the limiting case of a force
cpE(x) which vanishes in G for I x - t I > E and the total intensity of
which is given by the equation
HE
1
cpE(x)dx = 1.
~-E
Denote the associated deflection of the string by Ke(x, t); thus,
L[Ke] = (pK:)' - qKE = -cpE(x). Integrating this equation between the limits ~ - o and ~ + o, where o ~ E may be chosen arbitrarily provided the interval of integration remains in the fundamental domain G, we obtain
1
H6
~-6
(_!!:_
dx
(p dKE)
- qKe) dx
dx
=
.,.....1.
If we first take the limit as E -+ 0, assuming that KE converges to a
continuous function K(x, t) which is continuously differentiable except at x = t, and then let o approach zero, we obtain the relation
lim dK(x, t) 1z=H6
dx
z=~-6
.S-+O
=
--~
p(~)'
which characterizes the singularity of Green's function.
Reversing this heuristic discussion, we shall turn it into a
rigorous mathematical theory. We begin by defining a function
K(x, ~) as Green's function of the differential expression L[u], under
GREEN'S FUNCTION AND DIFFERENTIAL EQUATIONS
353
given homogeneous boundary conditions; we stipulate the following
properties:
(1) K(x, ~) is, for fixed~' a continuous function of x and satisfies the
prescribed boundary conditions.
(2) Except at the point x = ~' the first and second order derivatives
of K with respect to x are continuous in G. At the point x = ~
the first derivative has a jump discontinuity given by
dK(x, ~) 1x=e+o = __1_
p(~).
x-H
dx
(95 )
(3) K considered as a function of x satisfies the differential equation
L[K] = 0 throughout G, except at the point x = ~.
A continuous function which satisfies conditions (2) and (3), but
not necessarily the boundary conditions, is called a "fundamental
solution" of the differential equation L[u] = 0.
We shall prove the following relations, which seem plausible in
view of the above discussion: If ~(x) is a continuous or piecewise
continuous function of x, then the function
u(x) =
(96)
Jx
~)~(~) d~
1
K(x,
XO
is a solution of the differential equation
-~(x)
L[u] =
(97)
and satisfies the boundary conditions. Conversely, if the function u(x)
satisfies equation (97) and the boundary conditions, it can be represented by (96). To prove the first statement only the elementary
rules for the differentiation of an integral with respect to a parameter
are needed. Thus, because of (95), we obtain consecutively the
following equations:
u'(x) =
u"(x) =
Jxxo
1
K'(x,
i:
K"(x,
~)~Ct) d~,
~)~(~) d~ + {
1
K"(x,
+
K'(x, x - O)~(x) - K'(x, x
=
lx
XO
1
K"(x,
e)~(e)
~)~(~) d~
+ O)~(x)
de + (K'(x + 0, x)
K'(x - 0, x))~(x)
354
V. VIBRATION AND EIGENVALUE PROBLEMS
therefore,
pu"
+ p'u' -
s1
qu
=
1
(pK"
zo
+ p'K' -
qK)<p(~) d~ -
<p(x).
This completes the proof, since L[K] = 0.
To prove the converse we again use Green's formula (§1, (2b));
setting v = K we apply the formula to the two domains of integration Xo S x S ~and~ S x S X1. Formula (96), with x and~ interchanged, then follows directly from the discontinuity relation and the
boundary conditions.
If u and K are not subjected to the same homogeneous boundary
conditions (e.g. K = 0 at the end points x = x 0 and x = x1), we
obtain in the same way the more general expression for u
u(x)
=
1
%1
K(x, ~)<p(~) ~
zo
+ pK'u 1%1zo ;
for <p = 0 this expression furnishes the representation of the solution
of the boundary value problem for the homogeneous differential
equation L[u] = 0 in terms of the boundary values.
Green's function of a self-adjoint differential expression is a symmetric function of the parameter ~ and the argument x, that is,
K(x, ~)
=
K(~,
x).
This follows almost immediately from Green's formula (§1, (2b)),
if we substitute v = K(x, TJ), u = K(x, ~) and divide the domain of
integration into the three intervals Xo S x S ~, ~ S x S 71, 71 S x S X1 ,
treating each interval separately. The proof is completed by taking
into account both the discontinuity relation (95) at the points x = ~
and x = .,, and the boundary conditions. The symmetry of Green's
function expresses a reciprocity frequently occurring in physics: If the
force 1, applied at the point ~' produces the result K(x, ~) at the point x,
then the force 1 acting at x produces the same result at~2. Construction of Green's Function; Green's Function in the Generalized Sense. We construct Green's function for L[u] under the
prescribed boundary conditions: Consider any solution Uo(x) of the
differential equation L[u] = 0 which at x = Xo satisfies the given
boundary condition, e.g. vanishes. Then CoUo(x) is the most general
such solution. Similarly, let c1u1(x) be the family of solutions of
L[u] = 0 which satisfy the boundary condition at x = X1. There
GREEN'S FUNCTION AND DIFFERENTIAL EQUATIONS
355
are two possible cases: Either the two families of solutions are
distinct-the "general" case-or they are identical. In the first
case the functions Uo , u 1 are linearly independent, in other words,
1
u.ou~ - u~u1 ~ 0. In this case a curve of the first family can never
touch a curve of the second family in the fundamental domain (for,
if it did, the equation would be contradicted at the point of contact).
We can choose the two constants Co , c1 such that the point of intersection has a given abscissa x = ~ in the interval G and such that
the discontinuity of the derivative at this point has precisely the
value -1/p(~). In this fashion we obtain Green's function K(x, ~)
explicitly by means of the formulas
1
U = - - U1 (~)Uo(X)
c
In the second case, Uo and u 1 differ only by a constant factor.
Every solution belonging to one family also belongs to the other.
Thus in this case the function u.o(x) satisfies not only the condition
for x = 0 but also the condition for x = I, and the equation L[u] = 0
has a nontrivial solution u.o(x) that satisfies the-boundary conditions.
This can also be expressed by stating: "A = 0 is an eigenvalue of
L[u] + "Au = 0. Hence the above construction fails, and no Green's
function exists.
'"'The existence of Green's function is equivalent to the existence of
a unique solution of the homogeneous boundary value problem for
the differential equation L[u] = -cp(x) (cf. subsection 1). Therefore the following alternative exists: Under given 1wmogeneous boundary
conditions, either the equation L[u] = -cp(x) has a uniquely determined
solution u(x) for every given cp, or the homogeneous equati0n L[u] = 0
has a nontrivial solution.
Moreover, it will turn out that in the second case the nonhomogeneous equation L[u] = -<p(x) has a solution if, and only if, the
For, if c is a constant, then A= uou~ - u~u1""'
+ tl.p' = 0
from the given differential equation.
1
This is immediately clear.
c/p holds, as can be easily verified by deriving the equation pA'
356
V. VIBRATION AND EIGENVALU E PROBLEMS
solutions Uo(x) of the homogeneous equation L[Uo] = 0 are orthogonal to
the function <p as expressed in the orthogonality condition
z1
1
zo
<p(x)uo(x) dx = 0.
The orthogonality condition is seen to be necessary if we multiply
the differential equation L[u] + <p(x) = 0 by the function Uo(x),
integrate over the domain G, and apply Green's formula, taking
into account the boundary conditions. That the condition-is also
sufficient can be shown if we introduce in place of the Green's function a "Green's function in the generalized sense." We are led to this
function again by a simple heuristic argument based on intuition.
We recall: If A is an eigenvalue with an associated normalized eigenfunction u, then under the influence of an external force of the form
-tf;(x)ei'\l"'i:.t the string becomes unstable (resonance) unless the relaz1
tion
tf;(x)u(x) dx = 0 is satisfied. In the present case, where
l
zo
A = 0, this means instability under the influence of an external timeindepende nt force. In particular, the string is unstable when a point
force is applied at an arbitrary point. If the system is not to depart
arbitrarily far from its rest state when a point force is applied, then
the string must be ba~anced by a fixed time-indep endent opposing
force. This opposing force may be chosen arbitrarily , except that it
may not be orthogonal to the eigenfunction uo(x), since then it would
not prevent the excitation of the eigenfrequency zero. It is convenient to assume that this balancing force is in the symmetric form
tf;(x) = -Uo(X)Uo(~). Then the influence function K(x, ~) of a point
force acting at the point x = ~satisfies not only the boundary conditions but also, except at the point x = ~' the differential equation
L[K] = Uo(X)Uo(~),
and at x = ~it must satisfy the discontinu ity condition (95). These
conditions determine a function K only up to an arbitrary additive
function c(~)Uo(x). To single out a particular function K, we require
1:
1
K(x,
~)uo(x) dx
= _o
and call the function K so defined Green's function in the generalized
sense for the differential expression L[u]. Using the assumption that
GREEN'S FUNCTION AND DIFFERENTIAL EQUATIONS
357
L[u] is a self-adjoint differential expression, we find, as before, that
Green's function in the generalized sense has the symmetry property
K(x, ~) = K(~, x).
These considerations may be illustrated by a simple example, the
string with both ends free (see also §15, 1). Here Uo = const. is an
eigenfunction for X = 0; for the opposing force we take a force
which is constant along the entire length of the string.
Green's function in the generalized sense can be constructed in the
same way as the ordinary Green's function. We have only to prove
the fact: if L[u] = O·has a nontrivial solution Uo which satisfies the
boundary conditions, then L[v] = Uo(~)Uo(x) can have no such solution. In fact, if we multiply the latter equation by Uo(x) and integrate over the fundamental domain, taking account of the boundary
conditions, we obtain
UoW Zl u~(x) dx
1
=
zo
1Zl· v(x)L[uo] dx
=
0,
zo
Jz u~(x)
1
which contradicts the assumption that
zo
dx
~
0.
Green's function in the generalized sense serves the same purpose
as the ordinary Green's function. Note that the solution w(x) of a
differential equation L[w] = -cp(x) is determined only up to an
arbitrary additive function CUo(x); it can therefore be fixed by
means of the
~ondition
Jz
1
WUo dx =
0.
We can thus state the
zo
theorem: Let w(x) be a function which is orthogonal to u.o(x), satisfies
the boundary conditions, and has continuous first and piecewise continuous second derivatives. If w(x) and the piecewise continuous function cp(x) are related by the equation
L[w] = -cp(x),
then relation
z1
(98)
w(x) =
1
K(x, ~)cp(~) d~.
zo
also holds.
Conversely, the latter relation implies the former, if cp(x) is orthogonal
to u.o(x). This converse contains the second part of the theorem
stated above (p. 355).
358
V. VIBRATION AND EIGENVALUE PROBLEMS
The proof is similar to the corresponding proof for the ordinary
Green's function, but every function w(x) of the form (98) must be
orthogonal to Uo(x), since this is true of Green's function K(x, t).
As we have already seen, X = 0 can be at most a simple eigenvalue in the case of the second order differential equations considered
here. If, however, X = 0 is a multiple eigenvalue (for differential
equations of higher order), one can construct Green's function in the
generalized sense with an opposing force of the form
where Uo , u1 , · · · denote the orthonormal eigenfunctions belonging
to the eigenvalue A = 0.
3. Equivalence of Integral and Differential Equations. By means
of Green's function the eigenvalue problems discussed earlier can be
completely solved if we replace the differential equation by an integral equation. We consider a linear family of differential equations
(99)
L[u]
+ Apu =
(p(x)
?/l(x)
>
0),
depending on a parameter X; here ?/l(x) is a piecewise· continuous
function, p(x) a positive continuous function, and u, by assumption,
satisfies the given boundary conditions, say u = 0. If Green's function for L[u] exists under the given boundary conditions, the following
equation is immediately obtained from formula (94) by setting
~(x) = Apu - y/I:
(100)
u(x) = X
1:
1
K(x, t)P(t)u(t) dt
+ g(x),
where
g(x) =
-jz
1
K(x, t)J/!{t) dt
zo
is a given continuous function of x; equation (100) is equivalent to
(99). Hence, finding the desired solution u of (99) under the prescribed boundary conditions is equivalent to solving the integral
equation (100). To the homogeneous equation
(101)
L[u]
+ Xpu
= 0
corresponds the homogeneous integral equation
1
u(x) = X {
K(x, t)p(t)u(t) dt.
zo
GREEN'S FUNCTION AND DIFFERENTIAL EQUATIONS
35~
If we introduce
u(x)VP{X) = z(x)
as a new unknown function, multiply the integral equation by
Vp(x), and set K(x, ~) = K(x, ~)Vp(x)p(~), then the integral equation
(102)
z(x) = >.
l:e
1
K(x,
:i:o
~)zW d~
corresponds to (101). The kernel K(x, ~) of equation (102) is sym1
metric, since L[u] is self-adjoint. We can therefore apply the theorems on symmetric kernels investigated in Chapter III and obtain
results-partly contained in subsection 2-for the differential equation (99).
The following alternatives hold for the relation between the boundary value problem of the nonhomogeneo us differential equation (99)
and that of the homogeneous differential equation (101) under the
given homogeneous boundary conditions: Either, for fixed )., every
solutiort of the homogeneous differential equation (101) vanishes identically ~ is not an eigenvalue of (101)); then the nonhomogeneous
equation (99) has one and only one solution for arbitrarily chosen 1/f(x).
Or, for some value 'A = >.., the homogeneous equation (101) has a nontrivial solution Ui (Xi is an eigenvalue of (101) with the eigenfunction
ui); then a solution of the nonhomogeneous differential equation (99)
exists for ). = >.. if and only if the relation
l
:Sl
pUi1/f
dx
::i::
0
zo
holds for all eigenfunctions U,; associated with the eigenvalue ).i.
Moreover: There exists a sequence of eigenvalues 'A1 , >.2 ,
(An -+ oo )-with associated eigenfunctions U1 , ~ , • • • which form an
infinite system of functions satisfying the orthogonality relations
l:ei pUiUk dx
:eo
(i 'F- k),
= 0
l:ei pU~ dx
= 1.
zo
If; with Green's function K(x, ~) as kernel, we can represent a function
w(x) by a piecewise continuous function ~min the form
w(x}
= l:e
:eo
1
K(x,
~)~W d(~),
1 This symmetry and its consequences show the significance of the assumption that L[u] is self-adjoint.
360
V. VIBRATION AND EIGENVALUE PROBLEMS
then w(x) can be expanded in terms of the eigenfunctions into an absolutely and uniformly convergent series
w(x) =
t
z1
CnUn(x),
n=l
Cn =
J
zo
WpUn dx.
The set of functions which can be expanded in this way may be
more simply characterized. The equation L[w] = -<p(x) follows
from (94) because of the fundamenta l property of Green's function.
Conversely, if we consider any function w(x) which satisfies the
boundary conditions and has a continuous first and a piecewise
continuous second derivative, we can construct a corresponding
source-distribution <p(x) by means of the equation L[w] = ~<p(x).
Thus we obtain the result: Every function w(x) which satisfies the
boundary conditions and has ·a continuous first and a piecewise continuous second derivative may be expanded in an absolutely and uniformly convergent series w(x) = L':=1 CnUn(x).
This theorem implies immediately that the eigenfunctions form a
complete orthogonal system. For, any function continuous in G may
be approximate d in the mean to any desired degree of accuracy by a
continuous function which satisfies the boundary conditions and has
continuous first and second derivatives. Thus, in virtue of the
expansion theorem just stated, it may also be approximate d by a
finite combination of the form L':=1 CnUn(x).
The fact, noted earlier,1 that all the eigenvalues are positive (in
the terminology of the theory of integral equations, this means that
the kernel K(x, t) is definite) leads to a stronger form of the expansion
theorem. Since, in addition, K(x, t) is a continuous function of x a.nd
~'we can apply Mercer's theorem of Ch. III, §5, 4 and conclude that
the series expansions of the kernel
K(x, t) =
Vp(x)p(~) ~1 Un(X~~n(t)
(103)
or
K(x,
~) =
f:
n=l
Un(x)un(t)
An
converge absolutely and uniformly. This formula, explicitly relating
Green's function to eigenfunctions, is called simply the bilinear relation; for fixed ~ it represents a series expansion of a continuous func1
Cf. p. 294.
GREEN'S FUNCTION AND DIFFERENTIAL EQUATIONS
tion with a piecewise continuous first derivative.
combination
S = a1K(x, ~1)
361
If we form a linear
+ a2K(x, ~2) +
we obtain a continuous function the first derivative of which has the
preassigned jumps -a1/p(~1), -a2/p(~2), · · · at the prescribed points
~ 1 , ~2 , • • • ; this function can be expanded in an absolutely and
uniformly convergent eigenfunction series. Since from any function with piecewise continuous first and second derivatives we can
subtract a particular function S in such a way that the difference
satisfies the conditions of the above expansion theorem, we obtain
the result: A sufficient condition for the validity of the expan8'ion theorem
is that the first and second derivatives of the continuous function w(x)
should be piecewise continuous.
In this section we have assumed that Green's function for L[u]
exists, i.e. that X = 0 is not an eigenvalue of our differential equation L[u] + "Apu = 0 (subsection 2). If this assumption is not valid,
we replace the ordinary Green's function by Green's function in the
generalized sense; all considerations leading to the reduction of the
eigenvalue problem of equation (101) to an integral equation problem
hold. To insure the validity of the expansion theorem we must add
the condition of orthogonality to the eigenfunction Uo(x) associated
with A = 0. This condition, however, disappears completely from
the final formulation of the expansion theorem if we include the eigenfunctions associated with the eigenvalue "A = 0. The occurrence of
a zero eigenvalue is in no way pathological; this will be seen in connection with an approach to the eigenvalue problem by a method
based on the calculus of variations (cf. Ch. VI, §1).
In conclusion, we shall expand the solution of the nonhomogeneous
equation (99) with respect to the eigenfunctions. Corresponding to
the earlier scheme of §3, 3, which can now be justified by the expansion theorem, or directly by the integral equation theorem (Chapter
III, formula (56)) we obtain the solution
00
u(x) =
L 'YnUn(x)
n-1
with
This clarifies the fact that,
Cn
A - Xn'
z1
1
Un(x)t/l(x) dx.
=
zo
if X = Ai is an eigenvalue, equation (99)
'Yn
Cn =
z1
cannot be solved unless the orthogonality relation
1
zo
t/lui dx = 0 is
362
V. VIBRATI ON AND EIGENVA LUE PROBLEM S
satisfied. In physical terms: When the external force is in resonance
with an eigenvibration, ·a stationary state exists if and only if this force
performs no work on the system as the system moves in the given eigenvibration.
4. Ordinar y Differen tial Equatio ns of Higher Order. Ordinar y differential equations of higher order are not essentially different. We
restrict the discussion to a typical example, associated with the differential equatio n u"" - Xu = 0 or u'"' - 'Apu = 0, the homogeneous
or nonhomogeneous rod (cf. §4). As before, we introduce the influence function or Green's function K(x, ~)as the displacement of the rod,
in the state of equilibrium, under the influence of a point force acting
at the point x = ~and satisfying the prescribed boundar y conditions.
In the same manner as above we obtain the following typical conditions for this function:
(1) For every value of the parame ter ~ the function K(x, ~), together with its first and second derivatives, is continuous and satisfies
the prescribed homogeneous bounda ry conditions.
(2) At any point different from x = ~ the third and fourth derivatives with respect to x are also continuous. However, at x = ~the
following disconti nuity condition holds:
lim [K'"(~
E~O
+
~)
E,
-
K"'(~ -
E,
~)]
= -1.
(3) Except at the point x = ~' the differential equatio n
K""(x, ~)
=0
is satisfied everywhere.
The fundam ental propert y of Green's function can be stated as follows: Let u(x) be a continuous function which satisfies the bounda ry
conditions and has continuous first, second, and third, and piecewise
continuous fourth derivatives; let ~(x) be a piecewise continuous
function. If u(x) and cP(x) are connected by the relation
L[u]
=
u"" = -~(x),
then we have the represen tation
u(x)
and conversely.
=
1:i:i
K(x, ~)~(~) d~
zo
,
GREEN'S FUNCTION AND DIFFEREN TIAL EQUATION S
363
The eigenvalue problem of the more general differential equation
u"" - °'Xpu
= O,
the associated expansion theorem, and the theory of the nonhomogeneous equation
u"" -
Xpu, = -,Y{x)
may be treated like the corresponding questions in subsection 3, i.e.
by reduction to an integral equation with the symmetr ic kernel
K(x, ~) = K(x, t)Vp(x)p (t)· The result obtained is: There exists an
infinite system of eigenvalues X1 , X2 , · · · and associated eigenfunc tions
u 1 , u2 , · · · with the properties that the functions y'pu, constitute a complete orthogonal system and that any function w(x) which satisfies the
boundary condition s and has continuou s derivatives up to the third order
and a piecewise continuou s fourth derivative may be expanded in terms of
these functions in an absolutely and uniforml y convergent series. More1
over, Mercer's theorem implies that the bilinear relation
K(x,
~)
=
t
n-1
Un(x)unW
Xn
holds and that the expansio n theorem can be extended
third derivatives are only piecewise continuou s.
to functions whose
The existence proof and construction of Green's function and of
Green's function in the generalized sense offer no new difficulties.
They will be illustrate d by examples in §15.
5. Partial Differential Equations. In the case of partial differential
equation s with homogeneous boundary conditions, Green's function
can again be introduce d as the kernel of an equivale nt integral equation. As an example, consider the second order partial differential
equation
ll.v = -cp(x, y)
in the x, y-plane for a domain G under a homogeneous boundary condition, e.g. v = 0. This equation characterizes the deflection of a
stretched membran e in a state of equilibrium under the influence of a
time-independent force of density cp(x, y). As before, the solution
can be obtained with the aid of a Green's function K(x, y; ~' 11), repreCf. Ch. III, §5, 4. As in the problem of the vibrating string (cf. p. 360)
we can show that the kernel is positive definite.
1
364
V. VIBRATION AND EIGENVALUE PROBLEMS
senting the influe.nce at the point (x, y) of a point force acting at the
source point (~, 71). K and its derivatives of first and second order
must be continuous everywhere except at the point x = ~, y = 11, and
the differential equation AK = 0 must be satisfied. Furthermore , K
must satisfy the given homogeneous boundary conditions, and possess
a characteristi c singularity of a point force at the source point x = ~'
y = 71. The nature of this singularity is determined if we surround
the source point with a circle k of radius e, assume an external force of
density <{)E(x, y) vanishing outside k for which
Jl
<{)E(x, y)dx dy = 1,
and consider the Green's function K(x, y; ~' 11) as the limit (fore~ 0)
of that solution KE(x, y; ~' 11) of AK = -<pE which satisfies the given
boundary conditions. If we integrate the equation AK = -'<{)E over
the circle of radius o ~ e and apply Green's formula (5a) of page 280,
we obtain
1:r
KE ds = -1.
Here r = y (x - ~) 2 + (y - 17 ) 2 denotes the distance of the point
(x, y) from the point (~, 11), and sis the arc length measured on the
boundary K. We shall therefore subject the Green's function which
we wish to characterize to the condition
1:r
K(x, y;
~' 71) ds
=
-1.
This condition is satisfied if we require that, in a neighborhoo d of the
source point, K be of the form
K(x, y; ~' 71) = -
1
11" log r
2
+ 'Y(x, y; ~' 71),
where 'Y(x, y; ~' 71), together with its derivatives of first and second
order, is continuous in x and y. (Since log r is, for r ~ 0, a solution of
the potential equation, 'Y itself is a regular potential function.)
Reversing the order of this heuristic discussion, we define Green's
function K by means of the following conditions:
(1) Except at the source point (~, 71) the function K(x, y; ~' 11) and
its derivatives of first and second order with respect to x and y are
continuous. The function K has the form
K(x, y; ~' 71) = -
1
11" log r
2
+ 'Y(x, y; ~' 71),
GREEN'S FUNCTION AND DIFFER~NTIAL EQUATIONS
where -y(x, y; ~' 11) and its derivatives up to the second order are continuous.
(2) K satisfies the prescribed homogeneous boundary conditions.
(3) Everywhere except at the source point, the differential equation
.1K = 0 is satisfied.
Green's function so defined satisfies the symmetry condition
K(x, y; ~' 11) = K(~, 71; x, y).
The proof of this symmetry, which is another expression of the
physical reciprocity noted above, is again obtained from Green's
formula. We apply this formula for the functions K(x, y; ~' 11) and
K(x, y; ~', 11') to a domain which is obtained from G by removing the
circular disks k and k', each of radius E, drawn about the points (~, 11)
and (~', 11'), respectively. Taking the limit E --+ 0 and keeping in
mind the singularity property of Green's function, we obtain-since
the integral over the boundary r of G vanishes because of the boundary conditions-the symmetry formula in the form K(~', 11'; ~' TJ) =
K(~, 71; ~', 71').
As before, the fundamental property of Green's function is: Let
u(x,y) be any function which satisfies the homogeneous boundary conditions-say u = 0-and which is continuous and has continuous first
and piecewise continuous second derivatives in G. If
L[u] = .1u = -cp(x, y),
the relation
u(x, y) =
J£
K(x, y;
~' 71)cp(~, ,,,) d~ d'rf
holds. On the other hand, if cp(x, y) is any function which, with its first
derivatives, is continuous in G, then the function
u(x, y) =
J£
K(x, y;
~' 71)cp(~, 11) d~ d71,
which is continuous in G, has continuous first and second derivatives and
satisfies both the boundary condition and the differential equation
.1u
=
-cp(x, y).
Note that the differentiability assumption for cp(x,y) is more
stringent in the second part of the theorem than in the first. (For
ordinary differential equations, there was no such difference.)
366
V. VIBRATION AND EIGENVALUE PROBLEMS
The first part of the theorem follows again almost immediately
from Green's formula (5a). For v = K(x, y; ~' 11) we apply this
formula to the domain G - k, which is obtained from G by removing
a small circular disk k of radius E about the point (x, y); again we
denote the circumference of k by K. Since in the resulting domain
AK = 0, and the boundary integral over the boundary r vanishes,
we are left with
In the limit as
E
~
0,
1
1
u(iJK/iJn) ds approaches u(x, y) and
K(iJu/iJn) ds approaches zero, giving the desired result
u
=fl K'Pd~d71.
The second part of the theorem is proved most simply by means of
an artifice introduced by Riemann, in which the assumed continuity
of the first derivatives of 'P(x, y) 1 is used. We decompose the function
u(x, y)
=
Ji
K(x, y;
~' 11)'P(~,
11)
d~ d11
into two terms corresponding
to the decomposition K = -log r/2r
function, namely u = 1/; + x, where
21r1/;(x, y) =
x(x, Y) =
-Ji
Jl
+ -y(x,
'Pa, 71) log
-y(x, y;
y; ~' 11) of Green's
rd~ d.,,,
~' 11)'P(~, 11) dt d71.
Since the function -y(x, y; ~' 11) and its derivatives up to the second
order are everywhere continuous, we can at once form Ax by differentiation under the integral sign and obtain Ax = 0 since A-y = 0.
Hence, to calculate Au we need only compute Al/;. Again, by differentiating under the integral sign we obtain the first derivative 1/lz •
1 One cannot deduce the existence
of continuous second derivatives of u
merely from the continuity of 'P· The assumption made in the text is, however,
more stringent than necessary.
GREEN'S FUNCTION .AND DIFFEREN TIAL EQUATIO NS
Introduc ing polar ·coordinates r, fJ the integral
Ji
Jl
assumes the form
Ji <P(~,
7/) log
367
rd~ d.,,
<Pr log r dr dfJ; but if, before we introduce the
polar coordinates, we differentiate with respect to x, the integral
<P cos fJ dr dfJ, and the integrand remains continutakes the form
ous. If for the time being we set -log r/2r
obtain
"1z
Ji
=
=
S(x, y; ~' .,,), we
Sz<P d~ d.,,.
Since Sz = - SE , we may also write
"1z
= -
Ji
SE<P d~ d.,,.
Integrati ng this formula by parts, we may eliminate the derivativ e
SE and then once more differentiate under the integral sign. We
obtain
and
"1zz
= -
l
Sz<P d.,,
+
Jl
l
Se<P d.,, -
Jl
s"'°.,, d~ d.,,
Sz<PE d~ d.,, =
Jl
Se<Pe de d~.
Similarly, we find
"11111 = -
£
s""° d~ -
and hence
A"1 =
l :~
<P ds -
fl
(SE<PE
+ s""°") de d.,,.
If, on the right, instead of extendin g the double integral over the
entire domain G we integrate over a domain Ge obtained from G by
cutting out a small circular disk k of radius E and circumference K
about the point (x, y), then we can write
A"1
=
{
~ <P ds
lr "'"
- lim
e... o
ff
Jloe
(Se<Pe
+ S""°") de d.,,.
368
V. VIBRATION AND EIGENVALUE PROBLEMS
In this expression we transform the double integral on the right according to Green's formula and find, since A.S = 0 everywhere in G,
1
1-;-
J
J
as 'fJ ds as 'fJ ds + hm
.
.
-;-as 'fJ ds = hm
-as 'fJ ds.
un
r un
E-+0
IC an
E-+0
IC an
As we have already seen, the remaining boundary integral on the
right tends to -'fJ(x, y) as E ~ 0. This proves that I/; satisfies ''Poisson's equoJ,ion" Af = -'().
For Poisson's equation in three dimensions, A.u = -'fJ(x, y, z), and
for the associated eigenvalue problem of the equation
A.I/;
=
r
Au+ ~u = 0,
we obtain results which correspond word for word to those for the
two-dimensional case. In three dimensions, however, the singularity for Green's function is
1
1
41rT
411" V (x - ~)2 + (y - 17)2 + (z - t)2
and Green's function K(x, y, z; ~' 11, t) is of the form K(x, y, z; ~' 17, t) =
1/41rT + -y(x, y, z; ~' 17, t), where -y(x, y, z; ~' 17, t) and its derivatives
of first and second order are continuous. The function 1/41rT is itself
a fundamental solution of the equation A.u = 0 (cf. pages 353 and 377).
In the case of partial differential equations, the existence of Green's
function is, far more difficult to establish than in the case of ordinary
differential equations. General existence theorems will be proved in
Volume II by means of the direct methods of the calculus of variations. At this point we must either postulate the existence of Green's
function or limit ourselves to domains in which Green's function can
be represented explicitly. In the next section we consider such
domains. Once we have Green's function, however, the remaining
discussion is parallel to that for ordinary differential equations. We
consider here-for p > 0-the eigenvalue problem for the differential
equation
A.v + ~p(x, y)v = 0
under given homogeneous boundary conditions. As a consequence of
the fundamental property of Green's function, we at once obtain
from (104) the homogeneous integral equation
(104)
v(x, y)
=
X
Jl
K(x, y;
~' 17)p(~, 17)v(t, 17) d~ d17.
GREEN'S FUNCTION AND DIFFERENTIAL EQUATIONS
369
Let us introduce the symmetric kernel
K = KVp(x, y)p(~, 11)·
Evidently the function
u(x, y) =
v p(x, y)v(x, y)
satisfies the symmetric homogeneous integral equation
(105)
u(x, y)
= A
Jl
K(x, y;
~' 11)u(~, 11) d~ d17;
since these relations may be inverted, the eigenvalue problem of
equation (104) is seen to be completely equivalent with that of the
symmetric integral equation (105). This integral equation may be
treated by the theory of Chapter III; for, although the kernel becomes
infinite at some point of the domain of integration, the integral
Jl
K(x, y;
~'
11)
2
d~ d17 exists
and is continuous in the variables x, y.
Hence we are assured of the exisf,ence of the eigenvalues X1 , A2 , • · • with
An ~ oo f01· n ~ oo and of an associated system of eigenfunctions
v1 , v2 , · · · , assumed to be normalized and denoted by u 1 , u 2 , · · · .
According to the theorem on Green's function (cf. page 365) any function w(x, y) which has continuous first and second derivatives and
satisfies the boundary conditions may be represented by an integral
of the form
w(x, y)
fl
=
K(x, y;
~' 17)h(~, 11) d~ d17
in terms of the function h = -dw. Thus we obtain the result:
Every function w(x, y) which satisfies the boundary conditions and possesses continuous derivatives up to the second order may be expanded in
terms of the eigenfunctions in an absolutely and uniformly convergent
series w
=
L:':...
1 CnVn(X,
normalized eigenfunctions
y) with Cn
fl
=
pWVn dx dy.
Thus the
VP Vn form a complete orthogonal system.
Since Green's function becomes infinite, Mercer's theorem cannot
be applied here as it could in the case of ordinary differential equations. Therefore, in spite of the positive definite character of the
kernel, we cannot conclude that the equation
.t ) = ~
K( x,y,~,11
£...J
n-1
Vn(X,
y)vn(~, 17)
'
An
370
holds.
V. VIBRATION AND EIGENVALUE PROBLEMS
Our general theory proves only the weaker relation
lim
m-+co
ff
JJo
[K - t Vn(X, y)vn(~, 11)]2 dx dy
n=l
=
0.
An
The discussion of the general self-adjoint differential equation
pAv
+ p;JJ:i; +
p11v11
-
qv
+ >..pv = 0
is parallel to the previous discussion; it will therefore be sufficient to
point out the fact that the results also remain unchanged. The
only difference that should be mentioned is that now Green's function
must have the form
K(x, y; ~' 11)
a(x, y; ~' 11) 1
2rp(~, 11) og r
+ 'Y (x, y; ~' 11 '
1:
)
where 'Y(x, y; ~' 11) and its derivatives up to the second order are continuous in a neighborhood of the singular point (although in general
'Y need no longer be a solution of the differential equation) and where a
denotes a suitable function having continuous derivatives up to the
second order, such that a(~, 11; ~' 11) = 1 holds identically.
Similarly, for partial differential equations of higher order, the only
essential difference lies in the form of the singularity associated with
Green's function. If we consider, for example, the differential equation of a plate
AAv = -<P(X, y),
in two independent variables, then we must specify Green's function
not only by boundary conditions and the requirement AAK = 0,
but also by assuming that it has the form
K
= - 8~ r 2 log r + 'Y(x, y; ~' 11 ),
where 'Y(x, y; ~' 11) and its derivatives up to the fourth order are continuous. The reader will easily verify that the given singularity is
actually the correct one, i.e. that it corresponds to a point force. It
should be stressed, moreover, that the function r 2 log r is itself a
"fundamental solution" of AAv = 0.
In the case of a plate, passage to the corresponding integral equation
again shows~ A sequence of eigenvalues and associated eigenfunctions
which form a complete orthogonal system exists; every function which
371
EXAMPLES OF GREEN'S FUNCTION
satisfies the boundary conditions and has continuous derivatives up to
the fourth order in the fundamental domain G can be expanded in an
absolutely and uniformly convergent series in terms of these eigenfunctions.
§15. Examples of Green's Function
1. Ordinary Differential Equations.
Green's function for the ex-
pression
L[u] = u"
in the interval (0, 1) with the boundary conditions u(O) = u(l) = 0
is
for
(1. - ~)x
x ~ ~'
K(x, ~) =
{ (1 - x)~
for
x > E.
For the boundary conditions u(O) = 0, u'(l) = 0, Green's function
becomes
for
x ~ t,
K(x,
= {;
for
x > ~.
~)
For the interval - 1
:s;
~
x
+ 1 and the boundary conditions
u(-1) = u(l) = 0
we find
K(x, t)
=
-!{Ix - ~I
+ x~ -
1},
an expression which could also be obtained from the first example by
a transformation. On the other hand, in the interval 0 ~ x ~ 1,
for the boundary conditions u(O) = -u(l), u'(O) = -u'(l), we have
K(x, ~) =
-! Ix - t I + !.
Green's function for the differential expression
L[u]
=
xu"
+ u'
associated with the Bessel function of zero-th order Jo(x), for the
interval 0 :s; x :s; 1 and the boundary conditions u(l)
0, u(O)
finite, has the form
K(x,
~) =
t
for
x
~
t,
{ -log x
for
x
>
~'
-log
372
V. VIBRATION AND EIGENVALUE PROBLEMS
as may be easily ascertained by the general methods of the preceding
section. Green's function, under the boundary conditions u(l) = 0,
u(O) finite, for the differential expression
2
L[u) = (xu')' - n u
x
wr-
associated with the Bessel function Jn(x) (cf. eq. (28)), is given by
K(x,
~) = ~
K(x,
~)
=
(x~)" J
~ [ (Dn - (x~rJ
(x :,;
~)
(x > ~).
As a further example, consider the differential expression
L[u)
h2
2
((1 - x )u')' - - -2 u,
=
1 -
x
which for h = 0, 1, 2, · · · is associated with the Legendre functions
of zero-th order, first order, etc., respectively; the interval of definition is -1 5 x 5 +1, and the boundary conditions are: u finite at
both end points. We can immediately specify solutions of L[u] = 0
which are finite at x = -1 and x = 1, namely, c1[(l + x)/(1 - x)]h/ 2
and c2[(l - x)/(1 + x)t 12 , respectively. Combining these solutions
. by the rules of §14, 2, we obtain Green's function:_
K (x,
K(x,
~)
1 ( 1 + x 1 - ~)h'
= 2h 1 - x 1 + ~
~)
1(1+~1 = 2h 1 - ~ 1
x
+ x)h/
2
(x
S
~)
(x
>
~).
2
According to the general theory, this construction fails for h = 0,
since then the equation L[u] = 0 has the normalized solution u =
1/v'2 which is everywhere regular and satisfies the boundary conditions. Thus for h = 0 it is necessary to find a Green's function in
the generalized sense which satisfies the differential equation
L[u]
= t.
This function is given by
K(x, ~) =
where c = log 2 -
!.
-! log[(l
{-! log[(l
- x)(l
+ rn + c
+ x)(l
-
rn + c
(x
(x
S
~)
>
~),
373
EXAMPLES OF GREEN'S FUNCTION
Another simple example in which Green's function in the generalize d
sense occurs is the differential expression
L[u] = u"
for the interval -1 :::; x :::; + 1 with the periodic boundary conditions
u( -1) = u(l), u'(-1) = u'(l). Here again, there is a regular soluof L[u] = 0 which satisfies both boundary conditions
tion u = 1/
(corresponding to the physical problem of a string with both ends
free). Thus, we must construct Green's function in the generalized
sense from the differential equation
v2
'!.
2•
U " --
We easily obtain
K(x, ~) =
-! Ix
- ~I
+ !(x -
~)
2
+ j.
All these Green's functions serve as kernels of integral equations
whioh correspond to the respective differential equation problems.
We shall state explicitly the following bilinear formulas associated
with our examples:
!
t
2
2
(x
> t),
(x ;:; t),
f
!
11'"
n
(x ~ t),
(1 - t)x
{ (1 - x)~
sin n?rx sin n?r~
2
1r n=l
sin (n + !)?rx sin (n + !)?r~
(n + !) 2
·
n=O
{:
(x
> t),
furthermo re
K(x, t) =
t
(n
n-1
+ !)Pn(x)Pn(t)
n(n + 1)
where
-!log [(1 - x)(l +
K(x, t) = {
-! log [(1 + x)(l -
m +log 2 -
m+
l
log 2 - !
(x ~ t)
(x
> t).
Finally, we call particular attention to the Green's functions and
integral equations associated with the Hermite and Laguerre polynomials and orthogona l functions.
The differential equation (49)
u"
+ (1
2
- x )u +Xu = 0
374
V. VIBRATION AND EIGENVALUE PROBLEMS
of the orthogonal Hermite functions has the eigenvalue X = 0 under
the boundary conditions: regularity at plus and minus infinity. To
avoid constructing Green's function in the generalized sense, consider the value X = -2, which is certainly not an eigenvalue (cf.
page 328); accordingly, we construct Green's function for the differential expression
L[u] = u" - (1
+ x )u
2
under the boundary condition that the solution vanish at ± oo • To
obtain the general solution of the equation L[u] = 0, we note that
u(x) = e:r: 212 is a solution of L[u] = 0. Assuming the general solution
to be of the form u = we:r: 212, we at once obtain the equation
w"
+ 2w'x = 0
for w, which in addition to the obvious solution w = const. has also
the solution
W =
C1
l
s
cs
e-t' dt.
In this way, we find
Hence the particular solutions which vanish at x =
are given by
+ oo and x = -
oo
respectively. These lead at once to the following expression for
Green's function:1
(x
>
~).
1 Cf. R. Neumann, Die Entwicklung
willkiirlicher Funktionen etc., Dissertation, Breslau, 1912.
EXAMPL ES OF GREEN'S FUNCTIO N
375
The factor 1/Vr ensures the proper disconti nuity of the derivati ve
since the formula
100 e
1
-t2
y';
holds.
The differential equatio n L[u]
u(x)
=
=
dt
1
00
X
have the eigenvalues X = 2n
functions
+ Xu
1:
+2
=
K(x,
0 and the integral equatio n
~)uW
dt
(n = 0, 1, 2, · · ·) and the eigen-
e-z"12H,,,(x).
12
The Laguerre functions e-z L,,,(x) are solutions of the differential
equation
XU
, + ,+ (1z - x)4
u
u
+ Xu
0
=
for the eigenvalues X = n (n = 0, 1, 2, · · · ). We consider this
equatio n for the particul ar value X = _,.. 1, and define
L[u]
= xu"
+ u' -
G+ ~)u.
12
The equatio n L[u] = 0 has the particul ar solution ez . Assuming.
the general solution to be of the form
we obtain, as before, the expression
:£
W=C1
1
-t
~dt
cz t
for w, so that the two particul ar solutions which are regular at x = O
oo are given by
and vanish for x =
+
and
bez12 {oo e-t dt
l:
t
'
376
V. VIBRATION AND EIGENVALUE PROBLEMS
respectively. Green's function for the boundary conditions stated
in §10, 4, is constructed from these solutions:
K(x, ~) =
e<z+m2
ioo e~t dt
e<z+~>12
joo e-t dt
!
t
z
for
x ~ ~'
for
x
>
~·
No Green's function exists for the differential expression
L[u]
= u"
in the interval - oo < x < oo, with the boundary condition: u finite.
This corresponds to the fact that the homogeneous equation u" = 0
has the solution u = const. which is regular at infinity. On the
other hand, the differential expression
L[u] = u" - u
has the Green's function
and the singular integral equation
cp(x) =
~
2
loo
e-lz-~lcp(~) d~
-oo
obtained from this function has as its continuous spectrum all the
valuesX = 1+s2 ~1 with eigenfunctions (cos sx)/-Y;, (sinsx)/y;
(cf. §12). Here the bilinear relation is replaced by the integral
formula
_!_["°cos sx cos s~
7r
o
1
+ sin sx sin s~ ds = _![
+
s2
7r
00
o
cos s(x - ~) ds = ! e-lz-~l.
1 + s2
2
As an example of Green's function for a fourth order differential
expression, consider the equation L[u] = u"" in the interval
0 ~ x ~ 1, under the boundary conditions u(O) = u(l) = u'(O) =
u'(l) = 0 (corresponding to a rod clamped at both ends). We find
without difficulty
~) = x (~ ~
2
K(x,
2
l)
and a corresponding expression for x
(2x~ +
>
~-
x -
3~)
for
377
EXAMPLES OF GREEN'S FUNCTION
2. Green's Function for ~u: Circle and Sphere. We consider the
boundary condition, u = 0, and derive Green's function for the circle
and sphere. We use the fact that the circle and the sphere are the
geometrical loci of all points whose distances from two fixed points
P1, P2 are in a constant ratio. More precisely, let P1: (~, 71) or
2
2
y = 1 or
(~, ,.,, r) be any point in the interior of the circle x
2
2
having
z2 = 1 and let P2 be its reflection,
y
of the sphere x
2
2
2),
2
r2),
,.,'i.) (circle) or ~/(~
the coordinates UC~
2
2
2
2
2
r ) (sphere). If r1' r2 are the distances
r ), r/(f
,.,;(~
of the variable point P: (x, y) or (x, y, z) from the points P1, P2,
then the ratio r1:r2 is constant when the point P moves on the circumference of the circle or on the surface of the sphere, and the value of
2
r 2. Now we note
~2
,.,2 or
~2
this ratio is given by
that the functions -log ri/27r, -log r2/27r and 1/47rr1, 1/47rr2 are
solutions of .1u = 0, and that -log ri/27r and 1/47rr1, respectively,
have the proper singularities for a fundamental solution at the point
P 1 • Thus the functions
+ ,., +
+
+ ,., +
+ +
+ ,., ,,;er +
+ ,., +
y
v
+
K (x, y; ~' ,.,) = - -
1
27r
r1
log r2
+ ,., +
-~2 +
+ -217r log v_ 1
.,,2
and
K(x, y, z; ~, .,,, f')
1 ( 1
= 47r ~ - r2
ve +1 .,,2 + r2)
are the Green's functions for the circle and sphere respectively under
the boundary condition u = 0; for, these functions vanish on the
respective boundaries.
3. Green's Function and Conformal Mapping. For two independent
variables, it is helpful to use the function-theoretic fact that Green's
function is related to the conformal map of the domain G onto the
iy) be an analytic function which
unit circle. Let r = f(x
onto the unit circle in the f'-plane
conformally
G
domain
the
maps
in such a way that the point (~, ,.,) of G goes into the origin. Then
- log I f(x + iy) l/27r is the Green's function belonging to G. Thus
we have Green's function for all domains which can be mapped
conformally onto a circle. That these include all simply connected
bounded domains with piecewise smooth boundaries is Riemann's
1
fundamental theorem of geometric function theory .
+
1 Cf. Hurwitz-Courant, Funktionentheorie, pp. 389-423, in particular pp.
390-398, 3rd ed., J. Springer, Berlin, 1929.
378
V. VIBRATION AND EIGENVALUE PROBLEMS
4. Green's Function for the Potential Equation on the Surface of a
Sphere. A simple example of Green's function in the generalized sense
is given by the differential equation Ll*u = 0 (see §8 and §9, 1) with
the condition of regularity at all points of the surface except the
source point. Since the function u = 1/V4; is regular on the whole
surface, we must construct a Green's function in the generalized sense
which satisfies the equation Ll*u = 1/411". This function is obtained very easily by using the invariance of the expression A *u
under arbitrary rotations of the sphere. If we first put the source
point P1 of the Green's function at the north pole 8 = 0, we see
that the differential equation Ll*u = 1/411" is satisfied by the function -log [2 sin (8/2)]/211" which depends only on the coordinate 8.
Clearly, if p(8, cp; 81 , cp1) denotes the distance between two points
P: (8, cp) and P1: (81 , cp1) on the surface of the sphere, then from the
invariance of A *u under rotation it follows that
K(O, 'Pi 01, cp1) = -
2~ log ( 2 sin~)
is a solution of Ll*u = 1/411" which is regular at every point except
P = P1. Since this function, furthermore, has the proper singularity at P = P 1 , it represents the desired Green's function. If we
use K as the kernel of an integral equation
-211" Y(8, cp) = X
J£ log ( 2 sin~) Y(01,
cp1)
d81 dcp1,
we find that the only eigenvalues and eigenfunctions associated with
it are the (2n + 1)-fold eigenvalues X = n(n + 1) and the corresponding eigenfunctions Y = Yn(B, cp) defined in §9, 1.
5. Green's Function for Au
0 in a Rectangular Parallelepiped.1
Let the boundary planes of the parallelepiped be given by x = ±a/2,
y = ±b/2, z = ±c/2. As a natural generalization of the method
used in the case of the sphere (§15, 2), we shall find the Green's
function for the boundary condition u = 0 by constructing the lattice which corresponds to the original parallelepiped and has the vertices ((k + !)a, (m + !)b, (n + !)c) (k, m, n = 0, ±1, ±2, · · ·)
and reflecting the point (~, .,,, ~) repeatedly across the lattice-planes.
Thereby we obtain a system of points (ka + (-l)k~' mb + (-l)m.,,,
=
1 We are obliged to A. Ostrowski for the convergence proof and
the calculations of this section.
379
EXAMPLES OF GREEN'S FUNCTION
+ (-1rr).
We imagine that, at each of these points, a unit of
mass is concentrated which is positive if k + m + n is even and
negative if k + m + n is odd. We would conjecture that the potential of such a mass distribution is equal to zero in the lattice-planes,
since the contribution s of the individual mass units cancel each other
1
there. Thus we arrive at the following tentative expression for K :
(-l)Hm+n
1
L
L
(106)
K=- L
411" k~oo m~oo n-oo VN(k, m, n; ~' 7/, r; x, y, z)
nc
00
00
00
where
N(k, m, n; ~'
7/,
t; x, y, z)
= [ka
+ (-1)"~
+
[nc
+
2
- x]
+ [mb + (- l)m'TJ
(-1rr -
z]
- y]
2
2
•
Here, since the convergence can be at best conditional, we must first
discuss the order of summation; to this end, we shall in general denote the expression q;(k + 1) - q;(k), where q;(k) is any function of
k, by dJc'P(k). Then for fixed k and m, omitting the factor ( - l)Hm,
we can write the inner sum with index n in the expression for K as
N'(k, m) =
L
n=±l.±3,
dn
00 •
l
VN(k, m, n)
1
An VN(k, m, n)'
n=0.~4.···
since lim 1n 1.... N(k, m, n) = oo. We apply the same transformati on
to the sums with indices m and k; since lim1ml-+oo N'(h, m) = 0, as
will be proved shortly, we have
00
L
N"(k)
AmN'(k, m),
L
AmN' (k, m)
m~0.±2,±4,•••
m-±1,±3,···
and also
K ~ _!_
L
411'" k-±1,±3, •..
AtN" (k)
- _!_
411"
L
k-0,±2,±4, •••
AtN" (k),
because limlkl-+oo N"(k) = 0. Combining these results, we obtain the
transformati on
1
1
(107)
AmAnVN (k ,m,n)'
-LLL:Ak
K = ± 4
ft
m
11'" k
1
pp.
Cf. Riemann and Hatteridorf, Schwere, Elektrizitat und Magnetismus,
~iimpler, Hannover, 1880.
84-88, C.
380
V. VIBRATIO N AND EIGENVA LUE PROBLEM S
where each of the summati on indices runs over either all the even or
all the odd integers from - oo to + oo • The entire sum has the sign
+ if the summation runs over all the even integers an even number
of times; it has the sign - if the summati on runs over them an odd
number of times.
To prove our assertions it is sufficient to prove that the last sum
converges absolutely. For this purpose we estimate its general term:
(108)
IA> Am A• v'N(:. m, n) I
<
(di I k I
+ Ci) (d2 Im I + c2) (da In I + ca)
(yk2 + m2 + n2)1
<
c
(k2
+
m2
+
n2)2
for
x2
+ y2 + z2 < h,
~2
+ 11 + < h,
2
r2
C3 =
Ca(h),
c = c(h).
This estimate is obtained if we apply the mean value theorem of the
differential calculus three times and use the inequalit y relating the
arithmet ic and geometric means.
At the same time we also obtain uniform convergence in x, y, z,
~' 11, r if we sum only over k, m, n with k2 + m2 + n 2 > C4(h) such
that N(k, m, n) vanishes for none of the triples k, m, n.
2
For x + y 2 + z2 < h, ~ 2 + 11 2 + r 2 < h, k2 + m 2 + n 2 > C4(h),
the same consideration insures that all partial derivatives of the sum
(107) obtained by termwise differentiation converge absolutely and
that they converge uniformly in x, y, z, ~' ,,, r.
It is now clear that (107) is the desired Green's function; naturally ,
(106) and (107) are meaningful only if no N(k, m, n) vanishes. That
conditions (1) and (3) (§14, 5) are fulfilled requires no proof. To
show that condition (2) is satisfied, say in the plane x = a/2, we
make use of the represen tation
K
= _!_
47r
L
k==±l,±3, •••
A1; N" (k).
For x = a/2 the finite sums
L
k-±1,±3,·. •,±(l+l)
AkN"(k)
OF
EXAMPLES
381
GREEN'S FUNCTION
vanish because the individual terms cancel each other in pairs; hence
we obtain K = 0. In the same way it is seen that condition (2) is
also fulfilled on the other planes of the parallelepiped.
The sum (106) was represente d by Riemann as an integral over
certain 0-products. This Riemann representa tion may be derived in
the following manner: We write the equation
_ -1-2- ["" e-st2 dt -
v;.
(s
Vs
0
and substitute for s the expression N(k, m, n; x, y, z;
gives
~'
.,,, t).
>
0)
This
'"' '"'
'"' £..J
1
£..J Ak Am An ["" e-Nt2 dt.
_ 1- £..J.
~o
2'11'V '11' k m n
K --
If we could interchang e summatio n and integration , we would find
K =
(109~
'"' '"'
1_ r ["" £..J
£..J Ak Am An e-Nt2 dt
£..J '"'
2'11'V '11' 0 k m n
1
=
2 '11'.y;
1fdds
00
dt,
where the three factors under the integral sign are given by
!1
i:
=
(-l)lc
e-t2[ka+(-l) krzl2,
k-oo
f2
( - l)m e.-t2[mb-t(-l}m rtf]2 ,
~
=
£..J
n-=-oo
and may be expressed in terms of the theta function
n
UOO
(
z,
T
)
=
n (
u3 Z, T
)
=
~ eirv2,-e2irvz
£..J
11=-oo
We wish to prove formula (109); the main difficulty arises for
0, since the three series converge nonuniform ly in its neighborhood. We prove first that we can interchang e the order of integration and summatio n with respect to k:
1
f dda dt
'11'.y;
2
=
1
00
(110)
382
V. VIBRATION AND EIGENVALUE PROBLEMS
It is easy to justify interchanging the order of summation and integration from 1 to co • In fact, for the remainder of the sum fi we
have for t > 1, p > P(~, x) > 2 the estimate
I
L (- l)k e-t2[ka+(-I)k~-z]2 I < e-a2t2/4 L
jkj>P
e-a2t2fr2/2
jkj>p
<
-a2t2/4
~ L _!_ < ~e-a2t214 _I_
a2
lk I > P k2 . a2
p - 1
<
~ e-a2t2/4;
pa2
thus the integral from 1 to co converges to zero with increasing p.
On the other hand, f2 and fa clearly remain uniformly bounded in
the interval from 1 to oo.
To justify interchanging summation and integration from 0 to 1
it will be sufficient, according to a well-known theorem, to demonstrate the boundedness of the partial sums of the integrand. Now
each of the two sums .L:~o and .L:~1 into which fi may be decomposed is an alternating series whose terms, from a certain k on, decrease monotonically; k depends only on ~ and x, not on t. Hence
the value of any partial sum of either series lies between fixed bounds
for all t > 0. But a corresponding statement holds also for the partial sums of f2 and fa, which implies that f2 and fa themselves are
uniformly bounded fort > 0. Therefore we may apply the theorem
referred to and prove equation (110). A precisely analogous consideration also proves that summation with respect tom and n may be
interchanged with integration in the individual terms on the righthand side of (109). This completes the proof of equation (109).
We shall now represent K in terms of the function Boo. We have
_ -t2(z-~) 2 (
8oo
f1-e
2at i(x - ~) 4l! t
2
-
2 2
i)
,--
7r
7r
-e-121z+~>2 8 (
.
oo -
4b; i)
2
2at i(x
7r
+ ~) , -4a t i) ,
2 2
7r
2
.f
-
J2 -
11)'
e-121y-71)20ce ( -
-e-•'<•+.>'eoo( _ 2bt'i(~ + 11), 4b:'i),
fa =
i(: -
2
e-t2(z-t>2ooo( -
2ct
r)'
4c}!)
-e-''"+r>'ooo( _
2ct'i(: + i), 4c:'i).
383
EXAMP LES OF GREEN 'S FUNCTI ON
the
To the individ ual terms we apply the transfo rmation formul a for
theta functio n:
e-ri12/.,. _ I
800 ( Z, T ) =
V
1
.
-'tr
800
(Z
, -
-
If in additio n we set
taking the princip al value for the root.
q,
we find
/1
Ya; [ooo( - ";,; ~, ",J
,)
4 1
=
2at
(111)
=
k-oo
~ k2 (
y; 4.J
qz
-ta
k-1
Yr ~
2.
COS
k2
= ---£.. Jqz
at
k-1
ktr(X -
~)
a
-11'2/4c2t2
e
'
~ qxk2 e-k(z-H)r i/a]
4.J
k--oo
-
COS
k7r(X
+
~))
a
k1fX . ktr~
•
sm- sm- .
a
a
Analogous expressions are obtaine d for
lowing expression for K:
4
K = abc
=
-ooo( - "i; ~, 4:!1,)J
~ qzk2 e-k(x-E)r i/a) y'; [ 4.J
= --
1)
-
T
T
f 2 and fa , leading to the fol-
1t
00
0
. k7r~ · · ·
. ktrx sm
1 ~ ~ ~ sm
a
a
k-1 m-1 n-1
-3 £..J £..J £..J
. n7rf' e-(r2/4t2) [(k2/a2)+( m2/b2)+(n 2/c2) I dt .
sin-
c
2
Introdu cing 1/t = r as a new variabl e of integra tion in this expression, we obtain
. ntrtdr.
. ktrx
·sin -··· smc
a
This formul a is a valid substit ute for the expansion of Green's function in terms of the eigenfunctions
. n?rr
. ktrx_ ; k7r~
sm -a- s ma- · · · sm-c
oo
oooo
8
2
2
k2
L
L
K(x, y, z; ~' 11, t) = - b2 L 1 m=l
+ m + !!'.__
_
n= 1
a C7r k=
c2
b2
a2
384
V. VIBRATIO N AND EIGENVA LUE PROBLEM S
which is obtained formally by interchanging summati on and integration, but the convergence of which has not yet been proved.
As a simple expression for K, we find from (109), for r = 1/t2 ,
K=
32~bc f
{[
800 ( -
[ Ooo ( -
~' :::) z~
- r , 7riT)
4c
x ;;
2
-
800 ( -
(
x
ia ~' ::.)J···
+
7rir)J }dT.
Ooo - z~r , c2
4
6. Green's Function for 4u in the Interior of a Rectangle. Consider the rectangle R with one vertex at the origin and the others at
the points (a, 0), (0, b), (a, b). Let the source point be (~, 11), and
consider the point (x, y). If K(x, y; ~' 11) is the Green's function corresponding to the boundary condition u = 0, then K as a function
of x and y is a solution of Au = 0 in the interior of R, vanishes· on
the boundary , and becomes singular at the point(~, 11) like -log r/21r,
where r =
(x - ~) 2
(y - 71 ) 2• It seems reasonable, as in the
case of a parallelepiped, to construc t the lattice generate d by the
rectangle R, reflect the point (~, 11) repeated ly across the lattice
lines, and think of each of the resulting points as a source or a sink
of intensity 1, depending on whether it is obtained from (~, 11) by an
even or by an odd number of reflections across the lattice lines.
As before, we could form the potential X of the resulting mass
distribution by summing an infinite series. It is more convenient,
however, to make use of function theory and introduce the associated
analytic function ~(x + iy) = X + iY whose real part is X. Then
the function
v
+
f(X
+ iy)
=
e-2r(X+iY)
=
e-2rr,o(z+iy)
must have simple zeros and poles at (~, 11) and at the points resulting
from reflections. Now take sets of four adjoining rectangles of our
lattice and combine them to form rectangles of a new lattice. Then,
in every rectangle of the new lattice, f(x
iy) must have two simple
zeros and two simple poles, which are symmetr ic with respect to the
origin and congruent mod (2a, 2b), respectively:
+
Zeros:
(~,
Poles:
(-~,
11),
71),
( -~, -71),
(~,
- 71).
The simplest analytic function of this type is the elliptic function
which has these zeros and poles in the period-rectangles with vertices
385
EXAMPLES OF GREEN'S FUNCTION
(a, b), (-a, b), (a, -b), (-a, -b) and which may be represented in
terms of the associated u-function:
u(z - ~ - iTJ)u(z
u(z - ~ -t iTJ)u(z
f(z)
-t ~ -t
-t ~ -
iTJ)
iTJ) '
1
where
u(z)
w
=
z
I]' [ ( 1 -
= ka -t lbi
;.,
)e'
12
1
..,.' '""].
(k = 0, ±1, · .· ·)
(l = 0, ±1, · · · ).
If we substitute this representation in the expression for f(z) and
multiply factorwise, we find, setting exp {~TJi/w } = 1 for w = 0,
2
f(z) =
II [(z -t ~ w
(z
er = ~ -t iTJ, r = ~ -
+riTJ;
2w)(z 2w) (z -
k =
£-
r-
o, ±1,
2w) ehi/w2]
2w)
... ,
l =
o, ±1,
... ).
Here we have only to verify that the boundary condition is satisfied,
i.e. that f(z) has the absolute value 1 on the boundary of R. For
z = x = 9t~(z), the factor with w = 0 clearly has absolute value 1
and the factors corresponding to the remaining w may be arranged
in pairs corresponding to conjugate complex w in such a way that
the numerator of one factor is the complex conjugate of the denominator of the other factor of the pair. For z = x -t ib we multiply
first with respect to land then with respect to k. We can omit the
factor exp {~TJi/w2 } in the product taken with respect to l, since the
2
sum
1/w , taken over l for a fixed k, converges absolutely and has
a real value. We combine the remaining factors pairwise in such a
way that if one factor corresponds to w = ka -t lbi, the second
corresponds to w = ka - (l - 1)bi. It is then seen immediately
that the product of such a pair has absolute value 1. However, for
z = iy we first take the product with respect to l and, for I k I > 0,
combine any two such partial products which correspond to the
values ±k. We can then again omit the exponential factor, since
L l/w2, taken over l, converges absolutely and has a real sum. We
combine the remaining factors in pairs in such a way that one factor
corresponds to w = ka -t lbi, the other to w = - ka -t lbi. Then
every such product has absolute value 1. Finally, we take care of
L
1 II' denotes the product in which the factor corresponding
to w = 0 is
omitted.
386
V. VIBRATION AND EIGENVALUE PROBLEMS
the case z = a + iy by combining factors which correspond to
w = ka + lbi and w = - (k - l)a + lbi, and by multiplying with
respect to l. In this way we find for the desired Green's function
the representation
(
K x, y; ~'
(z = x
'1
) 1ffi {l
=
-
+ iy,
27r
e
t =
~
u(z -
og u(z -
+ i.,,,
t, w1, w2)u(z + t, w1, w2)}
f, w1 , w2)u(z
f =
~
- i.,,,
+ f, w1 , w2)
w1
= a, w2
= ib).
The Green's function just constructed can be expanded in terms
of the eigenfunctione (2/v'{ib) sin (kn/a) sin (m7ry/b) into a convergent series. 1 The expansion is given by
00
K(x, y; ~' .,,)
00
-4L L
ab?r 2 m-1
k=l
.
.
7r
•
k7r
sin k1r
- x sin
m b- y sm a
a
m2
k2
b2
~
.
sm m 7r
-b .,,
+ a-2
This is an example illustrating the validity of the bilinear formula,
which has not been proved in general. '
7. Green's Function for a Circular Ring. Consider a ring bounded
by two circles with centers at the origin; the product of their radii
is 1 (here the unit of length must be chosen in a suitable way). We
denote the radius of the inner circle k1 by q 112, and the radius of the
outer circle k 2 by q- 112, where 0 < q < 1. Then if c is the source point
(which at first we assume to be real and positive), if z = x + iy is
the point under consideration, and if both points lie in the interior
of the ring R, our problem is reduced to the following functiontheoretic problem: Determine an analytic function f(z) which has a
simple zero at c, is analytic elsewhere in R, and has modulus 1 on the
boundary of R. The desired Green's function is then to be obtained
from f(z) by
K(x, y; ~' .,,) = -
1
7r 9L {log f(z)}.
2
In order to find enough function-theoretic properties to enable us
to construct j(z) explicitly, we now attempt to continue f(z) beyond
1 The convergence is not absolute,
see V. A. ll'in, On the convergence of
bilinear series of eigenfunctions, Uspehi Matern. Nauk (N. S.) 5, No. 4 (38),
1950, pp. 135-138.
387
EXAMPLES OF GREEN'S FUNCTION
the two circles. To this end, we associate to each point z in R a
point z1 inside k1 so that zz1 = q. If z moves toward the boundary
of kl , then the same is true of z1 , and in fact z1 evidently approaches
the complex conjugate point. Now, because of the symmetry of the
assumptio!J.s, f(z) may be considered a real function, that is, a function which takes real values at real points and, more generally, conimplies
jugate complex values at conjugate complex points. This
2
z
when
1
f(zo)
I
value
that f(z)f(q/z) approaches the positive real
the
On
.
ki
circle
the
of
approaches a point zo on the circumference
other hand, j (z) has modulus 1 on the circumference of kl • Thus
for z on kl we have for f(z) the equation
f
(112)
{z)f(!)
= 1,
and this equation holds identically for all z. Similarly, reflection
across k2 leads to the second functional equation
!(•)!(~) =
(113)
1.
Since c is a simple zero of f(z), it follows from successive application
of these relations that f(z) has simple zeros at the points
<j1='c,
<j1=2c,
c,
and simple poles at the points
::1:6 -1
±1 -1
q
q c ,
c ,
... ;
thus it agrees in its zeros and poles with the function
(1 - ~) :-1fl (1 - cz) (1 - zc) .
(i - .!.)
II
21'
F(z) =
(1 -
c
21-
q
q
q2-i cz)
q2,_1
CZ
v-1
But for this function F(z) we have the following functional equations
of the type of (112) and (113):
F(z)F(~) =
1,
_!_2 '
= qc
F(z)F(_!_)
qz
as can be verified by a simple calculation. Thus constants a and b
can be determined in such a way that azbF(z) satisfies the functional
388
V. VIBRATION AND EIGENVALUE PROBLEMS
equations (112), (113) and has modulus 1 on the circles k1, k 2 , since
a, b turn out to be real constants. We find the values
b =
log c
log q ·
_l. _
2
Taking the negative sign for a, we obtain
f(z) = ql z-logc/logq(
•
~
11
_
C
•
IT (1
~)
11
Z
co "=l
IT (l
-
Z)(1 - ZC)
2v
q C
2v
q
_ q2v-1 cz) (l _ q2"-1 _!_)
~
P=l
This expression may be written in terms of the theta functions
fh (z) =
- i'C ql(e irz
- e-irz)
co
II
(l
- q2ve2irz) (l - q2ve-2irz) .
P=l
6o(z)
co
C
II (1
_ q2v-1e2irz)(l _ q2v-1e-2in)
,,_l
with
00
c = II c1 -
211
q
).
1'=1
If we set z = e2irv' c = e2ira' we find
.f(z) =
iz-2ira/logqf}i(v -
6o(v
a)
+ a)'
and the real part of log .f(z) naturally vanishes on k1, k2 also for complex c inside R; the problem is now solved.
§16.
Supplement to Chapter V
1. Examples for the Vibrating String. (a) The Plucked String. We
represent the solution for the case of a plucked string as a superposition of synchronous vibrations. At time t = 0 let the string be
pulled a lateral distance h from the x-axis at the point x = b, so
that the form of the string is linear from x = 0 to x = band from
x = b to x = 7r. Let the initial velocity be zero. Then the lateral
displacement u(x, t) has an expansion of the form
00
u(x, t) =
L an sin nx cos nt,
n=O
389
SUPPLEMENT
where
an
211f u(x, 0) sin nx dx
= -
r o
2h
= -;-
(lb b
0
. nx dx
x sm
lr
+
6
. nx dx )
rr -_ xb sm
. nb
b) sm •
2h
n 2 b(r -
Thus we find
u(x, t) = b"(
2h
r -
~·
L...J
b)
n=l
sin nb sin nx
cos nt.
n2
(b) Impulsive Excitation. The case in which the vibration of the
string is caused by an impulse applied to the str{ng in its rest position
in a neighborhood of the point x = b can be treated in an analogous
way. We obtain
co
L bn sin nx sin nt,
u(x, t)
n=l
and
21r
nbn = -
7r
Ut(X,
.
0) sm nx dx.
0
Now we must carry out a limit process, contracting the interval over
which the impact acts, but doing this in such a way that the integral
f.
u,(x, 0) dx = U remains constant.
In the limit we obtain:
bn = 2U sin nb
rn ,
~ sin nx sin nb . n t.
sm
u (x,. t) -_ 2 U L...J
rn
n=l
(c) Forced Vibrations. The general solution of the differential
equation of forced vibration
Utt -
Uzz
= f(x)
COS
nt
390
V •.VIBRATION AND EIGENVALUE PROBLEMS
with periodic external force has the form
u
= - 2-
lr
oo
cos nt
'1l"
L sin 11x
"-1
f(x) sin 11x dx
0
n 2 - 11 2
00
+L
sin 11x(a11 sin 11t +
P-1
If we let
21r
-
1(
0
b" cos 11t).
f(x) sin 11x dx = c",
we obtain for the corresponding integral, under the initial conditions
u(x, 0) = u,(x, 0) = 0, the expression
u(x, t) =
-t
sin 11x
"-1
c"
(cos nt - cos 11t).
n 2 - 11 2
Here the term
-c" :! sin 11x (cos nt - cos 11t)
n 2 - 11
generally dominates whenever n approaches the value v. We can
get the best idea of the behavior of this term by writing it in the
form
2c"
- -2
n2 -
v
+
.
.
. n - 11
sm vx sm n- - v t sm
- - · t;
2
2
we can think of this expression as the representation of a vibration
sin [(n + 11)t/2] with the variable amplitude sin [(n - v)t/2]. The
amplitude of the vibration becomes alternately strong and weak;
this gives rise to the phenomenon of "beats." In the limit as n--+ 11,
the term in question takes the form
c" .
.
-; sm vx sm vt· t
2,
and thus the amplitude becomes infinite with time.
2. Vibrations of a Freely Suspended Rope; Bessel Functions. Let a
homogeneous rope of length 1 and weight 1 hang along the x-axis,
and let the direction of the weight be opposed to that of the x-a}..'is.
391
SUPPLEMENT
We shall assume the rope is suspended at the point x = 1, so that the
free end lies at the point x = 0. Then if u is the displacement per1
pendicular to the x-axis, u satisfies the differential equation.
If we assume
u = q(t)cp(x),
we obtain the decomposition
~
=
->.
= (xc,o')'
cp
q
with the auxiliary condition: cp(l) = 0,
From this it follows that
I c,o(O) I <
oo.
cp(x) = cJo(2~),
where Jo(x) denotes the Bessel function of order 0 and the condition
Jo(2v) = 0 determines a sequence of eigenfrequencies v = v'A.
3. Examples for the Explicit Solution of the Vibration Equation.
Mathieu Functions. (a) Sector of a Circle. The vibration equation
a reprefJ
1, 0
r
Au + Xu = 0 for a circular segment 0
s s
s s
sented in polar coordinates is again solved by separation of variables:
u = f(r)g(fJ). Again, as in §9, we take u = 0 as boundary condition
and find the system of eigenfunctions
where Jmr/a denotes the Bessel function of index nr/a (cf. Ch. VII)
and the eigenvalues x~.m are determined by the transcendental equa.
tion J,nr/a(VXn,m) = 0.
(b) Ellipse. The solution of the eigenvalue problem for the ellipse
is obtained by introducing elliptic coordinates (see Ch. IV, §8, 3).
We find
AT
1
+ XT
T) + 'AT =
(iJ2T
- - - -o
at~
X2 at~
= -.-1 -
X1 -
2
Cf. Kneser, Integralgleichungen, pp. 39-43.
O·
'
392
V. VIBRATION AND EIGENVALUE PROBLEMS
the assumption T = U(t1)V(t2) leads to the equation
which is satisfied if and only if U and V are solutions either of the
differential equations
U"
or of
2
dU
dAt
=
-CAX1
+ µ)U,
V" = -(XX2
+ !(_1_ + _l_)dU =
2 A1 - ei
Ai - e2 dA1
+ µ)V
+
-AA1
µ
U
(A1 - ei) (A1 - e2)
and of the corresponding equation for V.
If we set
2A1 - e1 - e2
---ei - e2
2A2 - e1 - e2
e1 - e2
=
----- =
then u and v are real.
cosh u,
cos v,
We obtain equations of the form
2
-d U2 = -(A' cosh u
du
2
·
-d V2 = (A' cos v
dv
+ µ')U '
+ µ') V
'
where A' and µ' are constants. The solutions of these differential
equations, which are transformed into each other by the substitution
u = iv, are called functions of the elliptic cylinder or Mathieufunctions. 1
(c) Confocal Quadrilateral or Hexahedron. Heretofore we have
solved the vibration and potential equations by separation of variables for certain regions; these regions are either special cases or
limiting cases of a quadrangle or a hexahedron bounded by members
of a confocal system of cyclic curves or surfaces, respectively (cf.
§9, 3).
4. Boundary Conditions with Parameters. 2 We shall describe briefly
how certain boundary value problems with parameters in the boundi
2
Cf. Whittaker and Watson, A Course of Modern Analysis, pp. 404-428.
Cf. Hilbert, Integralgleichungen, pp. 77-81.
.
393
SUPPLEMENT
ary conditions can be reduced to integral equations. Consider, for
example, the differential equation du = 0, with the following boundary condition on the regular boundary curve r of a simply connected
domain G lying entirely in the finite part of the plane:
au
an +
AU
+
h(s)
= 0,
where n denotes the outer normal, A the parameter, and h(s) a given
function of arc lengths on r. Using the Green's function K(x, y; ~' fJ)
of the domain G whose normal derivatives vanish on the boundary,
we obtain the Green's formula:
u(~, 71)
=
l
[Xu(x, y)
+ h(s)]K(x, y; ~' f])
ds,
where the point (x, y) runs over the curve r. If we use the parametric representation x = a(s), y = b(s) for r, the values of
K(x, y; ~' fJ) yield a symmetric function K(s, u) of two varial>.les s, u:
K(s, u) = K (a(s), b(s); a(u), b(u)).
If in addition we let
u(a(s), b(s)) = <p(s),
l
K(s, u)h(s) ds = f(u),
then the above relation for u takes the form
f(u) = .<p(u) - X
l
K(s, u)cp(s) ds.
Only the solution of the first boundary value problem is needed to
determine u from <p(s); therefore, we have only to investigate this
integral equation, whose kernel becomes logarithmically infinite at
the single point u = s. The general theory is immediately applicable
to this kernel.
Analogous considerations hold for the general self-adjoint second
order differential equation of the elliptic type.
5. Green's Tensors for Systems of Differential Equations. The idea
which led us to introduce Green's function may be extended to problems involving systems of differential equations, for example, to the
394
V. VIBRATION AND EIGENVALUE PROBLEMS
problem of determining a vector u: (u1, u2, ua) from a differential
equation L[u] = -f, where f: (!1, f2, fa) is a given vector. By a
Green's tensor @ of the differential equation L[u] = -f associated
with prescribed homogeneous boundary conditions, for example u = 0,
we mean a matrix
@(x, y, z; ~' 11, t)
=
(
Ku
K21
K12
K22
Kia)
K2a
K31
Ka2
K33
with the following properties: the differential equation L[u] = -f
together with the boundary conditions is equivalent to the formula
u(x, y, z) =
ff!
@(x, y, z;
~' 17, t)f(~, 17, r) d~ d11 dt.
Here @f denotes the vector resulting from matrix multiplication of
the matrix @ with the vector f, that is, the vector with the components
K11f1
+ K12/2 + K1afa ,
K2if1
+ K~2 + K23fa ,
Kaif1
+ Kad2 + Kaafa.
Each column of the Green's tensor represents a vector ki, which,
except at the source point x = ~' y = 11, z = r, is continuous together
with its derivatives and satisfies the differential equation L[ki] = 0
and the boundary conditions. The nature of its singularity at the
source point becomes clear from its interpretation (e.g. in the case of
a single differential equation) as the influence function of a point
force acting at the source point x = ~' y = 7/, z = r. Green's tensor
satisfies the symmetry relations
K1i(x, y, z; ~' 7/, t)
= KiiU,
1/,
t; x, y, z),
Kik(x, y, z; ~' 1/, t) = Kki(~, 11, t; x, y, z),
whenever (as we shall assume) the differential expression L[u] is selfadjoint, i.e. results from the variation of a quadratic differential expression in the vector u and its first derivatives. The eigenvalue
problem for the differential equation L[u]
Xu = 0 may be solved
with the aid of Green's tensor in a manner completely analogous to
the ordinary case.1
+
1
Cf. Hilbert, Integralgleichungen, pp. 206-212.
395
SUPPLEME NT
6. Analytic
=
Continuation
of
the
Solutions
of
the
Equation
Au + lu 0. Let a solution of Llu + AU = 0 be given in a closed
domain G, which has the line-segment las part of its boundary. Let
this solution be continuous together with its derivative s up to the
second order and let the function u or the normal derivative au/an
vanish on l. We can reflect G across l, obtaining a new domain G',
and continue the function u into G' in the following way: If g' is the
mirror-image of the point g of Gunder reflection, let u(g') = -u(g)
when u = 0 on l and u(g') = u(g) when au/an = 0 on l. Then u is
an extension of u which is a continuous solution of Llu + AU = 0 in
the combined domain G + G' with continuous derivative s up to the
1
second order. Similar theorems may be stated for the equation
Ll!lu - AU = 0 of a plate. The assumptio n for the theorem may be
further weakened, as in the case of the reflection principle in function
theory; this will be indicated later.
0.
7. A Theorem on the Nodal Curves of the Solutions of Au+ lu
doa
of
interior
the
in
intersect
0
=
u
curve
If several brancltes of the
=
2
main of the x, y-plane in which u is regular, then the set of nodal
lines which meet at the point of intersection forms an equiangular system
of rays. The reader may prove this theorem by expanding the function u at the point in question in a power series.
8. An Example of Eigenvalue s of Infinite Multiplicity. For an
arbitrary plane domain, e.g. a circle, we consider the eigenvalue
problem of !l!lu - AU = 0 with the boundary conditions Llu = 0,
(a/an)!lu = 0. We easily obtain infinitely many eigenvalues Ah and
eigenfunctions Uh for this problem by noting that the functions
!luh = v" must be eigenfunctions f oc the clamped plate, as long as
!luh does not vanish identically . Thus we are led to eigenvalues
which coincide with those for the clamped plate; zero also occurs as
an eigenvalue of infinite multiplicit y. Namely, for A = 0 each of
the infinitely many linearly independe nt potential functions, regular
in G, satisfies the equation Ll!lu +AU = 0 under the given boundary
conditions.
9. Limits for the Validity of the Expansion Theorems. In stating
1
Cf. R. Courant, Beweis des Satzes etc., Math. Zeitschr., Vol. 1, 1918,
pp. 321-328.
2 It is not difficult to see that every continuous solution u with continuous
derivatives is a regular analytic function of x and y (cf. also Vol. II).
396
V. VIBRATION AND EIGENVALUE PROBLEMS
ou~ theorems on expansion in terms of the eigenfunctions of the differential equation
L[u]
+ Apu
=
0,
we have always made the assumption p > O. That this assumption
is essential is shown by the following example: In the equation
y"
Apy = 0, let p = 0 for an arbitrary subinterval of the fundamental domain. Then every eigenfunction must be linear in this
subinterval; hence the expansion theorem cannot hold for ''arbitrary''
functions.
+
References
Bocher, M., Uber die Reihenentwicklungen der Potentialtheorie. B. G. Teubner, Leipzig, 1894.
- - , Lec;ons sur les methodes de Sturm dans la theorie des equations differentielles lineaires et leurs developpements modernes. Gauthier-Villars,
Paris, 1917.
Courant, R., Zur Theorie der kleinen Schwingungen. Zeitschr. f. angew. Math.
u. Mech., Vol. 2, 1922, pp. 278-285.
Frank, Ph., and v. Mises, R., Die partiellen Differential- und lntegralgleichungen der Mechanik und Physik. F. Vieweg und Sohn, Braunschweig, 1925,
1927.
Hilbert, D., Grundziige einer allgemeinen Theorie der linearen lntegralgleichungen. B. G. Teubner, Leipzj.g and Berlin, 1912. (Cited as "lntegralgleichungen. ")
Hort, W., Technische Schwingungslehre. 2nd ed., J. Springer, Berlin, 1922.
Kneser, A., Die lntegralgleichungen und ihre Anwendungen in der mathematischen Physik. 2nd ed., F. Vieweg und Sohn, Braunschweig, 1922. (Cited
as "lntegralgleichungen. ")
Pockels, F., Uber die partielle Diffetentialgleichung du+ k 2u = 0 und deren
Auftreten in der mathematischen Physik. B. G. Teubner, Leipzig, 1891.
Rayleigh, J. W. S., The Theory of Sound. 2 vol~., 2nd ed., Macmillan and Co.,
London, 1894, 1896.
Riemann, B., and Hattendorf, K., Schwere, Elektrizitat und Magnetismus. C.
Riimpler, Hannover, 1880.
Weber, H., Die partiellen Differentialgleichungen der mathematischen Physik.
2 vols., 4th ed., F. Vieweg und Sohn, Braunschweig, 1900, 1901; 5th ed.,
1910, 1912.
Whittaker, E.T., and Watson, G. N., A Course of Modern Analysis. 4th ed.,
Cambridge University Press, Cambridge, 1927.
CHAPTER VI
Application of the Calculus of Variations
to Eigenvalue Problems
In the preceding chapter we saw that eigenvalue problems of differential equations are closely connected with those of quadratic
forms. As a matter of fact, these eigenvalue problems correspond
to the problem of transforming a quadratic form in infinitely many
variables to principal axes.
If, for example, we denote the potential and kinetic energies of a
one-dimensional continuum by
1...
(au)
p U = 1ax
2 o
1...
2
dx
T = 1
2 o
and
(au)
at
p -
2
dx,
we need only write. u = L::.1 f11(t) sin vx and expand p and p in
Fourier series; then we can consider the two expressions U and T as
quadratic forms in the infinitely many variables (coordinates) f11 and
j" , respectively. If an orthogonal transformation
Q()
i"
=
L: t""<j"
(v = 1, 2, · · ·)
p=l
of these variables into new variables q" and q" can be determined in
such a way that U and T take the form
then the numbers A11 are the eigenvalues of our vibration problem.
Since the eigenvalues of a quadratic form are characterized by simple
extremum properties, it seems reasonable to consider a similar characterization for the case of quadratic functionals. However, instead of treating these functionals as limits of finite quadratic forms
of a finite number of variables, we shall formulate and apply the
extremum properties in question directly without referring to such a
limit process.
397
398
VI. APPLICATION OF THE CALCULUS OF VARIATIONS
§I. Extremum Properties of Eigenvalues
1. Classical Extremum. Properties. Consider ·the eigenvalue problem of a self-adjoint second order partial differential equation
L[u]
+ ApU =
(puz)z
(1)
+ (pUr,)
qu
11 -
+ ApU =
(p
>
0
0, p
>
O);
here x and y are independent variables in the fundamental domain G,
bounded by one or several continuous curves r with piecewise continuous tangents. Let the boundary condition have the form u = 0,
or in general iJu/ an + uu = 0, where u denotes a piecewise continuous function of position on the boundary rand aI an differentiation
in the direction of the outer normal. 1 Typical quadratic functional
expressions for the variational eigenvalue problems are:
(2)
'.D[cp] = D[cp]
+
i
pu/ ds
with
(2a)
· D[cp] =
Ji
p(cp!
+ cp:) dx dy +
H[cp]
=
Jl
2
qcp dx dy
and
(3)
Ji pc/
dx dy;
the associated polar forms
'.D[cp, 1f] = D[cp, 1f]
D[cp, 1f] =
H[cp, 1f] =
fl
fl
+
P('Pzlfz
£
pu'Plf ds,
+
cp111f11 ) dx dy
+
Jl
q'(ft/I dx dy,
p'Plf dx dy
1
It should be stated that in general this boundary condition must be
interpreted in the "weak" sense; i.e·., the function is not actually required to
assume boundary values everywhere on r. This delicate point will be fully
discussed in Vol. II, Ch. VII in connection with the existence proof. Here,
where we assume the existence of the solutions, we need not specify the behavior of it at the boundary.
399
EXTREMUM PROPERTIES OF EIGENVALUE S
satisfy the relations
+ ~] = '.D[~J + 2'.D[~, 1/t] + '.D[i/t],
H[~ + ~] = H[~] + 2H[~, 1/t] + H[i/t].
'.D[~
+
r and
We require the argument function ~ to be continuous in G
to have piecewise continuous first derivatives in G.
Here piecewise continuity of a function i/t in G means: G may be
subdivided into a finite number of regions G1, G2, · · · , Gm by arcs
with piecewise continuous tangents such that i/t is continuous in each
closed subregion of G not containing any of these arcs in its interior.
If the boundary condition u = 0 is prescribed, then u is assumed
r and to vanish on r.
to be continuous in the closed region G
eigenfunctions U11
associated
the
and
A11
We obtain the eigenvalues
properminimum
following
the
from
(I)
of the differential equation
'.D[~l
expression
the
minimizes
which
function
ties: The admissible
differential
the
for
u1
eigenfunction
an
is
1
=
H[~]
under the condition
1
equation (I) and satisfies the natural boundary condition -a~/an
u~ = 0; the minimum value of ID is the .t;0rresponding eigenvalue. If
we impose not only the condition
+
+
H[~] =
(3a)
I
but also the orthogonality condition
H[~,
U1]
=
0,
then the solution is again an eigenfunction U2 of (I) satisfying the same
boundary condition, and the minimum value '.D[U2] = A2 is the associated eigenvalue. The successive minimum problems, '.D[~l = mininum subject w the condition H[~] = 1 and w the auxiliary conditions
(i = 1, 2, ... ' .,, - 1),
define the eigenfunctions u., of equation (1) with {he boundary coridition
u~ = O; the associated eigenvalue X, equals the minimum value
a~/an
'.D[u,].
We can omit the normalizing condition H[~] = 1 by minimizing the
quotients '.D[~]/H[~]; then the solution is determined only up to an
arbitrary factor of proportional ity.
If the boundary condition is u = 0, the same variational problems
+
1
See footnote 1 on previous page.
400
VI. APPLICATION OF THE CALCULUS OF VARIATIONS
define eigenvalues and eigenfunctions; but the boundary condition
cp = 0 is now added to the admissibility conditions. Then the boundary term£ pucp2 ds in '.D[cp] drops out automaticaJly.
A proof that our minimum problem does possess solutions with
continuous second derivatives will be given in Volume II, in connection with the direct methods of the calculus of variations. At present
we proceed on the assumption that minimum problems of the above
type possess solutions.
We have to show (a) that the solutions of the variational problem
are also eigenfunctions for our differential equation problem, and (b)
that they furnish all the eigenfunctions. The second assertion will
be proved in §3 by showing that the system of functions u1, u2, · · ·
obtained from the variational problem is complete. Statement (a)
could be derived from the general multiplier rule of Ch. IV, §7 but
will now be obtained directly.
Let us begin with the first variational problem, and assume that
its solution U1 satisfies the condition H[u1] = 1. If t is any function
which satisfies the same conditions as cp, and if e is an arbitrary constant, then for every value of e and for u = u1 , 'X. = 'X.1 we have
'.D[u
+ et]
;::: 'X.H[u
+ et]
or
2•{fl[u, !] - AH[u, !]
+ ; ('.!)[!]
- AH[!])}?: O;
these statements are equivalent in view of the relation '.D[u] =
>-.H[u]. The second inequality can be valid for arbitrary values of
e only if equation
(4)
'.D[u, t] - 'X.H[u, t] = 0
holds, i.e. if the first variation of the expression '.!) - 'X.H vanishes.
We now transform the expression '.I>[u, t] according to Green's
formula
'.I>[u, t]
= -
ff
Jlo tL[u]
dx dy
+
1
r
putu ds
+
1
r
pt au ds
an
(cf. Ch. V, §1); since the function t is arbitrary, we immediately
obtain equation (1) for u = u1 and 'X. = 'X.1. Considering the second
401
EXTREMUl\I PROPERTIES OF EIGENVALUES
0, we
minimum problem with the additional condition H[<P, u1]
obtain equation (4) for u = u2 and X = X2 at first only under the assumption that r satisfies relation
(5)
Now if .,, is any function having piecewise continuous derivatives of
second order, we can determine a number tin such a way that the
function t = .,, + tu1 satisfies condition (5)-we sett = -H[u1, rJ].
Moreover, we note that we can, in particular, substitute the function
r = U2 in equation (4) with u = U1 and A = A1 ; since 112 satisfies
the additional condition
(6)
we immediately obtain
(7)
If we substitute our function
= ~ , X = X2, we find
r = .,, + tu1
in equation (4), writing
u
ID[u, rJ] - >Jl[u, .,,]
+ t('.t)[u, u1]
- >Jl[u, u1)) = 0
or, taking into consideration equations (6) and (7),
(4a)
ID[u, 11] - >Jl[u, 11] = 0.
In other words, equation (4) holds also for arbitrary functions .,, or
t without regard to the auxiliary condition (5). From this it follows
directly, as above, that equation (1) is valid for u = ~and X = A2.
Continuing in this way, we conclude that the eigenvalue equation (4a)
holds generally for the solutions ui and the minimum values Ai . For
the solutions of the problem, which are normalized in accordance
with (3a), we have the relations
(8)
ID[ui]
= Ai,
ID[ui, uk] = 0
H[ui]
= 1,
H[ui, uk] = 0
(i
¢
k).
The eigenvalues thus obtained satisfy the inequality
(9)
since the class of functions <P admissible in the problem characterizing A.,, is a subset of the corresponding class admissible in the previ-
402
VI. APPLICATION OF THE CALCULUS OF VARIATIONS
ous problem characterizing Av-1 . Thus the minimum Av can be no
smaller than the preceding minimum Av-I.
Our variational problems produce an infinite sequence of eigenvalues and eigenfunctions of the associated differential equation
problem. Conversely, the eigenvalues and eigenfunctions associated
with the differential equation are obtained as solutions of our variational eigenvalue problems, as will be shown· in §3, 1.
2. Generalizations.1 It need hardly be pointed out that the other
eigenvalue problems considered in the preceding chapter can also be
treated from the standpoint of the calculus of variations. This is
equally true of single and multiple integrals and of differential equations of second and higher order. For example, the eigenvalue problem of the Sturm-Liouville equation
(pu')' - qu
+ 'Xpu
=
0
with the boundary conditions u'(O) - h1u(O) = u'(1r)
corresponds to a variational problem of the form
ID[cp] =
+ h2u(1r)
11( (pcp 12 + qcp2) dx + h1p(O)cp(0) + h2p(1r)cp(1r) 2 =
2
= 0
min.
without boundary conditions. All the homogeneous boundary conditions previously listed can be obtained by an appropriate choice of
hi and h2, if we consider the boundary conditions u(O) = u(1r) = 0 as
limiting cases for infinite h1 and h2 .
Even for Sturm-Liouville equations with singularities at end points,
the eigenvalues and eigenfunctions may be characterized by variational problems. It will suffice to give the formulation for Legendre
polynomials and for Bessel functions. We obtain the Legendre
polynomials by considering the free problem with
l
+I
H[cp) = _ cp2 dx;
1
here both end points are singular. The Bessel function Jo(x-y'A)
of order zero arises from the problem for which
H[cp] =
1xcp dx
1
2
0
1
See Coura.nt, 'Ober die Anwendung der Variationsrechnung ....
403
EXTREMUM PROPERTIES OF EIGENVALUES
with no boundary condition at x = 0, the Bessel functions of m-th
order with m ~ 1 from the problem for which
:J)[q>] =
J.' (x/
2
+ ~' q>
2
)
H[q>]
dx,
=
J.'
2
Xq>
dx
bo~ndary condition 'P(O) = 0. The boundary condition
0 at the regular end point x = 1 is associated with the
problem of a clamped membrane.
Analogous results are obtained for self-adjoint differential equations of higher order in more dimensions, such as the equation of the
vibrating plate,
with the
'P(l) =
(10)
ll.ll.u - AU = 0.
To treat the problem of a clamped plate (cf. Ch. IV, §10) we set
H[<P]
=
fl l
dx dy
and impose the condition
'P
= dcp =
an
0
on the boundary r of G. With these conventions the rest of the discussion and the formulas of subsection 1 remain entirely unchanged.
Other types of eigenvalue problems, not treated explicitly in
Chapter V, immediately fit into the scheme presented here. If we
recall that !H[cp] corresponds to the kinetic energy of our continuum
with mass density p, while !~[cp] represents the potential energy,
it is natural to construct mechanical models in which the mass is
not only distributed continuously over the domain G but concentrated at isolated points. For models of this kind we obtain (in one
dimension) the expression
(11)
which takes the place of H; here x 1 , x2 , • • • , x,,, denote given points
in the domain G and the numbers b,, are given constants. A quadratic functional of this form corresponds to the assumption that
masses of magnitude b,, are concentrated at the points X1, X2, • • • ,
x,,, • We shall assume throughout that these masses are non-nega-
404
VI. APPLICATION OF THE CALCULUS OF VARIATIONS
tive. Similarly, we can consider more general expressions of the
form
(12)
~[cp]
J. pcp
=
12
dx
G
+ J.
2
qcp dx
G
+t
a,,cp(xY.
11=1
For such problems we find eigenvalues and eigenfunctions ueing
precisely the same notation and argument as in subsection 1. These
eigenfunctions satisfy the differential equation
(13)
L[u]
+ °'Apu
= (pu')' - qu
+ °'Apu
= 0
except at the points X1, X2, • • • , Xh, where natural boundary and
jump conditions for the derivatives occur. We obtain these conditions immediately by forming the first variation. The eigenfunctions
of our problem, multiplied by yf,, are no longer orthogonal; instead,
they satisfy the conditions1
(14)
1. PUiUj dx + t
0
b11Ui(X11)u;(x11) = { 1
v=-1
G
for
for
i rt= j,
i = j.
A further example is presented by the expressions
(15)
and
(15a)
.p[cp] =
L
2
pcp dx
+
ll
k(x, y)cp(x)cp(y) dx dy,
where k(x, y) is a given symmetric function of x and y. For simplicity we shall assume that .p[cp] never takes negative values. In
place of the eigenvalue differential equation we obtain, by the method
of subsection 1, the integro-differential equation
(16)
(pu')' - qu
+ x(pU +
!.
k(x, y)u(y) dy) = 0
with the boundary condition u = 0, for example. The orthogonality
relations2 for the eigenfunctions of this problem become
1
a
pu,(x)u;(x) dx
1
+ alar k(x, y)ui(x)u;(y) dx dy
=
jol
"\.
ffor
or
~i ~- ~'J..
H. Kneser has suggested the term "weighted orthogonality."
They can also be considered as "biorthogonality relations" between the
system Ui and the system Vi if in addition to the eigenfunctions Ui(X) we in1
2
405
EXTREMUM PROPERTIES OF EIGENVALUE S
The
differential
of
problems
following remark, valid for all eigenvalue
equations, will be of importance later:
If G consists of a number of non-overlapping (open) domains G',
G", · · · , then the totality of eigenvalues and eigenfunctions of G consists of the eigenvalues and eigenfunctions of all the subdomains G',
G", · · · , where each eigenfunction is understood to vanish identically
in all except one of the separate subdomains.
This represents the self-evident physical fact that several separate
vibrating systems perform vibrations without interacting with each
other.
One way of proving the statement mathematica lly is to use the
definition of the eigenfunctions by means of the differential equation
problem. We need only remark that the eigenfunctions which are
defined for any one of the subdomains G', G", · · · and vanish identically outside this subdomain (together with linear combinations of
such eigenfunctions belonging to the same eigenvalue) are also eigenfunctions of G. Conversely, every eigenfunction of G must differ
from zero in at least one of the subregions. An alternative proof of
the assertion is based on the definition of the eigenvalues by means of
variational problems; proceeding step by step, it is easy to identify
the eigenvalues of the total domain with those of the subdomains.
4. The Maximum-m inimum Property of Eigenvalues. Again, as in
our discussion of quadratic forms (cf. Chapter I), we can define the
n-th eigenvalue and the associated eigenfunction without reference
to preceding eigenvalues and eigenfunctions.
We consider any one of the preceding variational problems, retain
the notation of subsection 1, and modify the problem by modifying
the conditions for the admissible functions <p. Instead of stipulating
H[<p, ui] = 0 (i = 1, 2, · · · , n - 1) we impose then - 1 modified
conditions
3. Eigenvalue Problems for Regions with Separate Components.
(i = 1, 2, · · · , n - 1)
troduce the functions Vi(x)
= pu.(x) +
l
k(x, y)ui(Y) dy.
In other words,
they can be written in the form
1
{o
u·v· dz == '\
~1
a ~ '
for
for
i ¢j,
i = j.
406
VI. APPLICATION OF THE CALCULUS OF VARIATIONS
where V1 , V2 , · · · , Vn-1 are arbitrarily chosen piecewise continuouf:l
functions in G. Under the conditions imposed, the integrals D[<P],
or more generally the functionals '.t>[<PJ, have a greatest lower bound,
which depends on the functions V1 , v2 , · · · , Vn-1 and will be denoted
by d{v1, v2, · · · , Vn-1}. We assert that the eigenfunctions Un and
the eigenvalues An, which were defined successively by variational
problems, can be characterized in terms of the modified problem by
the following theorem:
Given n - 1 functions v1, V2, · · · , Vn-1 which are piecewise continuous in G, let d{v1, V2, · · · , Vn-d be the greatest lower bound of the set
of values assumed by the functional '.t>[<P], where <Pis any function con-
tinuous in G with piecewise continuous derivatives which satisfies H[<P]
1 and the conditions
(17)
=
(i = 1, 2, · · · , n - 1)
Then An is equal to the largest value which this lower bound d assumes
if the functions v1 , V2 , · · · , Vn-1 range over all sets of admissible f unctions. This maximum-minimum is attained for u = Un and V1 = u1 ,
V2 = U2 , · · · , Vn-1 = Un-1 .
Under the boundary condition u = 0, the variational problem is no
"longer a free problem but is restricted by the condition <P = 0 on r.
To prove the theorem, we first note that by definition we have
d{v1, V2, · · · , Vn-d = An for Vi = Ui (1 ~ i ~ n - 1); then, we show
that for arbitrary V1, V2, · · · , Vn-1 we have d{v1, V2, · · · , Vn-d ~An.
We need only determine a particular function <P, satisfying the conditions H[<P, Vi] = 0 (i = 1, 2, · · · , n - 1), for which '.D[<P] ~ An .
For this purpose we determine a suitable linear combination of the
first n eigenfunctions <P = L :.1 CiUi with constant c1 , c2, · · ·, Cn •
The n - I relations (17) lead to n - 1 linear homogeneous conditions
on the n quantities c1 , c2 , · · · , Cn , and thus can always be satisfied.
The equation H[<P] = .L::.1 c~ = 1 is simply a normalizing condition
which determines a factor of proportionality. Since '.D[ui, uk] = 0
(i ¢ k) and '.D[ui] = Ai [cf. eq. (8)]
implies
n
ID[<P]
=L
i-1
C~Ai;
EXTREMUM PROPERTIES OF EIGENVALUES
407
because of
n
L: c~ =
1 and
An ~ Ai
(i
= 1, 2, · · · , n),
i-1
we have
Thus the minimum d{v1, v2 , · · • , Vn-il is certainly no greater than
An ; therefore, An is the largest value which this minimum can assume.
§2.
General Consequences of the Extremum Properties of
the Eigenvalues
1. General Theorems. Important conclusions can be drawn from
the maximum-minimum property by means of certain simple principles of the calculus of variations. The first of these principles
states: By strengthening the conditions in a minimum problem we do
not diminish the value of the minimum; conversely, by weakening the
conditions the minimum decreases, or at any rate does not increase.
The ~econd principle states: Given two minimum problems with the
same class of admissible functions cp, such that for every cp the functional
to be minimized is no smaller in the first problem than in the second,
then the minimum for the first problem is also no smaller than the minimum for the second.
The classical minimum definition of eigenvalues does not lend itself to the application of these principles when eigenvalues in different problems are compared; for, the classes of admissible functions
fail to coincide since the conditions of admissibility are not the same.
Under the maximum-minimum definition, however, the classes of
admissible functions agree, and our principles may be applied.
Consider for example any system, capable of vibrating, the
eigenvalues of which are determined by an eigenvalue problem of
the type treated here. Then any conditions of constraint under
which the system is required to vibrate may be expressed mathematically as auxiliary conditions imposed on the functions cp admissible in the variational problem. Whenever the conditions on cp in
the maximum-minimum problem are strengthened, the lower bound
d {v1 , V2 , · · · , Vn-1} increases, or at any rate does not decrease. Hence
the same is true for the maximum of these lower bounds, the n-th
408
VI. APPLICATION OF THE CALCULUS OF VARIATIONS
eigenvalue. Correspondingly, the value of the maximum-minimum,
i.e. of the n-th eigenvalue, is diminished or at least not increased
when the conditions on the functions <p are weakened.
Physically, this means:
THEOREM 1 : If a system which is capable of vibrating is subjected
to constraining conditions, the fundamental tone and every overtone
become higher (at least not lower) in pitch. Conversely, if restraining
conditions are removed, the fundamental tone and every overtone become
lower (at least not higher).
For example, in the case of a vibrating stretched elastic membrane,
the fundamental tone and all the overtones become higher if the
membrane is fixed not only along the boundary but also along curves
or at parts of the surface. On the other hand, the fundamental tone
and all the overtones become lower if the membrane is slit or, in
the case of a vibrating plate, if the material develops a crack. In
this case the conditions of continuity for the admissible functions <p
or their derivatives at the slit or crack are removed.
Mathematically, this principle leads to a number of important
general theorems on the distribution of eigenvalues. Theorems 2 and
3 concern the boundary condition u = 0 and compare the eigenvalue distribution of a domain with that of subdomains. Further
theorems make a corresponding statement for the boundary condition au/ an = 0. Other theorems involve more general boundary
conditions and compare the different spectra1 of the differential
equation for various forms of these boundary conditions.
THEOREM 2: Suppose that G', G", G"', · · · are a finite number of
non-overlapping subdomains of the domain G. Let A(>..) denote the
number of eigenvalues less than A of the differential equation L[u] +
>..pu = 0 for G with the boundary condition u = 0. Then the total
number of eigenvalues les~ than A for all the separate subdomains with
the same boundary condition does not exceed A(>..).
This theorem may also be expressed as follows: Under the boundary condition u = 0, the n-th eigenvalue An for the domain G is at most
equal to the n-th number A! in the sequence consisting of all the eigenvalues of the subdomains a<i>, arranged according to increasing magnitude and taken with their respective multiplicity.
The proof follows immediately from the following consideration:
Suppose that in the maximum-minimum problem which defines An ,
1• By
the term spectrum we mean, as before, the totality of eigenvalues.
GENERAL CONSEQUENCES OF EXTREMUM PROPERTIES
409
we stipulate a new condition for the functions <p, requiring them to
vanish at the boundaries of all the subdomains Gw and to vanish
in that part of G which belongs to no subdomain GCi>. Then, first,
in virtue of the fundamental principle stated above, the value of the
maximum-minimum is not diminished. On the other hand, the resulting maximum-minimum is precisely that which defines then-th
eigenvalue of the disconnected region consisting of the separate
domains G', G", · · · . In view of the remark in §1, 3, the new
value of the maximum-minimum is equal to>.!, and hence we have
An ~ >.!,as we wished to prove.
In particular, the theorem just proved leads to an important property of monotonicity of the eigenvalues associated with the boundary
condition u = 0.
THEOREM 3: Under the boundary condition u = 0 then-th eigenralue
1
for a domain G never exceeds the n-th eigenvalue for a subdomain of G .
A statement corresponding to that of Theorem 2 may be made
for the boundary condition au/an = 0.
THEOREM 4: Suppose that G', G", G"', · · · are a finite number of
non-overlapping subdomains which exhaust the domain G co;mpletely.
Let B(K) denote the number of eigenvalues less than K of the differential
Xpu = 0 for G with the boundary condition ou/on =
equation L[u]
0. Then the total number of eigenvalues less than K for all the separate
subdomains with the same boundary condition is at least as large as B(K).
We can also state this theorem in the following way: Let "! be the
n-th number, arranged in order of increasing magnitude, in the combined set of eigenvalues belonging to the subdomains a<i>, associated with
the boundary condition au/ on = 0, where every eigenvalue is to be taken
with its correct multiplicity. Then then-th eigenvalue Kn of the domain
G for the same boundary condition is larger than or equal to the number K !.
Here, too, the proof follows almost immediately from the first of
our general principles applied to the maximum-minimum problem
characterizing the n-th eigenvalue Kn for G. For, if we permit the
admissible functlons 'P in this problem to be discontinuous on the
boundary curves of the domains a<'1 in G (the discontinuities across
these boundary curves are at most finite jumps), we reduce the value
of the maximum-minimum, or at least do not increase it. On the
other hand, the modified maximum-minimum problem, according to
+
1
In fact, it is always smaller when we are dealing with a proper subdomain,
as can be ascertained easily by the method of §6.
410
VI. APPLICATION OF THE CALCULUS OF VARIATIONS
§1, 4, defines just the n-th eigenvalue associated with the natural
boundary conditions au/ on = 0 for the domain consisting of the
separate domains 0<•1 , that is, the value "! . This proves the relation Kn ~ "!.
We shall now prove some theorems concerning the relative behavior
of the spectra of the differential equation for the different types of
boundary conditions considered above.
THEOREM 5: Let An be the n-th eigenvalue of the dijferential equation
L[u]
Xpu = 0 for the domain Gunder the boundary condition u = 0,
and let µn be the n-th eigenvalue for the condition au/ on
uu = 0, or
more generally for the condition au/ an + uu = 0 on a part r' of the
boundary r, u = 0 on the remaining part r" of the boundary. Then
+
+
µn ~An•
This is proved by considering the maximum-mini mum problem
which characterizes the n-th eigenvalue µn of G, without boundary
conditions, as the maximum of the minimum of '.D[~]; we impose on
the functions ~ the further condition of vanishing on the boundary
r of G. Then the value of the individual minimum and hence also
of the maximum-mini mum is certainly increased or at least not
diminished. On the other hand, this new maximum-mini mum value
is clearly identical with An , since in view of the imposed condition
we now have '.D[~] = D[~]. Hence µn ~ An , as was asserted.
THEOREM 6: If, in the boundary condition au/an
uu = 0 on r,
the function u is either increased or diminished at every point, then
each individual eigenvalue can change only in the same sense.
This important fact is likewise a direct consequence of the maximum-minimum property by the second of the above-mentione d
principles. For, if we change the function u, the expression '.D[~]
changes in the same sense as u for each ~; hence the same is true of
its lower bound for given vi , and therefore for the maximum of these
lower bounds.
Theorems 5 and 6 exhibit a mutual relationship between the eigenvalues corresponding to the various types of boundary conditions.
If we change the function u at each point monotonically from 0 to oo ,
each individual eigenvalue µ increases monotonically from the value
which it had for the boundary condition au/ an = 0 to the value
which it takes for the boundary condition u = 0. In other words,
the theorem states that. of the boundary conditions considered, u = 0
+
GENERAL CONSEQUENCES OF EXTREMUM PROPERTIES
411
is the most restrictive and au/ an = 0 is the least restrictive if CT is nonnegative. That the limit of the eigenvalue µn for infinitely increasing
CT is indeed An may best be proved by investigating the nature of the
eigenfunctions more closely. Since this will not be done until later,
we refrain from carrying out the proof at this point (cf. Volume II).
In subsection 6 we shall see that this growth with increasing CT is
continuous. Investigation of the asymptotic distribution of the
eigenvalues for large n will show, furthermore, that in spite of the
indicated behavior of the eigenvalues, their asymptotic behavior for
n ---+- oo is independent of the boundary condition, and that accordingly the growth of the n-th eigenvalue due to the growth of the
function CT is negligibly small compared with the size of the eigenvalue for sufficiently large n.
The properties formulated in Theorems 5 and 6 suggest a simple
CTU = 0
physical interpretation. The boundary condition au/an
describes a boundary fixed by elastic forces, where the magnitude of
the constraining forces is determined by the function CT. The theorems state that as this elastic constraint increases in intensity each
eigenfrequency increases. The condition u = 0 represents the case
in which this force has become infinite, or in other words the boundary
is completely fixed.
Finally, the maximum-minimum property of the eigenvalues makes
it possible to investigate the dependence of the eigenvalues on the
coefficients of the differential equation and on the domain G.
ApU = 0, the
THEOREM 7: If, in the differential equation L[u]
every boundfor
coejfici,ent p varies at every point in the same sense, then,
sense. If
opposite
the
ary condition, the n-th eigenvalue changes in
sense,
same
the
in
everywhere
either of the coefficients p or q changes
boundthe
of
case
the
(In
sense.
same
this
in
every eigenvalue changes
CTU = 0, we assume CT ~ 0.)
ary condition au/an
First let p be changed everywhere in the same sense. Then for
every admissible function 'P the value of the expression ID[rp], therefore
also the lower bounds of these values for fixed vi and the maximum
of these lower bounds, then-th eigenvalue, change monotonically in
the same sense. If pis changed monotonically, resulting in the new
function p' ~ p, then for any admissible function 'P we have
+
+
+
ID[rp):
fl
prp2 dx dy
~ ID[rp):
fl
2
p'rp dx dy.
412
VI. APPLICATION OF THE CALCULUS OF VARIATIONS
Here we see that for fixed functions vi the lower bound of the left
side is no smaller than the lower bound of the right side. In forming
the lower bound of the latter quotient we have to substitute the
functions v: = vip/ p' for the functions vi in the auxiliary conditions,
since p has been changed to p' . If the system of all the functions
Vi ranges over the entire class of admissible functions then the system
of functions v~ does the same; it is, therefore, clear that the ratio of
the sizes of the maxima of the lower bounds considered is opposed
to that of the functions p and p' .
2. Infinite Growth of the Eigenvalues. In this subsection we shall
show that in the variational eigenvalue problems the eigenvalues An
become infinite for n - t oo. This implies, in particular, that each
eigenvalue has only a finite multiplicity, and that only a finite number of eigenvalues can be negative. The most important consequence of the unboundedness of the eigenvalues is, as will be seen in
§3, 1, the completeness of the system of eigenfunctions; this system
coincides therefore with the system of eigenfunctions of the differential equation.
To prove the unboundedness (here we shall not assume q > 0)
we denote by PM, qM, PM and Pm, qm, Pm the largest and smallest
values respectively of the functions p, q, p in G, and consider first
the boundary condition u = 0. If we replace the functions p, q, pin
1) and H by the constants Pm , qm , p M and p M , q M , Pm , we obtain
new eigenvalue problems with the eigenvalues
and
respectively,
and by Theorem 7 we have A~ ~ An ~ A ~ •
We ascertain first that the eigenvalues
are unbounded: In the
case of one independent variable, we can solve the associated differential equation eigenvalue problem explicitly in terms of trigonometric functions; the eigenvalues are the numbers (pm 112 + qm) / p M,
11 = 1, 2, · · · .
Since the eigenvalues
arising from the variational
problem are certainly contained in this sequence, our assertion An--+ oo
follows immediately.
We stated above that the eigenvalues resulting from the variational
problem coincide with the totality of the eigenvalues of the differential equation; assuming for the present that this assertion is true,
we have
A:
1
A:
A:
A:
x: = Pmn
2
+
qm,
2
A~ = pMn
+
qM,
PM
Pm
and thus we see that the quotient An/n2 remains between finite positive
bounds as n increases.
GENERA L CONSEQ UENCES OF EXTREM UM PROPERT IES
413
A:
for an arbitrar y domain G
In order to estimate the eigenvalues
the eigenvalues A! of
with
them
compare
we
ons,
dimensi
in several
ng all of G. We
containi
square
a
for
n
equatio
tial
differen
the same
n increases; the
as
infinite
become
these
..
that
§14
V,
Ch.
from
know
same is true for An, since by Theorems 3 and 7 we have
A!~ A: ~An.
We shall not extend this reasoning to different bounda ry conditions
since more precise estimate s of the asympto tic behavio r of the eigenvalues will soon show that they are unboun ded. However, we
sketch briefly an entirely differe nt-indi rect-me thod of proof which
nal problem
presupposes no knowledge of the solutions of the variatio
1
le.
preferab
in special cases, and thus is intrinsically
In the case of one indepen dent variable , we assume that our prob'lem has infinitely many eigenvalues X1 , X2 , • • • whose absolute values
all remain below some positive bound. The boundedness of
2 h2 p(x2)un(x2) 2
2
h1 p(x1)un(X1)
qu!) dx
An = [~ (pu~
2
+
+
+
:111
x2
and of
1
pu! dx immedi ately implies the boundedness of
:111
if the constan ts h1 and ~ are non-negative. (This restricti on can be
easily remove d with the help of the observa tions in subsection 5.)
We now make use of the following lemma: If for a set of functions
2
l dx are bounded, the functions
<1>' dx and
<P(x) the integral s
<P(X) are equicontinuous and uniformly bounded (cf. Ch. II, p. 58.)
Thus by the accumu lation principle (Ch. II, §2) we can select a uniformly convergent subsequence from the set of eigenfunctions Un •
If we again denote this new sequence by Un , it is clear that
limn,m-+ooH[un - Um] = 0. But on the other hand, from the orthogonality of the Un for n -:;t. m, we have
i
l
H[un - Um] = 2.
This contrad iction proves the theorem .
In the case of several (e.g. two) variables, the same reasoning holds
1 This method is due to F. Rellich, Ein Satz iiber mittlere Konverg enz,
Nachr. Ges. Gottinge n (math.-p hys. Kl.), 1930.
414
VI. APPLICATION OF THE CALCULUS OF VARIATIONS
on the basis of the following lemma by Rellich 1 (which will not be
proved here) :
Given a set of functions ~(x, y) defined in the domain G, we can select
a subsequence ~n for which
lim
n,m-.co
if the expressions
(~n - ~m) 2
{[
JJG
fl c~! + ~;)dx
dy
dx dy = 0
and
fl ~
2
dx dy are uniformly
bounded.
3. Asymptotic Behavior of the Eigenvalues in the Sturm-Liouville
Problem. In the case of the Sturm-Liouville problem, the maximumminimum property makes it possible both to determine the order of
magnitude of the n-th eigenvalue and to find its asymptotic value. Using
the transformation (20a) of Ch. V, §3, 3, we write the differential
equation (py')' - qy + Xpy = 0 in the form
(18)
z" - rz
+~
= 0
for the interval 0 ~ t ~ l, where r(t) is a continuous function and z
must satisfy new homogeneous boundary conditions which arise from
the original ones. We consider first the case y(O) = y(7r) = 0 or
z(O) = z(l) = 0. The eigenvalues are maxima of the minima of an
expression of the form
·
1
l
(z
+ rz2) dx
12
-in this subsection we shall assume the fact, to be proved later
(§3, 1), that the eigenfunctions and eigenvalues arising from the
variational problem coincide with those associated with th~ differential equation. If we omit the term rz2 in this expression, and thus
consider the integral expression
1
l
2
z' dx,
then, provided z satisfies the condition
l
[ z2 dx
1
See Rellich, op. cit.
=
1,
GENERAL CONSEQU ENCES OF EXTREMU M PROPERT IES
415
the new expression differs from the original one by no more than a
fixed finite bound rM (maximu m of the absolute value of r). Hence
the maximin ima of the first expression, i.e. the desired eigenvalues,
differ from those of the second by no more than r M • But the2 maxi2 2
minima of the second integral are just the eigenvalues µ 11 = n 7r /l of
z" + µZ = 0 for the interval (0, l); therefore, since limn ..... oo J.'n = oo,
we immedia tely have the asymptot ic formula
(19)
where, as before, 0(1) denotes a function which remains bounded as
n becomes infinite. Returnin g to our original notation, we find
21(11(1- )
2
2
lim~=
(19a)
p
o
1r
n-+oo An
!!.dx.
Precisely the same estimate may be derived for arbitrary boundary conditions, since the asympto tic behavior of the eigenvalu es of the equa-
tion z" + µz = 0 is independ ent of the boundary condition (cf.
also §4.)
Our asympto tic estimate is
4. Singular Differential Equations.
easily carried over to differential equation s with singularities. We
consider only Bessel's equation
XU II
+U +
I
2
(
m
X)
XA -
U =
0,
whose solutions are the Bessel functions J m (xv'A) . Here, with the
boundary condition s u(O) < oo and u(l) = 0, the eigenvalues X are
the squares Am,n of the zeros of J m (cf. Ch. V, §10, 1). In the case
m (xv'A)' with
m ~ 1 it is useful to consider the functions v =
the eigenvalue equation
vxJ
+ (A - 4m4x-~-=---!) v =
2
v"
2
0
and to character ize its eigenvalues by maxima of minima of the quotient D['P]/H[<p] with
D['P] =
11( ,2 +
0
'P
2
-
2) dx,
4m4 2 1 'P
x
H[<P] =
11
'P2 dx;
this replaces the character ization given in §1, 2. As a boundary condition we set <p(O) = <p(l) = O; since m ~ 1, we certainly have
416
D[<p]
VI. APPLICATIO N OF THE CALCULUS OF VARIATION S
~
£
1
<p 12 dx and thus An
~
n
2 2
7r •
On the other hand we obtain
an upper bound for An by restricting the admissibil ity conditions in
the following way: In an interval 0 ~ x ~ e to be determine d later,
we impose the condition <p(x) = 0 and majorize the second term in
the integrand of D[<p] by the constant
4m
2
-
42
E
11
0
1
2
c
<pdx=2 ·
E
This yields An~ n 27r 2/(1 - e) 2 + c/l. Now if we let e approach
zero with 1/n in a suitable way, e.g., if we set e = 1/vn, it follows
that limn ..... ooAn/n27r2 ~ 1. We thus obtain the asymptotic formula
for the zeros ~ of J m, which agrees with the formulas for the
nonsingula r problems. The same asymptoti c formula holds for other
conditions which we have considered, e.g. for the boundary condition
u'(l) = a.
This result may be immediate ly extended to the roots of the Bessel
function of zero-th order, in view of the relation J~(x) = -J1 (x)
(cf. Ch. V, §5, 5). Accordingly, the eigenvalues of Bessel's problem
with the boundary condition u(l) = 0 form = 1 are identical with
those for the boundary condition u'(l) = 0 form = 0 (apart _from
the first eigenvalue which is zero). Since Jo(x) has exactly one zero
between two consecutive zeros of J~(x)-as we may prove with the
aid of Rolle's theorem in the manner of Ch. VII, §2, 8-this justifies
the asymptoti c formula form = 0. 1
5. Further Remarks Concerning the Growth of Eigenvalues. (Occurrence of) Negative Eigenvalues. If, as we have assumed, the func-
tion u and the numbers hi, h2 in the variationa l problems of §1 are
non-negative,2 and the same is true of q, then clearly there can be no
negative eigenvalues. Now the considerations of subsection 2 show:
If the function q is not everywhere positive, at most a finite number of
negative eigenvalues can occur. This is also true when the f u:nction u or
the constants h1 , h2 assume negative values since here, too, the eigenvalues become infinite with increasing n.
1
2
See another proof form = 0 in §7, 8.
The function p, as well asp, is always assumed non-negativ e.
GENERAL CONSEQUENCES OF EXTREMUM PROPERTIES
417
To prove this, we consider, for the sake of brevity, a one-dimensional problem with the fundamental domain 0 ~ x ~ 1r and estimate
the negative terms associated with the boundary: If ~ denotes a
point in the interval 0 ~ x ~ t, we have
I y(O)
f
y(~) I = I
-
y' dx
1,
where t is to be chosen subject only to the restriction 0
Using the Schwarz inequality we find
<
t ~
1r.
y(~)I ~Vt~
ly(O) -
and hence, denoting the minimum of p by Pm ,
ly(O)I
Under the
c~ndition {.
~ such that y(~)
Thus we have
2
+¥kif
~ IYWI
2
py dx
py''dx.
= 1 there exists an intermediate point
~ 1/tpm, where Pm denotes the minimum of p.
I y(O) I ~ - }
·
V
tpm
+ •/
V
t
Pm
•
V
/1?: py'2 dx.
O
If the integral under the radical sign exceeds the fixed bound 1/?r:?,
we require t to satisfy
~=If py''dx.
In this case t falls in the interval 0 ~ x ~
1r.
Then we have
t =
y(O}'
~
c
If
py'' dx
1r;
otherwise we set
+ C1,
where c and c1 are constants independent of y(x). It follows immediately that since a similar estimate is valid for y(?r), every admissible function y satisfies the important relation
h,y(O}'
+
h,y(.-)
2
~
C,
11.·
py" dx
+
C,,
418
VI. APPLICATION OF THE CALCULUS OF VARIATIONS
where C1 , C2 are suitable constants.
Furthermore, we certainly have
and thus finally
'.D[y]
~
f py''
dx -
c. /
f
py'' dx -
f
1
a.~~
Since the eigenvalues associated with the integral
r
py'' dx -
c•.
py' 2 dx become
infinite, those belonging to '.D[y] also become infinite. Therefore
there can be only a finite number of negative eigenvalues.
In an analogous way we obtain for two-dimensional problems an
estimate of the form
(20)
I£
2
pu<p
ds
I~ c1 VJ D['P] I + c,,
from which the same results concerning the essentially positive character of the eigenvalues follow .1 We mention finally that entirely
analogous considerations show the unboundedness of the eigenvalues
for the general eigenvalue problems of §1, 2. 2
6. Continuity of Eigenvalues. Assume first that the function p is
changed to a function p', so that 0 < (1 - E)p ::s; p' ::s; (1
E)p
for positive E. Then by Theorem 7, then-th eigenvalue of the differential equation lies behyeen the n-th eigenvalues obtained by
replacing p by p(l - E) and by p(l + E), respectively, in the differential equation, i.e. by multiplying the n-th eigenvalue of the original
differential equation by the factors (1 - E)-1 and (1 + e)-1, respectively. If E is taken sufficiently small, then these two numbers lie
arbitrarily close to each other. Thus the n-th eigenvalue depends continuously on the function p.
The n-th eigenvalue also depends continuously on q. In fact,
from the relation p 2: Pm , 'where Pm is a positive constant, it foJlows
that
+
1 =
fi
p</ dx dy 2:
Pm
fi
2
'{>
dx dy.
1 See Courant, Uber die Eigenwerte bei den
Differentialgleichungen der
mathematischen Physik, in particular pp. 13-17.
2 See Courant, Uber die Anwendung der Variationsrechnung ....
GENERAL CONSEQUENC ES OF EXTREMUM PROPERTIES
Clearly, the integral
Ji
419
2
cp dx dy is uniformly bounded for all ad-
missible functions q;. This implies that for a sufficiently small
change in the function q the expression ~~cp] changes arbitrarily
little; in fact, it changes uniformly for all admissible functions cp.
The same is therefore true for the maximum-m inimum of ~[cp].
In a similar way the eigenvalues are seen to depend continuously
on the function u which occurs in the boundary condition. We may
again suppose that the expressions '.D[cp] in the variational problem
1
have a fixed upper bound. Then by relation (20) the boundary
2
integrals£ puq; ds have a common bound.
Thus if we chartge the
2
function u in the boundary integral £ pucp ds by a sufficiently small
amount, then the change in the integral is also arbitrarily small and
uniformly small. for all admissible cp. Therefore the same is true
for ~[q;] and for the maximum-m inimum of '.D[q;].
Continuous dependence on p follows in a similar manner.
Summing up, we have:
THEOREM 8: For all boundary conditions considered, the n-th eigenvalue of the differential equation L[u] + Xpu = 0 depends continuously
on the coejficients of the equation.
THEOREM 9: The n-th eigenvalue depends continuously on the f unction u which occurs in the boundary condition au/an+ uu = 0.
Finally, we investigate the continuity properties of then-th eigen• value, considered as a function of the domain G. It will be seen that
the n-th eigenvalue of a domain G', for corresponding boundary conditions, approximate s the n-th eigenvalue of the domain G arbitrarily
closely, if the domain G' approximate s G sufficiently closely. We
must, however, use a sujficiently strong concept of approximation· of a
domain G by another domain G'. Whenever normal derivatives occur
in the boundary conditions, it will not be enough to require that the
boundary of G' approximate the boundary of G pointwise. We shall
1 For instance, the upper bound may be the n-th eigenvalue for an arbitrary square lying entirely in the interior of G with the boundary condition
u = 0. For, Theorems 3 and 5 imply that then-th eigenvalue for G with the
original boundary condition is certainly no greater than the n-th eigenvalue
for such a square with the boundary condition u = 0. Thus we see that
fixing this upper bound for '.D(cp] cannot affect the solution of the maximumminimum problem.
420
VI. APPLICATION OF THE CALCULUS OF VARIATIONS
have to require, in addition, that the normals to the boundary of G'
approximate those of G. It can be shown, in fact, that under the
weaker definition of approximation the n-th eigenvalue need not be a
continuous function of the domain. 1
Approximation of the domain G by the domain G' in the stronger
sense is defined analytically in the following way:
Let G together with its boundary be transformed pointwise into
the domain G' together with its boundary by equations of the form
(21)
x' = x + g(x, y),
y' = y + h(x, y);
1
The following is a simple counterexample: Let L[<P] = !J..<P, p = 1, and let
G be a square of edge 1. Outside G we take a second square Ge of
side E, oriented parallel to G, which lies opposite one of the sides of G and at
a distance e from it. We then connect its interior with that of G by means
of a narrow strip S, perpendicular to the two squares, formed by two parallel
line-segments of length E at a distance .,, from each other. Let the domain
G' consist of the two squares G and G, together with the strip S. The first
eigenvalue of G' with the boundary condition au/an= 0 is zero, the associated
eigenfunction is u1 = const. Now if, for every E, the width 11 of the strip S
is taken sufficiently small, then the second eigenvalue of G' can also be made
G
9!
I
l9
-.--1
s
G.-
-.---
I
G'• G+ G.- + S
arbitrarily small. For, consider a function <P in G' which is equal to -1/E in
G, and to a constant c in G, and which falls off linearly in S from c to -1/E.
Let the constant c be chosen in such a way that the integral of <P over G' vanishes. If E is sufficiently small, c is arbitrarily close to 0. The integral 1'.)[<P]
over G' will be of the order 71/E 3• If, for example, we choose 11 = E4, this integral is arbitrarily small while the integral of <P 2 over G' is arbitrarily close to
1. Hence, as a consequence of the classical minimal property of the eigenvalues and eigenfunctions, the second eigenvalue of G' is certainly arbitrarily
small. If we permit E to approach zero, the second eigenvalue of G' will certainly converge to zero if 71/E 3 does. But the second eigenvalue of G is positive; hence it is not the limit of the second eigenvalue of G' although the
boundary of G' converges to the boundary of G.
421
GENERAL CONSEQU ENCES OF EXTREMU M PROPERT IES
here the functions g, h are continuo us and have piecewise continuo us
first derivativ es througho ut the domain, and both g, hand their first
derivativ es are less in absolute value than a small positive number E.
Then we say that the domain G is approximated by the domain G' with
the degree of accuracy E.
When E approach es zero we say that G' is deformed continuo usly
into G. We shall prove the following theorem:
the
THEOREM 10: For any of the boundary conditions considered,
convaries
0
=
}..pu
L[u]
equation
al
differenti
n-th eigenvalue of the
tinuously when the domain G is deformed continuously in the sense
defined above.
Proof: Consider a sequence of domains G' for which the above
numbers E converge to zero. We solve equation s (21) for x and y, set
u(x(s), y(s)) = T(s),
p(x, yt = p'(x', y') etc.,
cp(x, y) = cp'(x', y'),
+
and transform the two integrals which constitut e '.D[cp] into an integral
over the domain G' and one over its boundary r'.
In this way we obtain an eigenvalu e variation al problem for the
domain G, with coefficients differing arbitraril y little from the original
coefficients. Accordingly, continuo us dependen ce may be proved by
methods analogou s to those which led to Theorem s 8 and 9. We
shall carry out the proof in detail:
The integral D[cp] is transform ed into the integral
(22)
Ji,
2
1
{p [(c,c; (l + g:t) + cp; 1 h11)
1
+ (cp;'gy +
cp;,(1 + hu))
2
]
where M stands for the functiona l determin ant
(i + axag) (
M =
which is arbitraril y close to 1 if
over the boundary becomes
1
r
pucp2 dS
=
1
+ ah) _
ay
ag ah
ay ax
is sufficiently small.
E
f
r'
dS 1 ds' ,
p I T ( S ) cp12 -d
s
where ds' denotes the element of arc on the boundary
If in general we define
D'[if] =
rr {p(if! +if!) +
llo,
2
q1f
}
The integral
r' of G'.
f
2
dx dy, '.!)'[if] = D'[if] + r' PT(s)if ds',
422
VI. APPLICATION OF THE CALCULUS OF VARIATIONS
then the integrand in (22) differs from the integrand of D'[rp'] by the
factors M-1 and p/p' (which differ arbitrarily little from 1) and by
the additive terms in which rp~~, rp~~, rp~· , rp~·, and rp' 2, multiplied by
factors which converge to zero with E, appear. Using the inequality
2
j
JL "';. ~·
dx'
ay'
D[rp]
=
Is JL <"';~ + ~~) o.x' ay',
we obtain the relation
(1
+ f>) D'[cr;'],
where o denotes a quantity which approaches zero with E. This
notation will be used in the sequel, although f> will not always stand
for the same quantity. Since ds/ds' is arbitrarily close to 1, for
sufficient]y small E, we have
fr
p' r(s)cp' 2 dds, ds' = (1
s
+ f>) Jr pr(s)'(/
2
ds'
and therefore
~[~]
=
(1
+ o) 1)'[~'].
Furthermore, we must transform the auxiliary conditions (3a) and
(17) imposed on the functions ~ in §1. Thus we obtain the expressions
fl pc/ dx dy = fl, p'M- ~' dx' dy' =
fl prpvi dx dy = Jl, p' ~'v~ dx' dy' =
1
M-
2
1,
1
0
(i = 1, 2, · · ·, n - 1).
We multiply the function rp' by a constant factor and the functions
v~ by M- 1p' / p; these factors all differ arbitrarily little from 1 for
small E. The resulting functions~" and
satisfy the relations
fl,
JJG
v:
P'IJ
112
dx' dy'
=
1,
1
1
vi dx dy = 0
1111
[[
I
pep
We find that
ID[~] = (1
+ f>) ~'[~"]
(i = 1, 2, · · ·, n - 1).
GENERAL CONSEQUENCES OF EXTREMUM PROPERTIES
423
holds and that the function ~" satisfies the conditions of the maximum-minimum problem which characterizes the n-th eigenvalue for
G'; here the functions v7 in G' correspond to the functions Vi in G.
is just the set of adThe domain of variability of the functions
mitted functions over G'; it follows that the maximum-minimum of
the left side can differ from that of the right only by a factor which
approaches 1 as E approaches zero, which proves Theorem 10. The
above reasoning results in the following sharper form of this theorem:
COROLLARY TO THEOREM 10: If a domain G is deformed into a
domain G' by the transformation (21) in such a way that
v:
1::1 «, ltul «,
1:~1 «,
1:~1 «,
where E is any small positive number, then there exists a number 11,
depending only on E and approaching zero with E, such that for every n
and f qr any one of the boundary conditions in question the n-th eigenvalues µn, µ~for the domains G and G', respectively, satisfy the relation
I~-+~·
For the boundary condition u = 0, in which no normal derivatives
appear, the theorem on continuous dependence holds under less
restrictive conditions:
THEOREM 11 : For the boundary condition u = 0, the n-th eigenvalue
}\pu = 0 is a continuous function of the domain
of the equation L[u]
are no longer required to vary continuously
normals
the
though
even
G,
when the domain is continuously deformed.
In fact, we can always enclose the boundaries of any two domains
G and G', which are sufficiently close together (but for which the
normals at neighboring boundary points do not necessarily have
neighboring directions), between the boundaries of two domains B
and B' which are sufficiently close in the more restrictive sense.
Since by Theorem 3 then-th eigenvalue for the boundary condition
u = 0 is a monotonic function of the domain, the n-th eigenvalues
of G and G' must lie between those of B and B', which by Theorem
10 are themselves close together. This proves Theorem 11.
If we do not pass to the limit E ~ 0, the preceding considerations
furnish the following more general result:
If two domains G and G' are related to each other by a point trans-
+
424
VI. APPLICATION OF THE CALCULUS OF VARIATIONS
formation of the above type such that the absolute value of the functional
determinant is uniformly bounded, and if An and A: denote the n-th
eigenvalues of the domains G and G', respectively, then for sufficiently
large n the quotient An/A: lies between positive bounds which are independent of n.
§3.
Completeness and Expansion Theorems
1. Completeness of the Eigenfunctions. In connection with the
variational problems for the quotients 1)[cp]/.p[cp] considered in §§1
and 2, we obtained the relation
lim An
n-.oo
=
oo.
For the result it was essential that .p[cp] be positive definite, i.e.
that .p[cp] can assume only non-negative values and vanishes only if
the integrand cp vanishes. Using the above limit relation we shall
now prove the completeness theorem in a sharper form.
The system of eigenfunctions associated with the quotient 1)[cp]/ .p[cp]
is complete in the following sense: For any continuous function f and
any positive E, arbitrarily small, we can find a finite linear combination
of the eigenfundions such that
.p[j - Wn]
< E.
We obtain the best approximation, i.e. the smallest value of
with the Fourier coefficients
a, = c, =
.P[f -
wn],
.P[f, ui].
These satisfy the completeness relation
00
.P[f] =
(23)
L: c~.
i-1
Here, as in the case of arbitrary orthogonal functions, it follows
from relations (8) of §1 that the best mean approximation off with
respect to .P by a linear combination of the first n eigenfunctions,
that is, the smallest value of 4'[! - wn], is attained when ai = Ci =
.s)[f, ui] (the coefficients are independent of n). The relation
0 ::::;
4'[f - t
i=l
CiUi] =
4'(fl -
t C~
i-1
425
COMPLETENESS AND EXPANSION THEOREMS
immediately implies the convergence of the infinite series L7=1 c~ or,
more precisely, the Bessel inequality L7=1 c~ ~ S)[f].
To prove not only this inequality but the completeness relation
(23), we assume first that the function f satisfies the admissiqility
conditions of the variational problem. Then the function
n
Pn =
J-
CiUi
L
i-1
satisfies the orthogonality relations
(i = 1, 2, · · · , n)
(24)
and, by §1 equation (7), also the orthogonality relations
(i
(24')
=
1, 2, · · · , n).
From (24) and from the minimal property of An+1 , we find
(25)
On the other hand, ID[pn] is bounded, since
'.D[f) = ID
[t
CiUi]
1-=l
+ 2ID
[t
CiUi,
.-1
Pn] + ID[pn] ;
therefore, from relations (24') we have
'.D(f] = '.t)
[t
1-1
Ci Ui]
+ '.t)(pn)•
ciui] equals Li:.1 ).,-c~ and remains greater than a fixed
bound as n increases since only a finite number of eigenvalues can
be negative. Thus ID[pn] is bounded from above.
In virtue of relation (25) and the infinite growth of An+i ,
1)[Li.:!1
n
S){f) -
c~ =
L
i-1
~[pn] ~ 0
for
n~
oo,
which proves the completeness relation (23) and therefore the completeness of the eigenfunctions.
If the continuous function f fails to satisfy the admissibility conditions of the problem, we can approximate it by a function f* which
does satisfy these ·conditions in such a way that S)[f - f*] < E/ 4,
and then approximate f* by a function J! = L~=1 c~ui with the
property S)[f* - J!] < E/ 4. Then, since
S)[f -
J!J
=
~ff
- f*]
+ ~[f* - J!J + 2~[f -
J*, f* - J!],
426
VI. APPLICATION OF THE CALCULUS OF VARIATIONS
we have from the Schwarz inequality .s)[f - J!] < e and from the
minimal property of Pn, .s)[pn] < e. This proves the completeness
theorem for functions which are only continuous.
~he completeness of the system of solutions of the variational
problem implies that these solutions actually represent the totality
of eigenfunctions of the corresponding differential equation problem
(cf. the reasoning frequently used in Chapter V, e.g. page 301).
From the completeness relation (23) we easily deduce the more
general completeness relation for a pair of functions f and g,
00
(23a)
.P[f, g]
=
L .s)[f, ui].s)[g, ui].
i=l
By a slight addition to our argument, relation (23) can be supplemented by
00
(23b)
'.D[f]
=
L AiC~
i=l
provided '.D[f] is finite.
We assume first that a continuous L[f] = g exists, with .s)[g] finite.
Then using Green's formula, we write
'.D[f] = -.P[f, g].
Applying the completeness relation (23a) to the inner product 4)[f, g]
we obtain
Green's formula, together with (23a) and L[ui]
yields
=
-Xiui immediately
.s)[g, ui] = .s)[f, L[ui]] = -xi.p[f, uil,
Thus (23b) is proved, since 4)[f, ui] = Ci.
If L[f] = g does not satisfy the above assumption we obtain the
result by a suitable approximation off; the method is similar to that
used to prove (23).
2. The Expansion Theorem. In the case of a single independent
variable it is not difficult to derive from our result the theorems
on the expansion of arbitrary functions in terms of eigenfunctions,
thus supplementing the completeness theorem. This can be done
under essentially weaker assumptions than those in Chapter V.
427
COMPLE TENESS AND EXPANS ION THEORE MS
We shall prove: Every function f (x) which satisfies the admissib ility
conditions of an eigenvalue variational problem may be expande d in an
absolutely and uniform ly convergent series L':=1 CnUn in terms of the
eigenfunctions.
Un, it
Because of the completeness of the orthono rmal system
VP
f'
pfundx,
is sufficient to prove that the series Ln':.1 CnUn, where Cn =
converges uniformly (cf. Ch. II, p. 54). To prove this, we again
consider the function Pn = f - 2:,,!1 c,,u,, . As above, page 424, we
see that
n
'.D[pn]
=
ID[f] -
L c! X,,.
11=1
For sufficiently large n, say n ~ N, we have An+i ~ 0, and thus
IDlPnl ~ 0. Therefore the series :E,,:1 c!A,, converges, since its terms
are non-negative for v > N. Schwarz's inequal ity yields
k
(
~h CnUn(x)
)2
k
~ ~h c!Xn
2( )
co
00
2( )
~h u~nx ~ ~1 c!Xn ~h u~nx
k
.
Now, from Ch. V, §11, 3 we know that I un.(x) I < C, 2where C is a
constan t indepen dent of n. Since, by 2 §2, 2 and 3, An/n lies between
positive bounds, and since °2:':=1 1/n converges, L~=" u~(x)/Xn. is
arbitrar ily small uniformly in x for sufficiently large hand k, and thus
the same is true of L~h I CnUn(x) j. This means that the above
series converges absolutely and uniformly, which completes the proof
of the expansion theorem.
Our considerations And results remain valid even when singularities
occur, as in the case of Legendre and Bessel eigenfunctions. How.ever, our proof of the expansion theorem then holds only if we
exclude from the domain an arbitrar ily small neighborhood of the
singular points, since boundedness of the normalized eigenfunctions
has not been proved for such a neighborhood.
3. Generali zation of the Expansi on Theorem . The asympto tic
expressions for the Sturm-Liouville eigenfunctions (Ch. V, §11, 5)
permit us to generalize the expansion theorem just proved. In fact,
we can prove the theorem : every piecewise continuo1 us function defined
in the fundame ntal domain with a square-integrable first derivative may
the
We say that the derivativ e is square-in tegrable if the integral of
ntal
fundame
the
of
intervals
the
all
for
bounded
is
e
square of the derivativ
domain in which the function is continuo us.
· 1
428
VI. APPLICATION OF THE CALCULUS OF VARIA.TIONS
be expanded in an eigenfunction series which converges absolutely and
uniformly in all subdomains free of points of discontinuity; at the points
of discontinuity it represents (like the Fourier series) the arithmetic
mean of the right- and left-hand limits. (It should be remarked that
this theorem does not require that the functions to be expanded
satisfy the boundary conditions.)
As in §2, 3, we suppose the differential equation to be written in
the form (18),
+ Xz
z" - rz
= 0,
for the function z(t) in the interval 0 S t S l.
the series
G(t, T) =
f
Then we consider
Zn(t)z:(T)
n-1
An
in which Zn denotes the n-th eigenfunction of the above differential
equation, say for the boundary condition z = 0.
Application of the asymptotic formulas (70) and (71) of Chapter
V, together with formula (19), yields
•
2
G(t, T) = -
1f
oo
L
11-1
sm n
7f"
7f"
l t cos n l
n
T
00
+L
n~l
i/ln(l, T),
where i/ln(t, T) = 0(1/n2), so that G(t, T) differs only by an absolutely
and uniformly convergent series from
•
2
G*(t, T) = -
oo
L
1f n-1
=
sm n
7f"
l
t cos n
'Ir
l
T
n
! f ~n (sin n ~l (t + T) + sin n ?!:l (t
- T)).
1f n=-1
As we saw in Ch. II, §5, 1, this series has the following properties:
For fixed T (0 < T S l) it converges absolutely and uniformly for all
t of a closed interval which satisfy the conditions I t + T l > E,
I t - T I > E for E > 0. Since T > 0 and t > 0, this means that
the interval cannot contain the point t = r. Therefore, although
for t ~ T the series represents a continuous function, its sum has a
finite jump at t = T and at this point, again by Ch. II, §5, it is equal
to the arithmetic mean of the right- and left-hand limits.
ASYMPTO TIC DISTRIBU TION OF EIGENVA LUES
429
In the case of an arbitrary function which satisfies the above conditions, we remove the discontinuities by adding a suitable sum
:E a. G(t, Ti)
i
which is chosen (if necessary) in such a way that the boundary condition is satisfied. We obtain a function satisfying the conditions
of the general expansion theorem already proved in subsection 2;
this function may, therefore, be expanded in an absolutely and uniformly convergent eigenfunction series. However, as we have just
seen, the additiona l sum may itself be represented by an eigenfunction series with the properties stated in the generalized expansion
theorem (see page 427); thus we have proved this theorem for the
expansion in terms of the eigenfunctions of the differential equation
(18). If we transform the variables z, t back to y, x and the differential equation to the general Sturm-Liouville form, we immedia tely
obtain the expansion theorem for .the eigenfunctions Yn(x) of the
original differential equation ; for, except for constant factors, these
eigenfunctions are obtained by multiply ing the eigenfunctions Zn by
the same nowhere-vanishing function.
Asympt otic Distribu tion of Eigenva lues
The method of §2, 2 and §2, 3 for one independ ent variable can
also be used in the study of the asympto tic behavior of the n-th
eigenvalue for several independ ent variables. We obtain a result
significant for physical problems: the asymptotic behavior of the eigenvalues for differential equations with constant coejficients does not depend
on the shape but only on the size of the fundamental domain.
1. The Equation ~u + l.u = O for a Rectangle. In the case of a
rectangle of sides a and b, the eigenfunctions and eigenvalues of
du +AU = 0 are known explicitly (see Ch. V, §5, 4). In fact, for
the boundary condition u = 0 they are given-u p to a normalizing
factor-b y the expressions
§4.
7r2
and for the boundary condition
m'lr'Y
Z7rx
cos- cos b '
a
2)
(a2l2 + ~
au/ iJn =
(l, m = 1, 2, 3, · · · ),
0 by the expressions
(l, m = 0, 1, 2, 3, · · · ).
430
VI. APPLICATION OF THE CALCUI,US OF VARIATIONS
If the number of eigenvalues less than a bound A is denoted in the
first case by A (A) and in the second case by B (:\), then A (A) and
B(A) are equal to the number of integral solutions of the inequality
2
2
l
m
A
2+-b2:::;
a
r2;
here l > 0, m > 0 for the boundary condition u = 0 and l ~ 0,
m ~ 0 for the condition au/ an = 0. Simple asymptotic expresS'ions
for the desired numbers A(>.) and B(>.) may now be derived for
large >.. B(>.), for example, is precisely equal to the number of
lattice points with integral coordinates in the sector of the ellipse
x2
a2
y2
+ b2
A
=
r2
which lies in the quadrant x ~ 0, y ~ 0. For sufficiently large A
the ratio of the area of this sector to the number of lattice points
contained in it is arbitrarily close to I. If a unit square lying above
and to the right of each lattice point is associated with it, then the
region formed by these squares contains this sector of the ellipse;
however, if we omit the squares through which the ellipse passes,let the number of these be R(A)-then the region which remains is
contained entirely within this sector of the ellipse. We therefore
have between the areas the inequality
B(>.) - R(>.) :::; >. ~ :::; B(>.).
The arc of the ellipse contained in two adjoining boundary squares
has, for sufficiently large:\, at least the length I. Hence R(A) - I
is at most equal to twice the arc length of the quarter ellipse, and
this increases only with 0. We thus obtain the asymptotic formula
or
B(X)
~X
:!·
ASYMPTOTIC DISTRIBUTION OF El(}ENVALUES
431
More precisely, we can write
B('A.) =
_ h"
.,,.x + 8cv'A.,
ab
4
where c is a constant independent of X, and I fJ I < l. This formula
is valid for both boundary conditions considered; i.e., it also holds for
A (A), since the number of lattice points lying on the line-segments in
the boundary of the sector of the ellipse is asymptotically equal to
(a + b)-0/7r. If the eigenvalues are written in a sequence
'A. 1 , 'A.2, · · · ordered according to magnitude, we can calculate the
n-th eigenvalue asymptotically by setting A (Xn) = n and B(Xn) = n.
We obtain
or
+
l.u = O for Domains Consisting of a Finite
2. The Equation ~u
Number of Squares or Cubes. Now we consider the equation llu
Xu = 0 for a domain G which may be decomposed into a finite
number, say h, of squares Q1 , Q2, · · · , Qh (or cubes in the case of
+
three independent variables) of side a. Such domains will be called2
square-domains (or cube-domains). The area of G is then f = ha
3
(or its volume is V = ha ).
AE before fJ will denote a number between -1 and + 1, and c or
C a positive constant. When no misunderstanding seems likely we
shall denote different values of fJ and c or C by the same symbol
without explicitly distinguishing between them by means of indices.
For a domain G consisting of h squares, let A(X) be the number of
eigenvalues less than a bound X for the boundary condition u = 0
and let B(A) be the corresponding number for the boundary condition au/on = 0. Denoting by AQ 1(A), AQ 2 (A), · · · , AQ"(A) the
corresponding numbers for the subsquares with the boundary conditions u = 0, and by BQ 1 (A), BQ 1 (A), · · · , BQ"(A) these numbers with
the boundary conditions au/on = 0, we have from subsection 1
2
a2
1
(26)
BQ,(°'A) = :11'" 'A+ fJ ca-VX
AQ;(X) = 1r 'A + fJca-0,
4
(i = 1, 2, ... ' h).
432
VI. APPLICATION OF THE CALCULUS OF VARIATIONS
Combining Theorem 5 with Theorems· 2 and 4 (see §2) we have
AQ1(X)
+ · · · + AQh(X)
::; A(X) ::; BQ 1(X)
+ ··· + BQh(X).
Since the numbers AQi(X), BQ,(X) have the form given by equations
(26), we conclude
A(X)
=
fr x+ Ucay'A.
In other words, the following theorem holds:
THEOREM 12: For all boundary conditions considered, the number
A (X) of eigenvalues less than a bound X of the differential equation
.1u
Xu = 0 for a square-domain of area f is asymptotically equal to
f'A/4r; that is,
+
= _!_.
lim A('A)
f'A
(27)
4r
>.-co
More precisely, for all sujficiently large X the relation
4rA('A) _ 1 1 < __Q_
f'A
I
(28)
vx
holds, where C is a constant independent of X.
If Pn denotes the n-th eigenvalue corresponding to one of the
boundary conditions in question, this theorem or relation (28) is
equivalent to equation
(29)
Pn
4r
=y
n
_ 1+ Bcv
n,
where again -1 ::; 8 ::; 1 and c is a constant independent of n. To
see this one has only to set A(pn) = n in (28).
Theorem 12 remains valid even if the function u in the boundary
condition au/ an
uu = 0 assumes negative values. This will again
be shown with the aid of the remarks of §2, 5. First we note that
according to Theorem 5, the. n-th eigenvalue µn for the boundary
condition au/ an
uu = 0 can surely be no greater than the n-th
eigenvalue An for the boundary condition u = 0. Thus we may
assume from the outset that the expression
+
+
ID[~)
=
D[~] +
l
pul ds,
433
ASYMPTOTIC DISTRIBUTION OF EIGENVALUES
whose maximum-minimum is clearly µn, never exceeds the bound
An for any of the functions <p which are admissible in the variational
problem with the boundary condition u = 0.
Now by §2, 5 we have·
ll
pu,/ ds
.
I < c1v'I D[,;]l +
C2,
where c1, c2 are numerical constants; thus we have
D[<p] -
C1V I D[<,0] I -
C2
< '.D[<p] <
+ C1V I D[<,0] I +
D[<,0]
C2 •
From the assumption '.D[<p] :::; An it follows that
D[<p] -
C1V ID[<,0] I -
C2
< An ,
and this in turn implies that for increasing n, D[<p] can increase no
faster than An, or in other words that there must be a relation of the
form
D[<pJ
<
CaA11 ,
where c3 also denotes a constant. Since relation (29) holds for
Pn = An we have, under the assumptions made with respect to <p,'
D[<p] -
C4Vn :::;
'.D[<p] :::; D[<p]
+ C4Vn;
this relation is valid for the lower bounds of the expressions '.D[<pJ for
given functions v1 , v2 , • • • , Vn-I , and hence also for the maxima of
these lower bounds. The maximum-minimum of D[<p] is the n-th
eigenvalue for the boundary condition au/ an = 0, for which relation
(29) has already been proved. Therefore (29) follows directly for
the maximum of the lower bounds of '.D[<p], i.e. for then-th eigenvalue
µn with the boundary condition au/ an + uu = 0; this relation is
equivalent to the assertion of Theorem 12.
If there are three independent variables instead of two, the preceding discussion remains unchanged except for the expressions
AQ, and BQ, for the number of eigenvalues less than the bound A
under the boundary conditions u = 0 and au/iJn = 0, respectively.
We find, in fact, that
(26a)
A Q' (X) =
holds and obtain
1
r a3X3/2
6 2
+ Bea X,
2
B Q, (X)
=
1
3 3/2
r aX
6 2
+ Bea2X
434
VI. APPLICATION OF THE CALCULUS OF VARIATIONS
+
THEOREM 13: Consider the dijferential equation .c!lu
Xu = 0 for a
polyhedron of volume V consisting of a finite number k of congruent
cubes. For all boundary conditions in question the number A (A) of
eigenvalues less than a bound Xis asymptotically equal to VX312/61r 2, i.e.
.
!~
(27a)
A(A)
VA312
=
1
611'"2.
More precisely, for sujfici,ently large X we have the formula
2
(28a)
67r A(A)
VA 312
I
-
1
I < C ~'
_1
in which C is a constant independent of X. 1
3. Extension to the General Differential Equation L[u] + :l..9u = 0.
We shall now extend Theorem 13 to the general self-adjoint differential
equation (1). We assume that, by repeated halving of the length of
th.e edge a, the subdivision of the domain G into squares or cubes has
been refined in such a way that in the resulting subdomains the difference between the largest and the smallest values of the functions p
and pis always less thana given small positive number E. The function
q can have no influence at all on the asymptotic distribution of the eigenvalues, since the expression '.t)[~l, and with it its maximum-minimum,
changes by an amount which is bounded, namely, by less than I qM II Pm,
here qM and Pm have the same meaning as before. Accordingly, we
set q = 0 without affecting the asymptotic distribution.
We consider the case of a plane domain G consisting of a finite
number of squares. Let the number of squares again be h, and let
the lengths of their sides be a. A'(A) will denote the number of
eigenvalues less than a bound A of the differential equation L[u] +
'J\pu = 0 for the domain G; any of the boundary conditions considered
may be used, but for the condition au/ an
uu = 0 we must make
the restrictive assumption u ~ O. We denote the subsquares by
Q1 , Q2, · · · , Qh, and the associated numbers of eigenvalues of the
differential equation less than a bound A by A~ 1 (A), A ~ 2 (A), · · ·, A~h(A)
for the boundary condition u = 0, and by B~ 1 (A), B~ 2 ('A), · · · , B~h(A)
for the boundary condition au/an = 0. From Theorems 2, 4 and 5
we obtain
+
1 It is not in general possible to obtain a
sharper estimate of the error in
this asymptotic estimate of A(>.), since for both the square and the cube the
indicated order of magnitude of the error is actually attained.
435
ASYMPTOTIC DIS'l'RIBUTION OF EIGENVALUES
(30)
A~ 1 (A)
+ ··· + A~h(A)
:::; A'(A) :::; B~ 1 (A)
+ · · · + B~h(A).
Theorem 7 implies
<O
C~
B~,(X) S !!.-fi> Bo,(X)
A~,(X) ~ P~i> Ao,(X),
Pm
P.11
p~'>, p~i> the minima
by
and
where. by p~>, p~> we denote the maxima
(cf. subsection 2)
again
;
of the respective functions in the square-Qi
equation
differential
the
of
eigenvalues
the number of corresponding
A
by
denoted
and
(26)
equations
by
given
is
0
Au + Au =
0 ,(A) or
BQ,(A). For, substituting p~> for p and p~> for pin differential equation (1), we see by Theorem 7 that each eigenvalue becomes larger (or
at least is not diminished), and thus the number of eigenvalues less
than a fixed A decreases (or does not increase). On the other hand,
differential equation (1) goes over into the differential equation
(a")
Au
+ X ~~) u = 0,
whose eigenvalues are the eigenvalues of Au+ Au = 0 multiplied by
the factor p iJ> / p~'>. A corresponding argument holds if p:n•> is substituted for p ~nd pi;> for p.
Furthermore, since p and pare continuous functions we have
rr !! dx dy
JJap
h
=
a
2
h
( i)
~7T> + 0 =
~
i-1p
11
a
2
( i)
p ~) + 0
~
i=lPm
1
,
where the numbers Io I, Io' I may be made arbitrarily small by making the original subdivision into squares sufficiently fine, that is, by
taking a sufficiently small. As in subsection 2, we apply (30) and
find
A(X) =
:r
Ji~ dx dy + XO" + 8c0,
where I o" I also is arbitrarily small. This is equivalent to the following statement about the asymptotic distribution of eigenvalues:
THEOREM 14: For the differential equation L[u] + Apu = 0, under
any of the boundary conditions considered, the number A(A) of eigenvalues less than a given bound A for a square-domain G is asympto#cally
equal to
~ ({ !!. dx dy;
lla p
4r
in other words the relation
436
VI. APPLICATION OF THE CALCULUS OF VARIATIONS
. A(X)
hm
-
(31)
A
>t-"oo
= -1 L~ -p dx dy
4r a p
hol,ds.
As in subsection 2, we see that the original assumption u > 0 is
unnecessary.
Analogous considerations for three-dimensional space lead to
THEOREM 15: For the differential equation L[u]
>..pu = 0, under
+
any of the boundary conditions considered, the number of eigenvalues less
than a given bound>.. for a cube-domain G is asymptotically equal to
~' >.'
12
Jfl (~)"'
dx dy dz;
in other words, the relation
(32)
.
A(X)
~!! xa12
1
= 6r2
(ff
JJJo
(P)a12
p
dx dy dz
holds.
Finally, we mention that the arguments of the two preceding subsections may be applied also to a more general domain consisting of a
finite number of rectangles or rectangular parallelepipeds.
4. Asymptotic Distribution of Eigenvalues for an Arbitrary Domain.
With the boundary condition u = 0, the same asymptotic formulas
can easily be proved for an arbitrary domain G which can be approximated from within and without by special regions consisting only of
a finite number of squares and differing arbitrarily little in area.
However, we shall aim to establish at once the asymptotic laws for
arbitrary G and all the boundary conditions under consideration; for
this purpose a somewhat deeper analysis is required.
First we consider a plane domain G whose boundary has continuous
curvature and restrict ourselves to the differential equation
A!lu
+ >..u = 0.
We examine briefly the eigenvalues associated with this differential
equation under the boundary condition au/ an = 0, investigating the
number of eigenvalues less than a given bound for a few simple
domains.
Let G be an isosceles right triangle with sides of length a. Every
eigenfunction for the triangle is also an eigenfunction for the square
which is obtained by reflection across the hypotenuse under the same
ASYMPTOTIC DISTRIBUTION OF EIGENVALUES
437
boundary condition iJu/i)n = 0. For, it is immediately clear that any
eigenfunction may be continued into the reflected triangle if the
functional value of each original point is assigned to its reflection;
in this way the boundary condition <Ju/ <Jn = 0 is satisfied along the
entire boundary of the square. Then-th eigenvalue for the triangle is
therefore also an eigenvalue for the square; hence the n-th eigenvalue
for the square is certainly no larger than that for the triangle. In
other words, for the triangle, with the boundary condition <Ju/<Jn = 0,
the number of eigenvalues less than a given bound is at most equal to the
corresponding number for the square, that is, the number given by
formula {26).
Next, let G be an arbitrary right triangle of sides a and b, where we
assume b :::; a. Let side a lie on the x-axis, side b on the y-axis. By
means of the transformation ~ = x, fl = ay /b we transform the
triangle G into an isosceles right triangle G' with sides equal to
a. The expression D['P] then becomes
[(iJ'P) + b~ (~)
a,,, ] ~a d~ d,,,
2
2
D['P] =
ff
JJa,
a~
2
and the auxiliary condition H['P] = 1 becomes
while the additional auxiJiary conditions H['P, vi] = 0 of §1, 4 retain
their form under .this transformation. Thus, if we omit the insignificant constant factor b/a occurring in both integrals, we can characterize then-th eigenvalue Kn for the triangle as the maximum-minimum
of the integral
taken over G', where the maximum-minimum is to be understood in
the usual sense. Since a/b ~ 1, the relation
holds, and the maximum-minimum of the left side is therefore at least
as large as that of the right side, i.e. it is at least as large as then-th
eigenvalue for the isosceles right triangle G', and thus· a fortiori
438
VI. APPLICATION OF THE CALCULUS OF VARIATIONS
larger than the n-th eigenvalue of a square of side a. Therefore, under
the boundary condition au/ an = 0, the number of eigenvalues less than a
given bound for a right triangle with sides not larger than a is certainly
no larger than the corresponding number of eigenvalues for the square
of side a; hence it is no larger than the corresponding number for any
larger square.
Similarly, the number of eigenvalues less than a given bound for an
arbitrary rectangle is never larger than the corresponding number for a
square whose side is at least equal to the larger side of the rectangle.
These results, together with Theorem 4, show that it is possible to
obtain an upper bound for the number of eigenvalues less than a
given bound whenever the domain in question is composed of a finite
number of rectangles and right triangles.
~~
~
/'
I
[\
\
'
I
(
\
'\.....
__
J
L)
Figure 4
We now consider how the distribution of the eigenvalues is affected
by the boundary strip which remains when we approximate the
domain G by means of squares. First we define the boundary strip:
Let the subdivision into squares be so fine that, for every segment of
the boundary of G which is contained in one of the squares, the direction of the normal varies by less than a given small angle.,,, whose
size will be determined later. (This can be accomplished by repeated
halving of the side of the square.) Then, as in Figure 4, we can associate with the boundary r a number r of adjacent elementary domains
E 1 , E 2 , • • • , Er in the following manner: Each domain Ei is bounded
either by two perpendicular line-segments AB, AC of the partition,
whose lengths lie between a and 3a, and a segment BC of the boundary
(Figure 5), or by a line-segment AB of the partition, two line-segments, AC, BD of lengths between a and 3a perpendicular to AB,
and a segment CD of the boundary (Figure 6). We construct a bound-
439
ASYMPTOTIC DISTRIBUTION OF EIGENVALUES
ary strip S consisting of r domains of this kind, such that when this
strip is removed from G a square-domain is left, consisting of h
1
squares Q1 , Q2 , • • • , Q,, . The number r is evidently smaller than a
constant C/a, where C is independent of a and depends essentially on
the length of the boundary.
In order to obtain an upper bound for the number BB,(.'X) of eigenvalues less than some bound>. of the differential equation i.lu Xu = 0
for the domain Ei with boundary condition au/ an = 0, we must again
try to find a lower bound for the n-th eigenvalues. To this end, we
take any point on the curvilinear boundary segment of Ei and draw
the tangent through it. This tangent together with the straight
boundary segments of Ei forms a domain of the type AB'C' (Fig. 5),
+
c
B
A
B
Figure 5
Figure 6
e.g. if 11 is sufficiently small it forms a right triangle with sides
smaller than 4a, or else a trapezoid of the type ABC'D' (Figure 6) the
sides AC', BD' of which are also smaller than 4a; the shape of the resulting domain depends on the type to which E i belongs. We shall
denote the domains AB'C' and ABC'D' by E~. The domain Ei can
always be deformed into the domain E~ by a transformation of the
form (21), as discussed in §2. In the case of domains of the first
1 We leave it to the reader to carry out this construction: Divide
the boundary curve into a finite number of arcs which are of three kinds.
On arcs of the first kind the tangents should form an angle of at most 30° with
the x-axis, on those of the second kind one of at most 30° with the y-axis; on
arcs of the third kind the tangents should not form angles with either axis of
less than 20°. The end points of arcs of the first and second kind should have
rational absciss.te and ordinates, respectively. If the subdivision into squares
(on whose sides these end points lie) is sufficiently fine the construction is
possible.
440
VI. APPLICATION OF THE CALCULUS OF VARIATIONS
type, let A be the pole of a system of polar coordinates p, fJ, and let
f(fJ) be the equation of the curved segment BC, p = g(fJ) the
equation of the line B'C'. Then the equations
p =
(}' = (},
p'
=
p g(fJ)
j(fJ)
represent a transformation of the curvilinear triangle Ei into the
rectilinear triangle E~. For a domain of the second type ABCD, let
AB lie along the x-axis, let y = g(x) be the equation of the linesegment C'D' and let y = f(x) be the equation of the curved segment
CD. We then consider the transformation
x' = x,
Y
I
=
y g(X)
f(x) •
If we assume that the fundamental interval a is sufficiently small, and
therefore that the total rotation of the tangent on the curved segment
CB or CD is taken sufficiently small, then the transformations considered here evidently have precisely the form (21), and the quantity
denoted bye in (21) is arbitrarily small. It follows from the corollary
to Theorem 10 that the corresponding n-th eigenvalues for the domains
Ei and E~ differ only by a factor which itself differs by a small amount
from 1, uniformly for all n. Hence the same is true also for the corresponding numbers B11.(A) and B11~(A) of eigenvalues less than the bound
X for the boundary condition au/an = 0.
The domain E~ is either a right triangle with sides smaller than 4a
or a combination of such a triangle and a rectangle with sides smaller
.than 3a; it follows that if a is taken sufficiently small, the number
BB,(A) from some X on satisfies the inequality
(33)
BB.(X)
<
2
c1a X
+ c2aVX
where c1 , c2 are numerical constants, to be chosen suitably.
We can now deduce the laws that govern the asymptotic distribution of eigenvalues for the domain G. Let A(X), for any of the
boundary conditions considered, again denote the number of eigenvalues less than a bound X of the differential equation Llu + Xu = 0
for the domain G, where again, if necessary, we make the as1;mmption
u ~ 0. Suppose the plane is partitioned into squares of side a, inducing
a decomposition of the domain G into h squares Qi, Q2, · · · , Qh and r
boundary domains E1, E 2 , • • • , Er. For the square Qi the number of
441
ASYMPTOTIC DISTRIBUTION OF EIGENVALUES
eigenvalues less than }.. is again denoted by Ai(A) for the boundary
condition tt = 0 and by Bi(A) for the condition iJtt/iJn = 0. The
corresponding numbers for the domains Ei are denoted by As,(A)
and Bs,(A), respectively (we shall use only the numbers Bs,(A)).
From equations (26) we have
Ai(:~.) =
2
:1r A +
a01c1vX,
and, by (33),
= 8a(ca'Aa2 + aco1VX),
B.,(A)
where, as always, 81 , 82 , 83 denote numbers between -1 and
and c1, C2, ca, C4 are constants independent of a, i, and X.
From Theorems 5, 2, and 4 we find
Ai(A)
+ A2(A) + · · · + Ah(A)
~
B1(A) + ...
+ 1,
~ A(X)
+ Bh(X) +
Bs1CA) + ... + B.r(A);
furthermore, we have
2
2
81c1ha)
(ha
_ h
ha
Ai(A ) +···+A,.(A ) =-A+81c1havA=A - + - - - ,
V-A
47r
47r
B1(A) + · · ·
+ B,.(A) +
Bs 1(A) + · · · + Bsr(A)
2
=
=
ha
4
2
... A+ 82c2haVX +Bara ACa + 8a rac4 VX
>. [(::'
+ 8,c,ra') + (ha8,c, + raDac.)
.Jxl
2
Now ar < c6 ; therefore, for sufficiently small a, a r is arbitrarily
small and we have
I ha2 - JI < o,
no matter how small
relation
o is.
From these inequalities the asyftlptotic
lim 47rA(A) = 1
Af
x....co
follows immediately. For, we may choose the quantity a arbitrarily,
and by taking a sufficiently small fixed a, make the factors of>. in
442
VI. APPLICATION OF THE CALCULUS OF VARIATIONS
the above inequalities arbitrarily close to the value f /47r for sufficiently large A.
Even without the assumption u ~ 0, we obtain the same asymptotic
law by means of the arguments used at an analogous point in subsection 2. Summing up, we have
THEOREM 16: Under any of the boundary conditions considered, the
number A (A) of eigenvalues less than a bound A of the differential
equation du + AU = 0 for the domain G is asymptotically equal to
Af/47r; in other words,
. 47rA(X) _ l
l im~f - '
x-ao
I\
(34)
where f denotes the area of the domain.
In the proof, we first made the assumption that the boundary r of
G had no corners. However, both the result and the argument remain
essentially unchanged if a finite number of vertices is admitted.
The preceding argument remains valid if we deal with the more
general equation L[u] + Apu = 0 instead of the differential equation
du + AU = 0. As in subsection 3, we obtain
THEOREM 17: Under any of the boundary conditions considered, the
number A (A) of eigenvalues less than a fixed A of the differential equation
L[u]
+
ApU = 0 for G is asymptotically equal to
in other words,
4~ Ji
~
dx dy;
lim A(X) = _!__ ([ !!_ dx dy.
X-ao
A
47r JJo p
Considerations similar to those given here for the plane lead to the
following result for the eigenvalue problem in space:
THEOREM 18: Under any of the boundary conditions considered, the
number A(A) of eigenvalues less than a fixed A of the equation
du + AU = 0 for a space-domain of volume V is asymptotically equal
to X312 V /67r2; in other words,
(35)
THEOREM 19: The corresponding number for the more general
differential equation L[u] + Apu = 0 'is asymptotically equal to
x3/2 rrr ( )3'2
~ JJJo ~
6
dx dy dz; in other words,
ASYMPTOTIC DISTRIBUTION OF EIGENVALUES
(36)
. A(X)
---sT2
fun
>.-co A
=
1
62
1r
1~1
G
(P)P
-
443
3 2
'
dx dy dz.
We assume here that G has a boundary consisting of a finite number
of surface elements with continuous curvature, which do not come in
contact with each other but may form edges and vertices.
5. Sharper Form of the Laws of Asymptotic Distribution of Eigenvalues for the Differential Equation A.u + J..u = o. The asymptotic
eigenvalue laws can be formulated more precisely by establishing an
estimate of the error which arises when the expression A(X) is replaced
by its asymptotic value; we restrict ourselves to the differential
equation .!lu + AU = 0.
We need only approximate the domain G by elementary domains
composed of squares or cubes in such a way that these domains are
no smaller and no more numerous than necessary. First, let G be a
domain in the plane. We construct a sequence of approximating
square-domains: We begin with a subdivision of the plane into.squares,
each, say, of side 1, and suppose that, of these, ho squares Q~ , Qg , · · · ,
Q~ 0 lie entirely in the interior of G. Now let each square be decomposed
into four congruent squares each of side!, and let h1 of these squares,
Q~ , Q~ , · · · , Ql 1 , lie in the interior of G but not in the interior of
any of the squar~s Q~. Continuing in this way, after t steps we obtain
ht squares Qt , Q: , · · · , Q~, , each of side 1/2', which lie in the interior
of G but in none of the preceding squares. In accordance with the
preceding subsection, we proceed in such a way that the boundary
strip remaining after the t-th approximation consists of r subdomains
E1 , E2 , · · · , Er of the type defined there; here the number a is equal
to 1/2'.
By our assumptions on the boundary, the numbers hi and r satisfy
relations
(37)
r
< 2'c,
where c is a constant independent of i and t, determined essentially
1
by the length of the boundary .
Again we denote by A!(A) and ABm(A) the number of eigenvalues
less than a bound X for the domains Q! and Em for the boundary
condition u = 0, and by B!(A) and BBm(A) the number for the bound1 These inequalities mean that the perimeters of the approximating squaredomains and those of the residual boundary strips are not of a higher order of
magnitude than the perimeter of G.
444
VI. APPLICATION OF THE CALCULUS OF VARIATIONS
ary condition au/ an = 0. If the function u in the boundary condition
+ uu = 0 is non-negative, we have by Theorems 2, 4 and 5
iJu/ an
A(A) ~cm+ B~
+ · ·· + B2 + ·· · + (Bt + B; + ·· · + B~,)
A(X) ~ (A~+ A~+
· · · + A2 + · · · + (A;+ A;+ · · · +Al,).
0)
(38)
0)
By (26) and (33) the right side of the first of these inequalities is
equal to
1(
47r ho
ht
+ 22h1 +. h224 + . . . + 22i
+ r6c)
22t A
+ 81 C2( ho + 2h1 + 2°2h2 + ··· + 2tht + °2'r)-V 1 -A;
because
ho+
hi+
22
h,
+ 22t
+ 22tr =
• ••
r
j - 82C222e
and (37) holds, this right side has the form
L.(f + c11~. ·} + o,c.(t + 2)Vi
0
and we obtain the inequality
(B~
+ · · · + B2 + ·· · + (BE + · · · + BBr)
0)
1
(39)
~
f71" A + C ("
2t
4
+ t, r)
A ,
which holds for sufficiently large t, where as always C denotes a constant independent of A and t.
We choose the number t, which is still at our disposal, in such a
way that the two terms inside the parentheses are as nearly equal as
possible, i.e. t = largest integer ~ log X/log 4. Then for sufficiently
large A we have from (38) and (39)
(40)
A(A)
~
LA + CyX log A.
Precisely the same form is found for the lower bound of the expression A(A), with negative C.
EIGENVALU E PROBLEMS OF SCHRODING ER TYPE
445
We assumed up to now that if the function u occurs in the boundary
condition it never becomes negative. However, the argument of subsection 2 shows that in view of inequality (20) of §2, 5 the bounds
retain their form even without this restriction. Thus we obtain in
general the sharper asymptoti c law:
THEOREM 20: Under any of the boundary conditions considered, the
difference A(X) - fX/47r is, for X ---+ oo, of order not greater than
VXlogX.
The same argument, carried out for three-dimensions, leads to
THEOREM 21: Under any of the boundary conditions considered for
the problem of a domain of volume Vin space, the difference A(X)
2
VX312/67r is, for x ---+ 00' of order not greater than h log x.
Eigenvalu e Problems of the SchrOdin ger Type
In Ch. V, §12 we considered SchrOdinger's eigenvalue problem for
an infinite fundamen tal domain and studied the properties of the
associated spectrum. We shall now approach this problem from the
standpoin t of the calculus of variations ; this approach is significant
although its results are incomplete. Not only in the SchrOdinger
example, but also in other eigenvalue problems for infinite domainsproblems which cannot be solved by the separation of variables -we
shall see that the spectrum contains a countably infinite sequence of
increasing discrete negative eigenvalues. (It also contains a "continuous spectrum. ")
:Let the eigenvalue equation be
§5.
(41)
.1u
+ Vu +Au = 0,
with the condition that the function u(x, y, z) remain finite at infinity.
We suppose that the function V(x, y, z)-the negative of the potential
energy-is positive throughou t the space and vanishes at infinity in
accordance with the following inequalities, valid for sufficiently larger:
(42)
A< V < ~
r"'
,.tJ'
where A and B are positive constants and the exponents satisfy the
relations
0 < {j ~a< 2.
1
In addition, we permit V to become infinite at the origin, of an order
1 The following discussion remains applicable as long as V becomes singular
in this manner at a finite number of points.
446
VI. APPLICATION OF THE CALCULUS OF VARIATIONS
no higher than C/ r 'Y, where 0 ~ 'Y
point (x, y, z) from the origin.
If
J··· dg
<
2; r denotes the distance of the
stands for integration over the whole x, y, z-space,
then with the usual notation the variational problem leading to the
eigenvalue An and the eigenfunction Un may be written as follows:
(43)
J[~] = Jc~!+ ~= + ~; - v ~2 )
dg = max. min.
. with the auxiliary conditions
J~ dg
2
(44)
1
=
J~v., dg
=
0
(v = 1,2,
···,n -
1).
Here ~(x, y, z), along with its first derivatives, is assumed to be continuous and square-integrable over the whole space. We assume also
that
JV~ dg
2
exists; as before,
v~ ,
v2 , · · · ,
Vn-1
denote piecewise
continuous functions.
We prove first that this variational problem has a meaning, or in
other words, that under the given conditions the integral J[~] is
bounded from below. In order to do this we need only note that
v~<;+b
r
holds everywhere; by choosing the positive constant b sufficiently
large we may take the positive constant a to be arbitrarily small.
Hence
JV l
(45)
J~ ~ dg + b J~ dg.
2
dg :::; a
2
We now apply the integral inequality
(46)
which is proved as follows: The substitution t/; = ~
2
~z
+ ~11 + ~z
2
2
=
r1 (i/lz + "111 + i/lz
2
2
2)
1
- T2#r
vr yields
1
+ 4r3"'
2
,
447
EIGENVALUE PROBLEMS OF SCHRODINGE R TYPE
and therefore
J
(<Pz2
+ <P + <Pz
2)
2
11
dg ~ -
J
1r (i/I 2) r dg
2 2
l
+ 41 J~Vt
2
d g.
The first term on the right may be integrated explicitly; since
J
2
<P
dg
1
exists by hypothesis, it has the value zero. This yields the desired
inequality. With its help we obtain from (45)
J[<P]
~
j (<P! + <P! + <P!) dg -
(I - 4a)
and if, as we may, we take a
b,
< l, we find
Jf<P] ~ -b,
which proves that J[<P], and hence also the eigenvalues of (41), are
bounded from below.
In order to obtain upper bounds for the eigenvalues, we strengthen
the admissibility conditions of our variational problem by requiring
in addition that <P vanish identically outside a sphere KR of radius R
with center at the origin. According to our general principles, the
n-th eigenvalue vn(R) of the resulting problem for the sphere KR
satisfies the inequality vn(R) ~ An . On the other hand, it may be
estimated easily in terms of the eigenvalue µn(R) of the differential
equation i:lu + µu = 0 for the sphere KR with vanishing boundary
values. In fact, from assumption (42) we have V ~A/Ra in KR (for
sufficiently large R) and
f
JKB
(<P!
+ <P! + <P! -
V <P2) dg
{ (<P! + <P! + <P!) dg s JKR
RAa { <P2 dg.
JKR
Hence it follows immediately that
Vn(R)
A
S µn(R) - Ra•
2
But µn(R) = µn(I)/R , where the µn(I) are the eigenvalues for the
Uilit sphere, and we obtain
"A
<
n-
µn(l) _ A.
Ra
R2
In fa.ct, there must exist a sequence of values R1, R2, · · · , Rn , · · · such
1
2
dB-extended over the surfaces of the spheres with
that the integralsRn
the radii Rn-approach zero as Rn becomes infinite. First we integrate over
these spheres and then pass to the infinite domain in the limit.
1
-/'P
448
VI. APPLICATION OF THE CALCULUS OF VARIATIONS
Since a < 2, for given n the right side is negative when R is taken
sufficiently large.
Thus we have proved that our variational problems yield a sequence
of monotonically nondecreasing negative eigenvalues.
To prove that these eigenvalues approach zero as n increases, we
may estimate them (see also page 449) in terms of the eigenvalues Kn
of the special Schrodinger problem for V = c/r, for which they are
known explicitly and for which Kn tends to zero according to Ch. V,
§12, 4. We need only remark that an inequality V ~ a/r2
b/r +k
holds, where for sufficiently large b the positive constants a and k
may be taken arbitrarily small. Thus, using relation (46), we evidently have
+
An
~
(1-4a)Kn - k
if we take c = b/(1 - 4a). It follows that as n increases the eigenvalue An at some time exceeds the value - 2k; hence it converges to
zero, since k may be taken arbitrarily small.
The occurrence of a continuous spectrum of positive eigenvalues
is plausible if we consider the eigenvalue problem for the infinite
domain as the limiting case of the eigenvalue problem for a finite
domain, e.g. for the sphere KR with increasing radius R. In fact, the
n-th eigenvalue Pn(R) decreases monotonically as R increases, and it
can be shown that it approaches the n-th eigenvalue An for the
infinite domain. Every positive number is actually the limit point
of eigenvalues Pn(R); for, in the case of a finite domain there are
arbitrarily large positive eigenvalues vn(R), and if we let n increase
with R in a suitable manner we can approximate any positive number.
We can prove that the eigenvalues have a limit point at zero; the
method is similar to that used to show that the eigenvalues for a
finite domain become infinite. No explicit knowledge of the solutions
of a special problem is needed.
If the eigenvalues have a fixed negative upper bound, we can construct a sequence of functions 'Pi, 'P2 , • • • , 'P•, · · · for which (a) the
integrals D['P] =
J
('P!
+ "°; + 'P!) dg and H['P]
than a fixed upper bound, (b) the integral F['P] =
=
J
J
2
'P dg remain less
V 'P2 dg always re-
mains greater than a fixed positive bound, and (c) the orthogonality
relations F["°", "°"] = 0 (v =I= µ) are satisfied. Using a lemma (to be
449
EIGENVALUE PROBLEMS OF SCHRODINGER TYPE
proved below), we see from property (a) that a subsequence cp.,,,.
can be chosen from the functions cp.,, with the property F[cp,,,. - cp.,,m] --+ 0
as n, m --+ oo • But since F[cp,,,. , cp,,,,.] = 0, this would imply the
relation F[cp,,,.] + F[cp,,m]--+ 0, which contradicts property (b).
The sequence of functions cp,, is constructed as follows: We begin
with the above variational problem (43), which gives the first eigenvalue ).1 • We can find a function cp1 for which the two relations
(E
> 0)
and
H[so1] = 1
hold. We now tum to the variational problem (43), (44), which
furnishes the second eigenvalue ).2 ; if we impose the auxiliary
condition
we obtain as the minimum (by the maximum-minimum property)
a value which is certainly no greater than A2. Thus it is possible to
find a function cp2 such that
D['P2] - F['P2) ~ A2
+
E
while at the same time
H['P2) = 1,
Continuing in this way we obtain a sequence of functions cp1 ,
cp,, , • · · for which
(µ
=
1, 2, ... '
'P2 , • • •
JI -
1).
Now if all the numbers ).,, were less than the bound - 2E, then all
these functions would satisfy the inequality
(47)
From this inequality it follows, in the first place, that D[cp,,) remains
bounded; for, by (45), (46) we have
F[cp]
~
4aD[cp)
+ bH[cp]
450
VI. APPLICATION OF THE CALCULUS OF VARIATIONS
and therefore
(1-4a)D[~ ..]
:s;
b.
On the other hand, (47) implies that F[p11] ~ E. This proves that the
actually have the desired properties.
It remains to prove the lemma mentioned above: Given a sequence
of functions~ .. for which D[~J and H[~J are bounded, we can find a
subsequence ~ ..n such that the relation
functions~ ..
(n, m---> oo)
holds.
This theorem is a generalization of the lemma of Rellich mentioned
earlier (§2, 2) which made it possible to prove that the eigenvalues
for a finite domain become infinite. We now restrict ourselves to the
case in which the function V is regular at the origin. (If V is singular
at the origin of degree less than two, then similar estimates make it
possible to obtain the desired result.)
To prove the lemma, we exclude infinity by means of a sequence of
spheres Ki of radii Ri. In virtue of the earlier lemma (page 414) we
can select a subsequence ~In from the functions ~" for which
F[~In - ~Im] approaches zero provided that the integral is taken only
over the interior of the sphere K 1 • From this sequence ~1n we can
again select a subsequence ~2n for which the integral F[~n - ~2m]
taken over the sphere K2 approaches zero. We continue in this
manner obtaining sequences ~in for each i, and form the diagonal
sequence ~nn , which we shall denote by ~".. . We know that for this
sequence the integral F[~".. - ~"m], taken over any one of the spheres
Ki, approaches zero. In order to show that the same is true when
the integral is extended over the whole space, we have only to show
that the integral over the exterior of the sphere Ki is always less
than some bound independent of n and m which approaches zero as R
increases without bound. To do this, we remark that for sufficiently
large Rand for r ~ R we have assumed (42): V ~ B/I :s; B/Rf3,
and that therefore the integral taken over the exterior of the sphere
of radius R satisfies the inequality
B
4B
F[~11., - ~"m] :::; RB H[~lln - ~Jim] :::; RB '
This proves our assertion.
NODES OF EIGENFUNCTIONS
451
§6. Nodes of Eigenfunctions
It was possible to make precise statements with regard to the
general behavior of the eigenvalues; however the investigation of the
general properties of the eigenfunctions offers greater difficulties.
This is not surprising, in view of the large number of classes of
functions defined by means of eigenvalue problems. A few special
cases will be studied in the following chapter, while in the present
section we shall be concerned with a more general investigation of
eigenfunctions.
The nodes, i.e. those points of the fundamental domain G at which
some eigenfunction vanishes, are of particular interest (cf. Ch. V, §5).
In dealing with problems in one, two, three, etc. dimensions, we
speak of nodal, points, nodal curves, nodal surfaces, respectively; in
1
general we use the term nodes.
We remark that the first eigenfunction of an eigenvalue problem
can have no nodes in the interior of the fundamental domain (the
proof of this will follow directly from the theorem given below).
It must therefore have the same sign everywhere, and hence every
other eigenfunction orthogonal to it must have nodes.
It is possible to m_ake a number of general statements concerning
the position and density of the nodes. For example, consider the
AU = 0 with the boundary condition
differential equation Au
lies entirely in G and contains no
which
domain
a
is
G'
If
0.
=
u
nodal points of Un , we consider the smallest subdomain G" of G
containing G' and bounded by nodes of the function Un • For this
domain G" the function Un must be the first eigenfunction, An the
smallest eigenvalue. On the other hand, according to Theorem 3
the first eigenvaJue of G" cannot be greater than the first eigenvalue
'Y of G', and thus 'Y ~ An . For example, if G' is a circle of radius a
then 'Y = T2, where Tis the smallest root of the equation Jo(llT) = 0.
2
We thus have 'Y = k~.i/a where, as in Ch. V, §5, 5, we denote by
ko.1 the first zero of the zero-th Bessel function. We therefore obtain
2
a S k~.i/An; this relation tells us as much about the density of the
ne.t of nodal lines as can in general be expected. If we recall the
asymptotic relation An ~ 47rn/f of §4, we see that if n is sufficiently
Jarge every circle whose area is greater than k~.if/4n must contain
+
1 We postulate that for the differential equation under consideration the
nodes are piecewise smooth curves or surfaces and decompose the fundamental
domain into subdomains with piecewise ·smooth boundaries.
452
VI. APPLICATION OF THE CALCULUS OF VARIATIONS
nodal lines of the n.,.th eigenfunction. If instead of a circle we
consider a square of side a, we find correspondingly a 2 ::; 27r2/An •
The reader will be able to derive entirely analogous statements for
other problems in one or several variables.
In addition, it is possible to prove the following general theorem
concerning the nodes of an eigenfunction: Given the self-adjoint second
order' differential equation L[u]
"Apu = 0 (p > 0) for a domain G with
+
arbitrary homogeneous boundary conditions; if its eigenfunctions are
ordered according t,o increasing eigenvalues, then the nodes of then-th
eigenfunction Un divide the domain into no more than n subdomains.
No assumptions are made about the number of independent variables. 1
For simplicity we suppose G is a domain in the x, y-plane and the
boundary condition is u = 0. Let An be the n-th eigenvalue, i.e. the
maximum-minimum of the associated integral D[<P] under the prescribed boundary condition and the auxiliary conditions
Jl pql
fl
(48)
(49)
dx dy = 1,
P<PVi dx dy =
(i
0
=
1, 2, · · ·, n - 1).
We suppose that the nodes of the associated eigenfunction Un decompose the domain G into more than n subdomains G1, G2, · · · ,
Gn+1 , · · · , and define n functions W1 , W2 , • • • , Wn , such that Wi
coincides up to a normalizing factor with Un in the subdomain Gi and
vanishes outside Gi, and such that
fl pw~
dx dy = 1.
+
If we form a linear combination <P = C1W1
which itself satisfies the normalizing condition
fl Pl
dx dy =
C2W2
c~ + c~ + · ·· + c!
=
+ ··· + CnWn,
I,
we see at once, integrating by parts, that it satisfies the equation
D[<P]
= 'An,
1 See Courant, Ein allgemeiner Satz
zur Theorie der Eigenfunktionen
sel bstadj ungierter Differen tialausdriicke.
453
NODES OF EIGENFUNCTIONS
since Wi satisfies L[wi] + AnPWi = 0. Now since for any given functions vi the coefficients Ci can be determined in such a way that <p fulfills conditions (49) in addition to (48), then-th eigenvalue A: for the
domain G' = G1
G2
Gn of the same differential equation
with the boundary condition u = 0 can be no larger than An ; it is
precisely equal to An, since by Theorem 2, §2, 1, it also cannot be less
than An . From this it follows by Theorem 3 that for every subdomain G" of G which contains G' the n-th eigenvalue is precisely
equal to An . Let u <~>, u <;>, • · • , u <r:> be the eigenfunctions obtained
in this way, for an arbitrary number m of such domains G', G",
G"', · · · , o<m> each containing the preceding domain. If these
eigenfunctions are continued into the exterior of the corresponding
subdomain of G by the requirement that they vanish there identically,
then they form a system of m linearly independent1 functions which
are all solutions in G ofthedifferentialeqtiation L[u<~>]
AnPU <~> = 0.
We can determine a linear combination
+
+ · ·· +
+
<p
=
"Yiu<!>
+ · ·· + "Ymu<r;:>
with coefficients 'Yi , which do not vanish everywhere, in such a way
that the m - 1 conditions
Ji
p<pVi
dx dy
= 0
(i
=
1 2 · · · m - 1)
' '
'
are satisfied; since <p cannot vanish identically because of the linear
independence of the u<~>, we can normalize it according to (48) by
multiplying by a suitable factor. But then, because of the maximumminimum property of them-th eigenfunction, we must have
On the other hand, integrating by parts, we find
However, since limk-oo Ak = oo we have Am > An, for sufficiently
large m. We thus obtain a contradiction, which demonstrates the
1 It is immediately obvious that these functions
are linearly independent if
one remembers that u<!> cannot be identically zero in any subdomain of G<h>.
This fact, which in the case of ordinary differential equations follows from the
uniqueness theorem, is for partial differential equations a consequence of their
-elliptic character, to which we shall return in Volume II.
454
VI. APPLICATION OF THE CALCULUS OF VARIATIONS
impossibility of the above assumption of more than n domains G1 •
G2 , · · · . It need hardly be said that this proof is valid for any
number of variables. i
For the special case of the Sturm-Liouville eigenvalue problem
(py')' - qy + Xpy = 0, the general theorem just proved may be
strengthened in a remarkable way. Here we can show in addition
that the n-th eigenfunction divides the fundamental domain into no
less than n parts; thus we have the theorem: The n-th eigenfunction
for a Sturm-liiouville problem divides the fundamental domain into
precisely n parts by means of its nodal points. This is usually proved
by continuity considerations which we now sketch.
For simplicity we restrict ourselves to the differential equation
y"
Apy = 0. By y(x, "A) we denote a solution of this equation which
depends continuously on the parameter A and vanishes at x = 0;
we obtain the identity
+
y(x, Xi)y' (x, X) - y(x, X)y' (x, Xi) = (Xi - X)
If x =
~is
L" py(x, X)y(x, X1) dx.
a positive zero of y(x, "'A), it follows that
y((, Xi)y' (~, X) '== (Xi - X)
f
py(x, X)y(x, X1) dx.
Now let "'A.1 be greater than A and so close to A that the integral on the
right remains positive. Then y(~, "'A.1) and y'(~, "'A) must have the same
sign. If we assume that at x = ~ the f1Jnction y(x, "'A) changes from
negative to positive values and thus that y'(~, "'A) is positive-y'(~, "'A)
and y(~, "'A) cannot both vanish at the same time-then y(~, "'A.1) is
also positive. Since y(x, "'A.1) differs arbitrarily little from y(x, "A) for
sufficiently small "'A.1-A and must, therefore, pass from negative to
positive values in the neighborhood of x = ~' a zero of y(x, Xi) Jies at
the left2 of~ and we can state: As A increases continuously, all the zeros
of the function y(x, "'A) decrease. The first eigenfunction has zeros onJy
at the two ends of the fundamental domain. As A passes from the
first eigenvalue to the second eigenvalue, the second zero moves from
the right into the interior of the interval, continuing until the end
1 The theorem just proved may be generalized
as follows: Any linear combination of the first n eigenfunctions divides the domain, by means of its nodes,
into no more than n subdomains. See the Gottingen dissertation of H. Herrmann, Beitrage zur Theorie der Eigenwerte und Eigenfunktionen, 1932.
2 The fact that y and y' can never vanish at the same
point implies thnt
zeros neither appear nor disappear between 0 and I; as X increases.
455
SUPPLEMENTARY REMARKS AND PROBLEMS
point of the interval becomes a third zero of the function, etc., and
1
thus the theorem is evident.
We were able to prove this fact because we were dealing with an
ordinary differential equation. In the case of eigenvalue problems of
partial differential equations, arbitrarily large values of n may exist
for which the nodes of the eigenfunctio ns Un subdivide the entire
fundamental domain into only two subdomains. Simple examples of
this2 are offered by the equation "1u + 'X.u = 0 for a square: 0 s x s 7r,
In ~his case, it is easily seen that the eigenfunctio ns
7r.
y
0
µ. sin x sin 2ry belonging to the eigenvalues X =
y
sin
sin 2rx
is a positive constant sufficiently close to 1) possess
(whereµ.
1
4r2
only a single nodal curve. Figures 7 and 8 show how these nodal
curves arise from slight variation of a system of curves for the case
r = 12.
§7. Supplemen tary Remarks and Problems
1. Minimizing Properties of Eigenvalues. Derivation from Completeness. The completenes s of the eigenfunctio ns obtained in our variational problems has been used to prove that they constitute the
s s
+
+
1 We can avoid continuity methods by basing the proof on the following
. theorem, which is not limited to one independent variable: Let u be a solution
of L[u] + Xpu = 0 which is twice continuously differentiable in a closed region
B. If u vanishes on the boundary r of B without changing sign in the interior,
p.pv = 0 where p. > X, then v must change sign
and if v is a solution of L[v]
in B. (Naturally we exclude the case in which u or v vanishes identically in
B.) We prove this at once by concluding with the aid of Green's formulasay for the case of two independent variables-th at
+
fl
(vL[u] - uL[v]) d:r; dy
=
(µ - X)
Ji
puv d:r; dy
=
£~
v
ds,
where o/iJn denotes differentiation with respect to the outer normal. Without
loss of generality we may assume that u and v take positive values in B; since
we have au/on ~ 0 on r, the expression on the right in the above equation is
not positive, while the middle expression would have to be positive if v did not
change sign in B.
If we apply this result to the Sturm-Liouvi lle problem with vanishing boundary values, we see that of two eigenfunction s, the one having the larger number
of zeros must belong to the larger eigenvalue. For, an interval between two
suitably chosen zeros of the eigenfunction with fewer zeros must contain as
a proper subinterval such an interval determined by two zeros of the other
eigenfunction . Since the first eigenfunction has no zeros in the interior, the
n-th (which can have no more than n - 1) must accordingly haven - 1 zeros
in the fundamental domain.
2 See A. Stern, Bemerkungen iiber asymptotische s Verhalten von Eigenwerten und Eigenfunktio nen, Dissertation, GOttingen, 1925.
456
VI. APPLICATION OF THE CALCULUS OF VARIATIONS
totality of solutions of the corresponding differential equations. Conversely, we may (e.g. in the case of the trigonometric and Legendre
functions) start with a complete system of functions which are
solutions of the eigenvalue problem for a differential equation. We
shall show that such systems of functions coincide with the systems
defined by means of the extremal properties: Suppose we are dealing
with the differential equation
L[u]
+ Xpu
= 0
for the two-dimensional domain G and the boundary condition
u = 0. Let the eigenfunctions of the differential equation problem
be u1 , u2 , · · · and the associated eigenvalues ">..1 , X2 , · · · . Consider
Figure 8
Figure 7
any function "' with continuous first derivatives and piecewise continuous second derivatives in G which vanishes on the boundary r and
satisfies the conditions
(50)
(51)
fl p</
fl
dx dy = 1,
P'fJ'Ui
dx dy = 0
We now show that 'P satisfies the inequality
D['P]
~
'Xn.
(i = 1, 2, · · ·, n - 1).
SUPPLEMENTARY REMARKS AND PROBLEMS
457
For, because of the boundary condition <p = 0, Green's formula yields
f£cpL[<p) dx dy;
D[<p] = -
in addition, the completeness relation (cf. formula (23a), page
applied to the functions <p and L[<p]/p, leads to the relation
426)~
(52)
where
'Yi
=
Jl
pcpUi dx dy.
It follows from (52), with the aid of Green's formula and the relation
L[ui] = -A,pui, that
00
(53)
D[<p] =
L Ai 'Y~ •
i-1
Now, since (51) implies
for i = 1, 2, · · · , n - 1,
'Yi = 0
and since from (50), together with the completeness relation, we
have
00
L: 'Y~ =
i-1
1,
it follows immediately that if the Ai are ordered according to increasing
.
magnitude, we have
Moreover, as shown earlier, by a simple calculation we obtain
D[un] =An,
which means we have proved the minimizing property of the n-th
eigenfunction with respect to the function <p. The same considerations may be applied if we assume only that the functions <p are continuous and have piecewise continuous first derivatives; for, a function
of this kind, together with its derivative, may always be approximated
by a function of the class defined above in such a way that the cor-
458
VI. APPLICATION OF THE CALCULUS OF VARIATIONS
responding integrals D[cp] differ by an arbitrarily small amount (in
this connection, see the remarks in Ch. IV, §3, 7).
2. Characteriz.ation of the First Eigenfunction by Absence of Nodes.
The first eigenfunction has been characterized by the property that
it does not vanish. We shall now use an important method introduced by Jacobi to investigate this property (method of multiplicative variation).
We limit ourselves to the equation
Au - qu
+ Xu
=
0.
We wish to prove: If there exists a solution u of this equation which
vanishes on the boundary r of a domain G but nowhere in the interior,
then
'.l)[cp] =
fl
(cp!
+ cp; + qcp
2
)
~ >.
dx dy
fl
2
cp dx dy
for all admissible functions cp, where the equality holds only for
cp = const X u. We suppose every such function cp to be represented
in the form
cp = U11,
which is possible, since u does not vanish in G; we have
'.l)[cp] =
fl
2
[u (11!
+ 11;) +
+
2uUx1111z
2UUu111111
+ (u! + u;)r,2 + quV] dx dy.
(112 ) 11 and integrating by parts,
By observing 21111z = ( r,2)z , 2111111
we obtain
'.l)[cp] =
fl [u (11! + 11!) 2
uAu11
2
+ quVJ
since all the integrals over the boundary vanish.
differential equation for u, we find
'.l)[cp]
=.Ji [u
2
(11!
dx dy
Makillg use of the
+ 11!) + >.uVJ dx dy
~
>.
fl
uV dx dy = >.
fl
2
cp dx dy,
SUPPLEMENTARY REMARKS AND PROBLEMS
459
const.,
where the equality holds only for 11:11 = 1111 = 0, i.e. for 11
q.e.d.
3. Further Minimizing Properties of Eigenvalues. The reader may
prove the following theorem: The problem of minimizing the integral
expression
D[v1 ,
V2 , • • • · , Vn]
= '.tl[v1]
+ '.tl[v2] + · ·· + '.D[vn],
in which we admit all systems of n mutually orthogonal normalized
functions with piecewise continuous derivatives in the fundamental
domain G, is solved by the functions Vi = ui or by any system of functions which arise from these functions through an orthogonal transformation. Here the functions u1 , u 2 , • • • , Un stand for the first n
eigenfunctions for the domain. The minimum of D[v1 , v2 , · · · , vnJ
An .
A2
is equal to the sum of the first n eigenvalues, A1
The following theorem can also be proved: Let v1 , V2 , • • • , Vn-1 be
continuous functions in G, and let d { v1 , v2 , · · · , v,._1} be the lower
bound of the integral expression '.tl[cp] where, in addition to satisfying
the usual continuity conditions, cp is subject to the single auxiliary
condition
+ + ·· · +
ff.
~l (JI. p<f'V; dx dy)'
pq} dx dy -
1.
Then then-th eigenvalue An equals the maximum of d {V1, V2 , • • • , Vn-l}
and is assumed for v1 = u 1 , V2 = U2 , · · · , v,._1 = u,._1 ; cp = Un •
This formulation is interesting since it uses only one quadratic
auxiliary condition and does not require linear conditions; however,
the auxiliary condition has a form which is unusual for isoperimetric
problems. It wiJl be left as a problem for the reader to carry over
this formulation to the corresponding elementary .problem for quadratic forms.
Other formulations of the eigenvalue problems, useful in many
applications, will be given in connection with the differential equation
du + AU = 0 with the boundary condition u = 0:
H[cp] =
Jiu cp dx dy =
2
min. max.
subject to the auxiliary conditions
D[cp] =
Ji
(cp!
+
cp!) dx dy = 1,
D[cp, vi] = 0
(i = 1, 2, · ·., n - 1),
460
VI. APPLICATION OF THE CALCULUS OF VARIATIONS
from which one sees immediately the significance of the formulation
as a maximum-minimum problem.
Another equivalent problem is that of making the expression
a maximum-minimum under the same auxiliary conditions, where we
again require the functions 'P to have continuous first and piecewise
continuous second derivatives.
4. Asymptotic Distribution of Eigenvalues. (a) For the differential
equation AAu - AU = 0 of the vibrating plate, with the b~undary
conditions u = 0 and au/an = 0 (clamped plate), we have the
asymptotic relation
from which
follows. Here, a~ earlier, A (A) is the number of eigenvalues less
than the bound A, An denotes the n-th eigenvalue, and f the area of
the plate. We may state, therefore: The n-th eigenvalue of the
clamped plate, as n increases, is asymptotically equal to the square of
the n-th eigenvalue of the clamped membrane. In particular, it
depends only on the size and not on the shape of the plate. An
analogous statement holds in three dimensions. 1
(b) Derive the laws of asymptotic distribution of eigenvalues for
the Sturm-Liouville equation (see the results of §2, 3) and also for
ordinary differential equations of fourth order, using the methods
of §4, 3.
(c) Derive the laws of the asymptotic distribution of eigenvalues
for the case of elliptic self-adjoint differential equations arising from
an arbitrary definite quadratic variational problem.
5. Parameter Eigenvalue Problems. Carry out the solution of twoparameter eigenvalue problems (see the Lame problem in Ch. V,
§9, 3) using methods of the calculus of variations.
1
See Courant, tJber die Schwingungen eingespannter Platten.
SUPPLEMENTARY REMARKS AND PROBLEMS
461
Eigenvalue probcondition (cf. Ch. V,
aid of methods of the
equation Au = 0 and
to minimize an integral
6. Boundary Conditions containing Parameters.
lems with the parameter in the boundary
§16, 4) may also be solved easily with the
calculus of variations. For the differential
the boundary condition au/ an = Xu, we have
of the form
Ji
(cp!
+ cp;) ax ay
where the integral of cp2 over the boundary satisfies the condition
£
2
cp ds
= 1
and suitable linear auxiliary conditions are imposed. Further development of this idea is left to the reader.
If G is the unit circle, the solutions of this problem are given by
the potential functions rn cos nfJ, rn sin nfJ; the eigenvalues are An = n.
In the general case, it is easily seen with the aid of the methods
developed in the present chapter that An is of the order of n. It
therefore follows from §3, 1 that the eigenfunctions are complete with
respect to the expression S)[cp] =
£
2
cp ds.
In other words, the bound-
ary values of the eigenfunctions considered as functions of s form a
complete system from which, in turn, it may be concluded that
every potential function which is regular in G may be approximated in
the mean by its eigenfunctions.
7. Eigenvalue Problems for Closed Surfaces. The eigenvalue problem of the Laplace spherical harmonics is an example of a problem
on a closed surface, for which the condition of regularity over the
entire surface replaces boundary conditions. The methods of the
present chapter show that these eigenvalue problems are closely
related to minimum or tQ..maximum-minimum problems for a quotient
'.D/S); '.D is a quadratic expression formed with the derivatives of cp
and S)[cp] is a positive definite quadratic expression which does not
contain derivatives and has the closed surface as the domain of
integration. This theory may be carried over to other quadratic
differential expressions on closed surfaces.
8. Estimates of Eigenvalues when Singular Points Occur. In §2, 4
\Ve treated singular points in connection with Bessel eigenvalue
462
VI. APPLICATION OF THE CALCULUS OF VARIATIONS
problems; Bessel functions of zero-th order required special treatment using particular properties of the Bessel functions. This will
no longer be necessary because we shall now develop a more general
method.
We consider the eigenvalue problem associated with the expressions
with no boundary conditions at x = 0, and with the boundary condition cp(l) = 0 at x = 1. After introducing Vx cp as new dependent
variable, we find without difficulty the estimate An ~ n 27r 2 for the
n-th eigenvalueAn of our problem. Thus for the number A(X) of eigenvalues less than A we have A(X) ~ y'X/7r.
In order to estimate An from below, and thus find an upper bound
for A(X)-this is our specific objective here-we choose an arbitrarily
small positive number E between 0 and 1 and note that A (A) ~
B1(X)
B2(X), where B1 and B2 denote the number of eigenvalues less
than A of the expressions
+
E
D1
=
1 xcp
12
dx,
0
and
1
D2
=
1
E
12
Xcp dx,
respectively. Here we no longer require that the function cp be
continuous at the point x = E, so that in both cases the point x = E
occurs as a free end point. Using the methods developed in this
chapter we find the asymptotic relation B2(X)/y'A ~ (1 - E)/7r for
B2(X); it remains to estimate B1(X). We majorize H1 by Hi and
minorize D1 by Di:
Using an obvious notation, we have B1(X) < Bi(X). On the other
hand, we can write down the eigenfunctions and eigenvalues of this
new eigenvalue problem explicitly. To do this, we map the interval
0 ~ x ~ E onto the interval -1 ~ ~ ~ 1 by the transformation
463
SUPPLEMENT ARY REMARKS AND PROBLEMS
+
~)e/2. For the eigenfunctions we obtain the Legendre
(1
2
1) / ~ •
polynomials. in ~, and for the eigenvalues the numbers n(n
~)VA, where ~ approaches zero with
e(l
B{(">.)
Hence B 1 (">.)
we obtain almost immediately
results,
our
up
increasing ">.. Summing
the asymptotic relation
x =
s
s
+
+
1
. A(A)
hm_h"= r
>.-+ao v A
since E could be chosen arbitrarily small.
9. Minim.um Theorems for the Membrane and Plate. Of all clamped
membranes or plates with given perimeter or area and given
constant density and elasticity, those with the circle as boundary
have the lowest fundamental tone. (For proofs, see the first paper
cited below for the case of a given constant perimeter, and the works
2
of G. Faber1 and E. Krahn for the case of a given constant area.)
10. Minimum Problems for Variable Mass Distribution. The reader
may· prove the following theorems:
The fundamenta l tone of a stretched string of given uniform tension
along which a given mass has been distributed is lowest when the
entire mass is concentrated at the midpoint.
Prove the analogous results for the membrane and plate.
11. Nodal Points for the Sturm-Liouville Problem. Maximumminimum Principle. The theorem of §6, which states that the n-th
eigenfunction divides the fundamental domain into n parts by means
of its zeros, can also be proved on the basis of the following considerations. 3 Given a string capable of vibrating, if n - 1 arbitrarily
chosen inner points are fixed, the fundamenta l tone of the resulting
system consisting of n independent strings is identical with the
lowest of the fundamenta l tones of the subsystems (see §1, 3). The
fundamental vibration of the decomposed system is then the fundamental vibration of the subsystem in question while the remaining
subsystems are constrained to the rest position. If we vary the
prescribed nodal points, the fundamenta l tone of the decomposed
1 G. Faber, Beweis, dass unter allen homogenen Membranen von gleicher
Flii.che und gleicher Spa.nnung die kreisformige den tiefsten Grundton gibt,
8.-Ber. Bayr. Akad. Wiss. (math.-phys. Kl.), 1923, pp. 169-172.
2 E. Krahn, tiber eine von Rayleigh formulierte Minimaleigen scha.ft des
Kreises, Ma.th. Ann., Vol. 94, 1925, pp. 97-100.
3 See K. Hohenemser, Praktische Wege zur angenii.herten Schwingungsberechnung ela.stischer Systeme, Ingenieur-Arc hiv, Vol. 1, No. 3, 1930, pp. 1-24.
464
VI. APPLICATION OF THE CALCULUS OF VARIATIONS
system reaches a maximum when then subsystems all have the same
fundamental tone. For, if two neighboring subsystems had different
fundamental tones, we could raise the fundamental tone of. one and
lower that of the other by moving the nodal point which they have
in common so that both subsystems would have equally high fundamental tones. Now, in the extremal case considered, the fundamental vibration of the decomposed system may be represented by a
continuously differentiable function which represents an eigenfunction
of the original unrestrained system associated with the vibration
frequency in question and vanishing at then - 1 points corresponding
to the maximum fundamental tone. Hence: If a string is fixed at
n - 1 points, and we wish to choose these points in such a way that the
fundamental tone of the resulting system is as high as possible, we find
that the solution is given by an eigenfunction of the original system having
n - 1 zeros at inner points. If we call the eigenvalues thus obtained
µn and the associated eigenfunctions Vn, then µn+i ~ µn since there
exists an interval, determined by two suitable neighboring zeros of
Vn , which contains the interval between two zeros of Vn+i as a proper
subinterval, and since a reduction in the length of the interval
produces a rise in the fundamental tone. (See page 455, footnote).
If, as before, we denote the eigenva]ues of the string, ordered with
respect to increasing magnitude, by An, we see that we always have
µn ~ An since the µn must certainly be contained among the An . On
the other hand, the restriction of a prescribed nodal point is only a
special or limiting case of a linear auxiliary condition such as we have
considered in §1, 4 for the variational problems defining the eigenvalues An. If we limit ourselves to such special auxiliary conditions
the maximum-minimum, i.e. the number µn , cannot be greater than
the maximum of the minimum when arbitrary linear auxiliary conditions are admitted, i.e. not greater than An • Hence µn ~ An , and
therefore, in view of the above result, we have µn = An. This completes the proof of the theorem on the zeros of Sturm-Liouville
eigenfunctions.
References
Courant, R., Beweis des Satzes, dass von allen homogenen Membranen
gegebenem Umfanges und gegebener Spannung die kreisformige den tiefsten
Grundton besitzt. Matq. Zeitschr., Vol. 1, 1918, pp. 321-328.
- - , Uber die Eigenwerte bei den Differentialgleichungen der mathematischen
Physik. Ibid., Vol. 7, 1920, pp. 1-57.
REFERENCES
465
- - , Uber die Schwingung en eigenspann ter Platten. Ibid., Vol. 15, 1922, pp.
195-200.
- - , Ein allgemeiner Satz zur Theorie der Eigenfunkt ionen selbstadjun gierter
Differential ausdriicke. Nachr. Ges. Gottingen (math.-phys . Kl.), 1923,
Session of July 13.
-·-,Uber die Anwendung der Variationsr echnung in der Theorie der Eigenschwingung en und iiber neue Klassen von Funktionalg leichungen. Acta
Math., Vol. 49, 1926, pp. 1-68.
Kneser, A., Die Integralglei chungen und ihre Anwendung en in der mathematischen Physik. 2nd ed., F. Vieweg und Sohn, Braunschwe ig, 1922.
Liouville, J., Memoire sur le developpem ent des fonctions ou parties de fonctions en series dont les divers termes sont assujetis a satisfaire a une meme
equation differentiell e du second ordre contenant un parametre variable. J.
de math. pures et appl., Ser. 1, Vol. 1, 1836, pp. 253-265. Ibid., Vol. 2, 1837,
pp. 16-35, 418-436.
Weyl, H., Das asymptotisc he Verteilungs gesetz der Eigenwerte linearer
partieller Diff erentialglei chungen (mit einer Anwendung auf die Theorie
der Hohlraums trahlung). Math. Ann., Vol. 71, 1912, pp. 441-479.
- - , Uber die Abhii.ngigkeit der Eigenschwi ngungen einer Membran von
deren Begrenzung . Journ. f. d. reine u. angew. Math., Vol. 141, 1912, pp.
1-11.
- - , Uber das Spektrum der Hohlraumst rahlung. Ibid., pp. 163-181.
- - , Uber die Randwertau fgabe der Strahlungst heorie und asymptotisc he
· Spektralges etze. Ibid., Vol. 143, 1913, pp. 177-202.
Richardson , R. G.D., Das Jacobische Kriterium der Variationsr echnung und
die Oszillationseigenscha.ften linearer Differential gleichungen zweiter Ordnung. First communica tion, Math. Ann., Vol. 68, 1910, p. 279. Second
communica tion, Ibid., Vol. 71, 1911-12, p. 214.
- - , Uber die notwendige n und hinreichend en Bedingunge n f iir das Bestehen
eines Kleinschen Oszillations theorems. Ibid., Vol. 73, 1918, p. 289.
In an unpublished manuscript , submitted in 1914 to the editors of Mathematische Annalen in the form of an abstract, Richardson treated the eigenvalue
problem of elliptic differential equations with results similar to some of those
of the present chapter. In particular, Richardson 's paper dealt with the
behavior of the eigenvalues and the coefficients in the boundary condition
with respect to the growth of the domain and their independen ce of the coefficients of the differential equation; he also treated theorems concerning the
zeros of the eigenfunctio ns.
CHAPTER VII
Special Functions Defined by Eigenvalue
Problems
§1. Preliminary Discussion of Linear Second Order Differential Equations
In the present chapter we shall investigate certain classes of
functions which have already been defined, Bessel functions, Legendre
functions, and general Laplace spherical harmonics. Our point of
view will be somewhat more general than in the preceding chapters;
we shall permit the independent variable to assume complex values
and use the methods of function theory. Furthermore, we shall
consider not only the functions mentioned but also the totality of
solutions of the corresponding differential equations. We assume
it to be known that: linear differential equations of this type, even
in the case of a complex independent variable z = x + iy, possess two
linearly independent solutions, the general solution is a linear combination of these solutions with constant coefficients, and all solutions
are regular analytic functions of z except at certain fixed singular
points determined by the coefficients. Many important classes of
functions can be defined as solutions of such linear differential
equations with analytic coefficients.
To obtain solutions of a linear differential equation
L[u]
+ µu
=
0
in the form of an .integral representation, the method of integral transformation is often useful; we start with a general outline of this
method. In place of the unknown function u(z) we introduce a new
unknown function v(t) of the complex variable r = ~
i11 by means
of equation
+
(1)
u(z) =
l
K(z, f)v(f) df,
where the transformation kernel K(z, t) (assumed analytic in each
466
PRELIMINA RY DISCUSSION
of the complex variables) and the path of integration C are to be
suitably determined. The differential equation takes the form
L
(L[K]
+ µ.K)v(t) dt
= O;
here the differentiation process L refers to the variable z and it is
assumed that the process L is interchangeable with integration .
Now we specify K by subjecting K(z, t) to a partial differential
equation
L[K] = A[K]
where A[K] is a linear differential expression containing differentiation
only with respect to the variable r. Integratin g by parts we eliminate these derivatives. The above integral takes the form
l
K(z, t)(B[vl
+ µv) dt;
here B[v] denotes the differential expression adjoint to A[v] (see Ch.
V, §1). In addition to this integral, a term depending on the boundary occurs, which can be made to vanish by a suitable choice of the
·path of integration. If the partial differential equation for K, which
can be chosen in various ways, as well as the transforme d equation
B[vl
+ µ.v = 0
can be solved explicitly in such a way that the above assumptions
hold, then this method leads to the solution u(z) in the indicated integral form.
In analysis such integral transforma tions occur in various forms.
For example the kernel
K(z, t) = e'r or eirf
produces the Laplace transformation, the kernel
K(z,
r)
= (z -
rrz
the Euler transformation if the path of integration is suitably chosen.
§2. Bessel Function s
First we shall discuss Bessel's equation
(2)
468
VII. SPECIAL FUNCTIONS DEFINED BY EIGENVALUE PROBLEMS
and all its solutions, considering both z and the parameter X as complex quantities.
1. Application of the Integral Transformation. We subject equation
(2) to the transformation (1). Substitution in the differential equation yields
L
2
(z K ••
+ zK. + z2K
- X2K)v(f) df = 0.
We now require that K satisfy the equation
+ zK. + z2K +Kn
2
z K ••
= 0,
for which the function,
K(z, f)
=
e±iuint
is a single-valued regular solution throughout the z- and f-planes.
Equation (2) becomes
L
(Krr
+
X K)v(f) df = 0
d!
+
2
or, integrating by parts,
L
K(z, r){v"
+ X v}
2
l :r
{Krv - Kv'} df
= 0.
Since the transformed equation v" +
X2v = 0 has the solutions e±i~.r, we
have only to choose the path of integration in a suitable way. We notice
that on the vertical portions of the
paths denoted by L1 and L2 in Figures
9 and 10, the real part of -iz sin r is
-ioo
negative for fR.e (z) > 0 and approaches
- oo exponentially with increasing Ir I .
Figure
10
Figure 9
Therefore, if we set K(z, !) = e-izsinr,
the expression Krv - Kv' tends to zero in both directions on L1 and
L2, and we obtain the integrals
-J{+ioo
(3)
.Z+ ioo
469
BESSEL FUNCTI ONS
as the two solutions of equatio n (2), the so-called Hankel functions.
0
It can be easily verified that these integrals converge for ffi.e. (z) >
ion.
derivat
and satisfy the assumptions necessary for their
are
2. Hankel Functio ns. The Hankel functions Hi(z) and H~(z)
0.
>
(z)
9ie
ne
half-pla
defined by the integrals (3) only in the right
cally:
However, it is easy to continue them analyti
If, for fixed z = x + iy, we set
J(t) = -iz sin r
X
+ i'At,
=a+ ib,
then
ffief(t)
= y
sin t cosh 71
+ x cost sinh 71
9m.f(t) = -x sin t cosh.,, ~ y cost sinh 11
-
bt - a11,
+ at -
b71.
If, instead of 0 and -11", we take the abscissas to and -11"-to for the
1
vertical portions of the path L1 , the integral /,, e <r> dt taken over
Lt
the new path L~ remains convergent for those z for which
y sin to - x cos to < 0
holds, i.e. for all z in the half-plane bounde d by the line
y sin to - x cos ~o = 0.
Both paths of integra tion may be used in that part of this half-plane
's
which also lies in the half-plane x > 0. It is clear from Cauchy
ing
remain
the
In
result.
same
the
give
they
integral theorem that
c
part, however, the integra l over the new path furnishes the analyti
we
way,
riate
approp
an
in
Ii,
H~(z).
n
functio
the
continu ation of
let to run throug h an unbounded sequence of positive and, similarly,
throug h a sequence of negative values, we gradually obtain the
complete analytic continuation of the function H~(z); i.e., we obtain
a Rieman n surface with a branch point at the origin the order of
which depends on X.
For ~o = -11"/2 the horizontal segment of the path of integration
vanishes, and for H~(z) we obtain the integral
-iff>./2
l( ) - e- - . .ZH>
1f''I,
loo e
-oo
~
i•cosh11->.1 u71,
·
470
VII. SPECIAL FUNCTIONS DEFINED BY EIGENVALUE PROBLEMS
which represents the function in the upper half-plane .'1m (z)
>
0.
If we permit z to become infinite in the sector
o~
arg z
~
7r -
o,
the integrand tends to zero over the entire path of integration; hence
the function Hi(z) also tends to zero since the integral converges
uniformly in every subset ffni (z) ~ p > 0. Similarly, we find the
function H~(z) tends to zero as z approaches infinity in the sector
- 7r
+ o~
arg z
~
-o.
We therefore have the result:
The Hankel function Hi(z) tends to zero when the variable z tends to
infinity in a sector o ~ arg z ~ 7r - o of the upper half-plane. The
Hankel function H~(z) tends to zero when z tends to infinity in a sector
- 7r + o ~ arg z ~ - oof the lower half-plane. 1
From the behavior of the.Hankel functions at infinity, it follows that:
neither the function Hi(z) nor the function H~(z) vanishes identically,
and they are linearly. independent for every A.
To prove this, we show that, as I z I increases, the functions H~(z)
and Hi(z) increase without limit on the positive and negative imaginary axes, respectively.
We obtain a representation of H~(z) which converges on the positive
imaginary axis by taking the values - ~o and 7r + ~o for the abscissas
of the portions of the path L~ , where ~o is any number in the interval
0
< ~o ~ 7r/2. Since the integrals
J/<r>dt over these portions
converge to zero with increasing y, we need only investigate the
remaining integral
substituting~ = ~'
+ 7r/2, this integral becomes
(r/2)+h
1
0
cosh b~ e11008 ~ cos a~ d~ .
1 This statement
is true only for the initial branches of the functions H~ (z)
and H~(z); the remaining branches are obtained as linear combinations of the
original branches, and do not exhibit the described behavior.
471
BESSEL FUNCTIONS
But this integral increases without limit as y ~ oo ; this is immediately1
clear if 1 a I ~ 1 and can be seen by somewhat more precise estimates
iflal>'L
The discussion is analogous for the function H~(z) and the negative
imaginary axis.
The linear independence of the functions H~(z) and Hi(z) implies
that the Hankel functions yield the totality of solutions of Bessel's
equation. For, every solution may be represented as a linear combination
We observe, in addition, that the Hankel functions H~(z) and
H~(z) are determined uniquely, up to a factor not containing z, by their
beliavior at infinity and by equation (2). For, if there were two linearly
independent solutions of Bessel's equation having the described
property,_ say for the upper half-plane, then every solution-an d,
in particular, Hi(z)-woul d have to have this property. · But this
contradicts the fact just proved that I H~(z) I increases without limit
on the positive imaginary axis.
Finally, we consider the Hankel functions, for fixed z ~ 0, with
respect to their dependence on the parameter X. Since the integrand
in (3) depends analytically on X and the integrals converge uniformly
in every finite X-domain, it follows that the Hankel functions are
analytic functions of X; in fact, they are integral transcendental f unctions.
3. Bessel and Neumann Functions. The solutions of equation (2)
which are real for real X and z are of particular interest in physics.
In order to find expressions for them, we write
(4)
+ iN,.(z),
H~(z)
J,.(z)
Hi(z)
J,.(z) - iN,.(z),
1
Let ~o be chosen in such a way that 7r/2 + ~o is an integral multiple of
7r/2a. Then the integral in question becomes
nr/2a.
1
cosh b~ e11008 Ecos a~d~
0
=
~[''cos e{~' (-!)' cosh Ke+.
i) .. ~·c•/•HH'</21} d~.
Here the first term of the sum dominates as y increases, since. the exponent
cos(~/a) islargerbyatl east the amount 1 - cos(7r/2a) than anyofthefollo wing
exponents
cos:(~+"
i);
this term increases with y beyond all bounds.
472
VII. SPECIAL FUNCTIONS DEFINED BY EIGENVALUE PROBLEMS
where
(5)
+ H~(z))
= !(H~(z)
J'A(z)
is called the Bessel function of index X and
(5')
N">-(z)
;/H~(z)
=
the corresponding Neumann function.
1
1
2
2
1
2i
•
1
H~(z))
-
Since the determinant
i
-2
-2?
of the substitution is different from zero, the functions J">-(z) and N'A(z)
are linearly independent for all X.
For real z and real X the Hankel functions H~(z) and H~(z) are
complex conjugates of each other. For, if we replace t by - t in the
representation
where
L1 is the reflection of L1across the real axis, we have
since the path obtain
L1 is the same as L2 taken in the negative sense, we
H '>I(- z) -_ --1
1r
f
)
e-izsint+•)f d1-~ -_ H2(
'>- z .
L1
Hence, J">-(z) is the real and N"A(z) the imaginary part of the Hankel
function H~(z) for real X and z, and therefore J"A(z) and N'A(z) are real.
The function H~">-(z) (11 = 1, 2) is a solution of Bessel's equation for
every value of X for which H~(z) is a solution, since only X2 appears
in the differential equation. The functions H~(z) and H~"A(z) cannot
be linearly independent because, by subsection 2, they show the
same behavior at infinity.
473
BESSEL FUNCTIO NS
In fact, if we introduce the new variable of integration the representation
l ( ) H ->.
z - --1
f
1r
L1
r
-1('
in
e-iuin r-iAr d,.~'
we at once obtain the relation
(6)
H!..>.(z)
=
e'"A.,, H~(z)
and, by means of a similar calculation,
(6')
For the Bessel and Neuman n functions of negative index one obtains
}
) + e-i>.rH2(
>.Z
1'.Z
=e iArHl(
J ->. ()
(7}
2
z
and
(7')
) +e
>.Z
=e Dt.rHl(
N ->. ()
2i
z
)
-1">.rHt(
>.Z.
'
unlike the Hankel functions, they are not linearly dependent on the
functions J>.(z) and N>.(z), respectively, for every X, but only for those
values of X for which the determi nant
1 eDt.r e-i>.r
I
4 1
1
I
·
=;sin Xr
vanishes, in other words, only when Xis an integer n.
we have
(8)
In this case
J--(z) = (-1rJ,. (z)
and
(8')
The general solution of equation (2) can therefore be represented
in the form
474
VII. SPECIAL FUNCTIONS DEFINED BY EIGENVALUE PROBLEMS
when A is not an integer n.
sum
On the other hand, if A = n we use the
It will be seen later, however, that even in this case Nn(z) can be
calculated easily from J n(z) and J _n(z) (see subsection 9).
4. Integral Representations of Bessel Functions. If we add the
integrals (3) for H~(z) and H~(z), the paths of integration over the
negative imaginary axis cancel and in the right half-plane fR.e (z) > 0
we obtain the representation
(9)
for Jx(z), where Lis the path indicated in Figure 11.
_ .,.;, 00
In particular, if A = n is an integer the in8
z.,.t,oo
.
tegrals over the vertical
portions of the path
L also cancel, because of the periodicity of the
l
integrand. It follows that
(10)
0
-.JC
.1E
Figure 11
(10')
or, since the real part of the integrand is an
even function and the imaginary part is odd,
11r
J n(z) = -
1(
0
cos (z sin r - nt) dt.
By means of these integrals J n(z) is defined for all z. We see that
the Bessel functions with integral index are regular and single-valued in
the whole plane and hence are integral functions.
The representation (10) implies also that J n(z) is the n-th Fourier
coefficient in the Fourier expansion of
(11)
eiuinr
=
00
L
J nCz)einr
-oo
with respect tor. This expansion can be considered as the definition
of the function J n(z) for integral n by means of a generating function
eiasinr .
475
BESSEL FUNCTIONS
For real z and
r,
(11) yields the relations
00
cos (z sin r) =
L: J n(z) cos nr,
-oo
00
sin (z sin r)
=
L: J n(z) sin nr,
-oo
which also hold for complex z and
If we observe that
r.
we obtain
00
cos (z sin r)
= Jo(z)
+ 2 L: J~n(z) cos 2nt,
1
(12)
ao
sin (z sin r) =
in particular for
r=
cos z
2 ~ J2n-1(z) sin (2n - l)r;
.
1
r/2 we have
= Jo(z)
- 2J2(z)
+ 2J.(z)
-
+ ... ,
sin z
If the new variable of integration
f'
= e-it is introduced in (9), it follows
that
(13) J.,.(z)
-oo
= 2~
{ e•<r-r-•>1 2r-"--l df,
-00
~
1
where L is the pa.th shown in Figure
Figure 12
12. It extends on both sides of the
=
r
point
the
to
up
axis
real
negative
1
-1 and then circles the origin along the unit circle.
1 We could have obtained this representat ion directly, on the basis of the
method outlined in §1, by requiring the transformat ion kernel to satisfy the
differential equation
z1K., + zK, + z1K - t(tK,), = 0,
12
which is solved bythefunc tionK = e•cr-r-•> • The transformed 1 differential
equation is then [t(tv)']' - >.Iv = O; it has the solution v = r±~- •
476
VII. SPECIAL FUNCTIONS DEFINED BY EIGENVALUE PROBLEMS
For integral X = n the integrals on the straight portions cancel, and
we have
(14)
Thus J n(z) is the n-th coefficient in the Laurent expansion
QC)
e·<r-r-1>12
(15)
= I: J nCzkn.
-ao
This expansion could also have been employed in the definition of
J n(z) for integral n.
If we make the transformation
assumption of real z
find
(16)
>
J >. (z)
r = 2v/z in (13)-at first under the
0- taking the same path of integration we
=
__!_. (~)>.
21f?, 2
fe
L
f1-a2/4v v-<>.+I) d v.
But since the integral on the right converges for all values of z,
(16) furnishes a representation of the Bessel functions for all z.
In particular, we see that the quotient J >. (z) / i is an integral function of
z for every X.
5. Another Integral Representation of the Hankel and Bessel Functions. Another integral representation for the Bessel functions is
obtained by considering the differential equation for Jx(z)/i and
applying the Laplace transformation. (Indeed, it seems reasonable to
expect that in this way we shall arrive at particularly simple results,
since Jx(z)/i is a single-valued function of z.) For this purpose, we
introduce in the equation
2
u"
1 +
+ z-u'
( 1 - }..- ) u = 0
2
z
the new variable w(z) by means of
u = wz,>.
from which we obtain
(17)
zw"
Writing
w(z) =
+ l)w' + zw = 0.
+
(2X
l
K(z, t)v(t) dt,
477
BESSEL FUNCTI ONS
we obtain
l
{zKzz
+
(2A
+
+
l)K,
zK}v(f ) df = 0.
In the special case of the Laplace transfo rmatio n we have
K,
=
fK,
Kr = zK,
2
and hence zKu = f Kr; we therefo re obtain
l { +f
(1
2
)Kr
+ (2>. + l)fK}v (f)
=
-
l
df
K(z, f) {(1
+ f )v'
2
!!_ (Kv(l + f)) df
+ jcar
-
(2X - l)fv} df
= 0.
in
Thus the differential equatio n is solved if v(f) and C are determ ined
such a way that
2
(1 + f )v'(f) - (2A - l)fv(f) = 0
and e'r v(f)(l
+f
2
)
takes equal values at the ends of C.
2>. -
v'(f)
v(f)
=
1
From
1
+f r
2
we have
Thus
w(z) = c
l
e•t {1
+ f)>.-i/
2
df
or, if we introdu ce if as a new variabl e of integra tion, include
1 2
tion
i(-1/ - ' among the constan ts, and denote the path of integra
by C, we have
w(z) = c l
ei•tcf - 1)>.-l/ 2 df.
In order to find an admissible path of integra tion, we first constru ct
the
the Riema nn surface of the integra nd which is one-va lued on
478
VII. SPECIAL FUNCTIONS DEFINED BY EIGENVALUE PROBLEMS
Riemann surface of log (r -:- 1) + log (r + 1) = X· If we cut the
r-plane along two rays parallel to the positive imaginary axis, starting
at r = + 1 and r = -1 and ending at r = +i 00 , we obtain a simply
connected domain in which a branch of x is uniquely determined by
the value of the imaginary part of x on the positive
real axis of the r-plane. we define two branches
B1 and B2 of x such that the imaginary part of
x is zero on B1 and equals 27ri on B2 for real positive
values of t. A path crossing either of t~e branch
cuts from the right-hand side to the left-hand side
leads from B1 into B2 .
-1
+1
Now let C1 and C2 denote paths lying in the
Figure 13
branches B1 and B2, respectively, such that C1 encloses the ray starting from r = + 1 and C2 encloses
the ray starting from r = - 1 without intersecting either of these
rays (see Figure 13). Instead of introducing rays through t =
± 1 which are parallel to the imaginary axis, we can introduce rays
making an angle a with the positive real axis (where 0 < a < 271'")
and then define C1 and C2 as before.
The integral w(z) converges on C1 or C2 for those z for which
fR.e (izt) tends to - oo along the ray. At the same time the expression
Kv(f ._ 1) =
1)>.+t/ 2 ei•C
<r -
tends to zero at both ends of the path of integrati~n; in other words,
w(z) is a solution of (17). If the ray makes the angle a with the
~-axis, this will be the case when
y cos a
+ x sin a
>0
and thus when z = x + iy lies in one of the half-planes bounded by
the line y cos a + x sin a = 0. However, we can continue the
integrals analytically, as in subsection 2, by letting a run through an
unbounded sequence of positive values and a similar sequence of
negative values in a suitable way. In particular, if for both paths we
choose a = 71'"/2 as in Figure 13, both integrals converge in the right
half-plane fR.e (z) > 0. If we rotate the path C1 so that it lies along
the positive real axis, then the corresponding integral converges in
the upper half-plane and tends to zero as z increases without limit
in the sector
(O
<
8
< r/2).
479
BESSEL FUNCTIONS
According to the observation of subsection 2, the integral must
therefore coincide, up to a factor independent of z, with H~(z)/i.
We find
and similarly
H~(z) = <Liz°''
f
e"C(f -
Cs
1r·-
1 2
'
dt.
The coefficients a 1 and a2 , which can depend only on A, are equal
but have opposite signs. This follows-at first for real A-from the
remark that the Hankel functions are complex conjugates of each
other for real A and z (see subsection 3).
1
We find for real values of A and z:
H~2'(z) = a2i'
1
ei•r(f -
Ci
1»·-
1 2
'
dt.
This can be verified by observing that for large values of I r2I in the
proper branches of Bi. and B2 , the imaginary part of log (! -1) on
the four parallels to the imaginary axis in Figure 13 is approximately
(i.e. for I r I --+ oo and for r approaching the branch cuts from different sides), from left to right, -?ri, ?ri, -?ri, ?ri. Therefore we have
) _ Hm
Hu>(
_ -ti2
<2>(.)Z
A
>.ZH>
'
a1
that is ti2 = a1 • Now we shall show that °'2 = - a1 by proving that
Since a 1
a1 and °'2 are pure imaginary if X is real and X > - ! .
of X.
values
all
for
0
=
°'2
a
X,
on
analytically
1
and °'2 depend
and
0
>
y
iy,
=
z
taking
by
seen
be
can
imaginary
That a1 is pure
from
axis
real
positive
the
along
path
a
into
C
path
the
by turning
1
r = 1 to r = + oo • Then
+
H~1> (iy)
= a1(l -
e-2ri<>.-112>) (iyY"'
From the formula
-ri>./2
e
H ,..l(.iy ) -- --.71'1,
lao e
lxi e-Yr
-11cosh,,->.71
(r2 -
1)'·-112
dt.
d.,,
-co
1 In order to insure convergence on the positive real axis, we take C1 and C2
parallel to the imaginary axis (see Figure 13).
480
VII. SPECIAL FUNCTIONS DEFINED BY EIGENVALU E PROBLEMS
we find that
b(X)
a1 (X) cos rX = - .
ri
where b(>-.) is real and positive for real X > -!. For such X we
have, therefore, d1 = -a1 so that °'2 = -a1.
Now since by subsection 2 the Hankel functions and, as we see
immediate ly, also the integrals
eizra·2 - 1/·- 112 dt depend an-
J
alytically on X, the coefficients a1(X) and a2(X) are
analytic functions of X. In other words, the rela~
tion a1 = -°'2 = c holds generally.
Figure 14
If we add the two integral representa tions for
the Hankel functions, we can deform the resulting
path of integration into the figure eight ~ of Figure 14, which
circles 1 in the positive sense and -1 in the negative sense. We
obtain a representa tion of the Bessel functions
~
+
J'A(z)
=
ci
J. eizl"(f -
l)'A-l/ 2 dt,
which, for X ;:z6 n + ! (n = 0, ±1, · · · ), is valid in the entire z-plane,
since ·the path of integration lies· in the finite part of the plane.
To determine the constant c, we compare this representa tion with
the integral representa tion (16) and find for z = 0 the relation
c
J.
(t
2
-
~ 2 ~i
l)'A-I/ 2 dt =
l e"'v~H
dv.
As will be seen in the next subsection , the integral on the left has the
value
1(f •
1)>--112 dt = 2 ·
" r(x
+
r(!)
.
1)r(! - X)
In order to evaluate the integral on the right, we consider the integral
1
--:
21r't
f e"v
t-1
L
cl v,
in particular for positive real values of t. Since this integral represents an analytic function of t, it is sufficient to reduce it to known
analytic functions for such t.
481
BESSEL FUNCTIONS
Under the assumption t > 0, we can contract the unit circle to the
origin; since the exponent t - 1 is greater than -1, the integral is
convergent up to the origin. By
o
-oo
Cauchy's theorem, the value of
Figure 15
the integral does not change if,
instead of integrating over L, we
integrate from - oo to O below the real axis and then integrate from
0 to - oo above the real axis (see Figure 15):
-1.
2ri
f
ev t-l av = -1.
11
2rt
L
Lco
0
ev t-l av
11
+ -1.
1-
2rt o
ev t-l av
11
(t
above
below
the real axis
If we set v = -w, the first integral becoJUes
-1.
2ri
1°co w
t-1 -<t-l)ri
e
1""
w t-1 e-<t-I)rie-w dw,
e-w( - dw) = -1.
2ri o
the second becomes
d ).
t-1 (t-l)ri -w(
e -w,
-1. 1co we
2rt o
thus their sum is
1
00
w t-1 e-w( etri - e-tri) dw.
-1.
2rt o
Since etri - e-tri = 2i sin 7rt and, by definition,
00
1
0
w'-1e-w dw = r(t),
.
the sum of the integrals has the value
sin 7rt r(t).
7r
From the supplementary theorem for the gamma function,
r(t) r(l - t) - sin7r 7rt ,
it follows that
sin 7rl r(t)
7r
1
r(l - t) ·
>
0).
482
VII. SPECIAL FUNCTIONS DEFINED BY EIGENVALUE PROBLEMS
We have, therefore,
2-'1lr-i
fv
L
e dv
t-1 v
1
= -r(_l___t_)°
Hence for the constant c we find the value
c
=
1 1
2ri 2>-
r(! - >.)
r(!)
and finally obtain for J>.(z) the representation
J,(z) =
(18)
r~~~:i) (~)'
J. e"t(f -
i)'-112 dt.
This representation holds for al1 'A except 'A = n + ! , where n is an
integer ~ 0.
Corresponding formulas are found for the Hankel functions:
H~(z)
~ r(! --;
=
ri
(18')
If fRe. ('A)
sentation
(19)
> -i,
(20)
T,
>.)
(:.)>.
2
J
eizt (r2 -
1)>--112. dt,
C1
one can derive from (18) the very useful repre-
1
J>.(Z) - f(!)f(A
Setting r = sin
f(2)
+ !) ( 2z )
>. L+l
1
~
2>.-1/2
dt.
> -!,
we find, for fRe. ('A)
1
( :_)>.
J>.(z) = --- 1-- -- 1
r(2)r(>. + 2)
2
i•t
e (1 - r)
L+r/2 cos (z sin T) (cos
2
T) >.
dT.
-r/'2
6. Power Series Expansion of Bessel Functions. One can obtain a
power series expansion for J(z)/z\ which is single-valued and analytic
in the entire z-plane, on the basis of elementary considerations. As
in Chapter V, we could make the substitution
co
u(z) =
i
L
a"z"
0
in the differential equation (2) and successively determine the coefficients a,,. However, in the framework of our present approach
483
BESSEL FUNCTIONS
the power series expansions will be obtained from integral representations.
We begin with the integral representation (18) and expand the
function ei•t in its power series; to be able to apply (18) we must
assume that Xis not of the form n + ! (n = 0, ±1, ±2, · · ~ ).
Since this series converges uniformly in every finite r-domain, we
can integrate term by term and obtain
Jx(z) =
t
z"
X) (~)>.
1
2 n-o n!
2rzr(-2)
r(i. -
In evaluating the integral
f. r"(r
i""
1
ref - 1)>.- 112 dt.
11
2
-
2
l)x- 1' df", we note that we
deal with analytic functions of X and that it is therefore sufficient
to determine these functions for all X such that 9la (X) > 0. In
fact, in this c~e we can deform the path of integration into the
interval -1 ~ X ~ 1, traversed in both directions.
For the integrand we find the value
eri(>.-lf 2> r"(l -
r/-
1 2
'
above the real axis,
e-ri(>.-l/ 2) r"(l - r)>.-l/ 2 below the real axis,
and therefore
l
ref - 1)"-
1 2
'
df"
=
L:
-2i sin r(X - !)
rc1 - f)>.-
112
df".
The integral on the right vanishes for odd n; for even values of n
we find
L
f"(r 2
-
1)).-l/ 2 dr
=
2
Using the transformation r
1
f"(f - 1)>.- 112 dr
=
4i sin r(X
=
f"(l - r 2)>.-I/ 2 .dr.
u this becomes
2i sin r(X
.
11
+ !) {
+ !)
1
1
u"- 112 (1 - u)>.- 112 du.
0
The integral on the right is an Euler integral of the first kind.
the well-known relation
B ·cp, q)
= 11 p-1(1 0
x
x
)q-1 d
x
=
r(p)r(q)
q)
r(p
+
From
484
VII. SPECIAL FUNCTIONS DEFINED BY EIGENVALUE PROBLEMS
we obtain
l r n(f -
1Y'-1' 2 dt
2
= 2i sin ?r(A + !)B(n +
· ·
(' + 1) r(n +
2ism
?r "
2
r(A
=
!, A +
+Unr(A++1) U .
But r(x)r(l - x) =~,hence f(A +!)sin ?r(A + !) ~
sm ?rX
Therefore we have
r fn(f -
la
1»·-1/2 dt
=2
i
1r
r(! -
r(n
u
+u
+ A+
A)f(n
r
(1
1r
2 -
) •
A
1)
and in particular, for n = 0,
r <r2 -
la
1)>.-1/2 dt
= 2 .
n r(A
+
rm
l)r(l - >.)
.
Substitution of the values obtained in our series for Jx(z) yields
J,(z)
= rti)
(~)' ~ ~;:j~ z"' r(:~ ~ ~ 1)
or, since
we have
(z)
00
z)>.
-1r
( 2 ?;0 ---ni-- 2
2
+1A + 1).
If A is not an integer, none of the coefficients 1/r(n + A + 1) vanish.
(2 l)
J.,.(z) =
(
n
r(n
Howe~er, if A is an integer, then
1
r(n +A+ 1) =
1
A
for
o
n + A+ 1
~ 0,
1
+ A)! for n + A + 1 > 0.
The above assumption A ¢ n + ! is seen to be unnecessary for
r(n
+ + 1)
- (n
the .validity of the expansion (21), since the series (21) converges
uniformly also for the values A = n + ! and, as we have already
seen, Jx(z) depends analytically on A.
485
BESSEL FUNCTIONS
The series in (21) converges for all z and therefore Jx(z)/i is an
integral transcendental function, unless it is a polynomial or a constant.
But the latter is impossible, since r(n +}.. + 1) is finite for negative
integral}.. in all but a finite number of cases. Hence the coefficient
of z2n cannot vanish, and the series for Jx(z)/i always has infinitely
many nonvanishing terms.
It is immediately apparent from the series expansion (21) that
J x(z) is real for real values of }.. and z, since the gamma function has
real values for real arguments.
7. Relations Between Bessel Functions. Now that we have derived
power series expansions and integral representa tions for the Bessel
functions, we shall develop a number of general properties of Bessel
functions with the aid of the integral representation. We have equation (16)
J >. (z) = 2_!__. (~)>..
2
rt
fv
-(>.+l) v-z2/4v
e
L
d
v,
in which L denotes the path of integration of Figure 12, page 475;
therefore
->..- -
Jx(z) _ 1 1
.
2x -2
1f''l,
Z
f
L
-<>.+1> v-z2/4v
e
v
d
v.
2
We differentiate with respect to z , taking the derivative on the
right side under the integral sign in a formal way,
J >. (z)
__!__
2
d(z
)k
=
z>-
.!__ _!__
2>- 2ri
f
V -(Hl) ( -
l)k
ev-z2/b
4v
L
dv
•
Differentiation under the integral sign is permissible since the inequality I v I ~ 1 holds on the path L and hence, for I z I ~ h, the
function
a uniformly
is uniformly bounded. The right side is therefore
2
•
z
of
function
analytic
an
convergent integral of
When the last equation is multiplied on both sides by z'>'+k another
Bessel function is obtained on the right, so that we have relation
(22)
_!:__ J>.(Z) = (-
d(z 2)k
z>..
!)k J>.+k(z)
2
z>..+k
486
VII. SPECIAL FUNCTIONS DEFINED BY EIGENVALUE PROBLEMS
or, in a different form,
In particular, for k = 1 we have
(23)
in other words, the recursion formula
(24)
which for A
0 assumes the special form
=
= -
Ji(z)
The cases A = we have
d.lo(z).
dz
! , A = ! will be investigated further.
z)-112
J_ 112 (z) = ( 2
?;o n!r(n + !) (z)2n
2
+ !)
!)(n - j) ... i·lr(!)
oo
(
-1)"
and, since
r(n
= (n -
=
(n - !)(n - i) · · ·
this yields
J-i<z>
=
1
00
2-
(-1)" 2n
:L -<
)' z
1TZ n-o 2n .
=
i·!v;,
12
-
1TZ
cos z.
Using formula (23) for A = -!,
d
dz
.f-112(z)
z-112
we have
!!:_ .. ~ cos z
dz
'Y ;.
=
-
'V/~ sin z =
i
1f'
-
J 112 (z)
z-112
By (21)
487
BESSEL . FUNCTIO NS
or, in other words,
(25)
dividing, we obtain
{_-112(z)
J112(z)
= cot z.
-!, X = !
Thus the Bessel function s for X =
simply by trigonom etric functions.
From the formula s
J A(Z)
()
J -AZ
we obtmn, for X =
may be expressed
+ H~(z) ,
=
H~(z)
=
l( ) i>.r
H AZe
2
+ H2{ )
-1">.r
AZe
2
!,
and
_ i(H~12(z) - H~12(z))
J -1/2 (z) 2
or
-i.J ( ) _ Hb2(z) -2 H~12(z) •
-1/2 z , Adding the two expressions we obtain
Hf12(z) = J112(z) - i.J-112(z)
(26)
= . 12
'Y
(sin z - i cos z)
'JrZ
subtract ing,
(26')
. /! (sin z + i
H~12(z) = Ji12(z) + i.J-112(z) = 'V
'WZ
=
cos z)
·12 -··
i
-
'JrZ
e •
488
VII. SPECIAL FUNCTIONS DEFINED BY EIGENVALUE PROBLEMS
These formulas show again that the relation between the Bessel,
Neumann, and Hankel junctions is similar to that between the sine,
cosine, and exponential functions. This analogy will be apparent.
also in connection with the theorems on the distribution of zeros which
will be derived later (see subsection 8).
If we apply relation (22)
_!_ J.,.(z)
d(z2)k
zA
!)k
J'A+k(~
2
zHk
= (-
!,
derived on page 485 to the case X =
_!_ . /~sin z =
d(z 2)k
'V
11'
(-
!)k
h+112(z)
2
z
zH 1 12
'
it follows that
k
Jk+1 12(z) = (-1)
(2z)H 112
.y;.
dk
sin z
d(z2)k -z-;
in other words, every Bessel function Jk+1;2(z) may be expressed as a
rational function of trigonometric functions and of z, multiplied
byvz.
A different recursion formula is obtained by differentiating under
the integral sign in the relation
JA(z)
= __!_,
2rz.
1· e'<r-r-1>12r-A-1 dt.
L
This yields
J~(z) = ~
4ri
(27)
{J e'<r-r-1>12r-x dt - J e•<r-r-1>125->.-2 dt}'
L
L
J~(z) = !{J>.-1(z) - J>.+i(z)}.
If we subtract from this the expression
J~(z) = ~z JA(z)
it follows that
(28)
- J,.+i<z),
489
BESSEL FUNCTION S
The last relation can also be written as
1
2>i.
1
J>.-1(z) _ 2>i.
J>.(Z) - Z - J>.(Z) = Z - 2A + 2
1
- z - - 2A
J>.+1(z)
+4 _
z
J>.-i(z)/ J>.(z) is thus represen ted by an infinite continue d fraction;
however, we cannot investiga te the question of its convergence here.
If we multiply through by z, the continue d fraction assumes the form
z J >.-1 (z)
= 2A
J>.(z)
(29)
For A =
- _z_2_
2 2A
+
z2
2
A
+4 _
! this reduces to
t
J-112(z) _
()-zc o z
z-J
1/2 z
(30)
which represents an infinite continued fraction for cot z, known in the
1
18th century and used by Lambert to prove the irrationa lity of r,
for which purpose he set z = r / 4.
The following functional relation holds for Bessel functions of integral
index n:
J n(a
(31)
+ b)
cO
L:
=
J,,(a)J _,,(b).
•--co
The proof is obtained directly by considering the generating
function ei<a+b>sinr = eiasinr. eibsinr. We have accordingly
'j:,~ J .(a + b)e'•t = .f~ (i:~ J,(a)J~.(b) )•'•t,
from which the assertion follows.
There is a generalization of this formula which, for n = 0, reads:
Jo(Va 2
(32)
1
+ b + 2ab cos a)
2
co
=
Jo(a)Jo(b)
+2L
J ,,(a)J_11(b) cos va.
1
J. H. Lambert, Memoire sur quelques proprietes remarquab les des quan-
tites transcend entes circulaires et logarithm iques, Acad. sc. Berlin, Meni., Vol.
17 (1761), 1768, pp. 265-322, in particular p. 269.
490
VII. SPECIAL FUNCTIONS DEFINED BY EIGENVALUE PROBLEMS
To prove it, we make use of the integral representation (10) and
write the product J .,(a)J _.,(b) as a double integral,
With a little manipulation this can be brought into the form
;r L:
Jo(v a2
+ b + 2ab cos
2
a)
e-i•« da,
which proves relation (32).
Finally, we note that a functionf(r), subject to certain assumptions,
can be represented with the aid of Bessel functions in a way similar to
that in which it can be represented (according to Fourier's integral
theorem) in terms of exponential functions (see Ch. II, §6 and Ch. V,
§12). Let f(r) be continuous and piecewise smooth, and _let
lao r l f (r) I dr <
Then for every integer n and r
(33)
f(r)
=
fo
> 0, f(r)
00
s ds
oo.
is represented by formula
L'° tf(t)J.,ht)J"(sr) dt.
This formula is derived in the following way: We set
x = r cos O,
y = r
sin fJ
and consider the function
g(x, y) = f(r)einfJ'
which, under the above assumptions on f(r), is certainly continuous
and-except in the neighborhood of the origin-has piecewise continuous derivatives. If we apply Fourier's integral theorem for two
dimensions (see Ch. II, §6, 2) to g(x, y) and interchange the order of
integration of the two inner integrals-whic h requires justificationwe obtain
g(x, y) = 41 2
r
Lao Lao e•<iu:+tiu> du dv Lao Loo g(~, 71)e-iCuE+ti71) d~ d71.
oo oo
ao
~
We now introduce polar coordinates for the variables of integration
~' 11; u, v:
491
BESSEL FUNCTIO NS
~
= 8 cos a,
fl = 8
v = t sin {J,
sin a,
and obtain
f(r)inB = _!__2
471"
1
00
t dt [
0
11'
eirccoa(JJ--B>
= t cos fJ,
u
dfJ
1·
sf(s) ds
["'
e'nae-ido o•<-{J)
da.
..
0
11'
If we make the substitu tion
a -
fJ = a' - !
2
this expression becomes
f(r)ein8 = einll 1«>t dt
4'JI" 2 0
111'e-irtsinfl'+in{JI d{:l
1.,sf(s) ds LT e-idsinal+inal da',
0
-ir
since the exponential functions are periodic.
f(r) =
11'
The asserted relation
1"° tJ n(rt) dt 1., sf(s)Jn(st) ds
follows immedi ately if we apply formula (10) and integrat e with
respect to a' and fl'.
Instead of justifyin g the inversion of the order of integrat ion, a
proof similar to that for the Fourier integral theorem can be given
for the integral formula (33). We employ relation
f(r) = lim
,,_'°
(34)
Pi:(s, r)
={
1°
sf(s)Pw(s, r) ds,
0
tJn(st)J ,ht) dt,
which holds for every piecewise smooth function f(r) vanishing at the
origin; a denotes any positive number a > r > 0. This relation
corresponds exactly to the Dirichle t integral of Chapter II and is
proved in a similar way. We shall show that; under the assumption
that the integral L~ r I f(r) I dr exists, the integrat ion with respect
492
VII. SPECIAL FUNCTIONS DEFINED BY EIGENVALUE PROBLEMS
to scan be extended to infinity.
(35)
P"(r,s)
For, the identity
= s~
{sJ,hr)Jn-r1(vs)
- r2
- rJ,hs)Jn-r1(vr)},
valid for r ¢ s, which is obtained from (36) of the next subsection
using recursion formula (24), implies that, for fixed r ~ 0, P" (r, s)
tends to zero with increasing s uniformly in v (e.g. using the asymptotic
expansions of the Bessel functions for large arguments-see Ch. V,
§11, 2 or Ch. VII, §6, 2). Hence the integral
ib
sf(s)P 11(r, s) ds
also becomes arbitrarily small, uniformly in v and b, for sufficiently
large a. Our assertion-relation (33)-follows.
8. Zeros of Bessel Functions. Finally, we shall derive some theorems co~cerning the zeros of the .Bessel functions 1•
The Bessel function Jx(z) satisfies the differential equation
J~'(z) + ~ J~(z) + ( 1 - ~:)Jx(z)
=
0.
If we set
z = 6t,
~i
=
+
(
const.
¢
0,
we obtain
1 Jx(~il
I
)
Jxfl (~1l ) +~it
2
X ) Jx(~1t) = 0.
1 - ~~t2
Similarly, for
~2 =
const.
¢
0,
we have
J~ 1 (~2t) + ~~t J~(~2t) + (1
- ~~:2) J>.(~t) 0.
=
If we multiply the first of these equations by ~~tJ>.(~2t), the second
by -~~tJx(~ 1 t), and add the resulting expressions, we obtain
1
Cf. the related results in Ch. VI, §2, 4.
493
BESSEL FUNCTIONS
t(~~J~' (t1t)J >.(~2t) - ~iJ~' (~2t)J,.(~1t))
+
(t1J~(~t)Jx(t2t) - ~2J~(t2t)Jx(t1t))
+ Ct~
- t:)tJ,.a1t)J,.(~2t)
= o.
The first two terms taken together are equal to the derivative of the
function
with respect to t. If this function exists over the interval (0, 1), we
obtain, integrating the last equation from 0 to 1,
I!
t(t1J~Ct1t)J>-Ct~t) - t2J~(t2t)J>-Ct1t))
+ (t~ -
(36)
·Since
J,.~)
z
ti)
f
tJ>.(t1t)J>.(t2i) dt = 0.
is an entire function of zit is not difficult to show that
= t2>.+2g(t),
of t. Therefore, if X > -1, this ex-
t(~1J~(t10Jx(~t) - t2J~(t2t)Jx(t1t))
where g(i) is an entire function
pression vanishes at t = 0 and (36) leads to
(t1J~(t1)Jx(t2) - t2J~(~)Jx(~))
(37)
From this equation, conclusions may be drawn about the distribution of the zeros of Jx(z) (see Ch. VI, §6).
Lett be a zero of Jx(z) which is different from zero. We seq1 = t,
~ = l, where l denotes the complex conjugate oft· Thus ti and ~2
coincide only for real e.
Let X be real, so that Jx(z) takes on real values for real z. The
coefficients of the power series (21) are real; hence if Jx(l;) vanishes,
thenJ,.(l)a lsovanishe s. If in equation (37) we set Jx(t1) = h.(t2) = 0
the expression in large parentheses vanishes, and the second term
becomes
(~2
-
f)
ft
I
J,.(~t) 1
2
dt =
o.
494
VII. SPECIAL FUNCTIONS DEFINED BY EIGENVALUE PROBLEMS
It was assumed that t
identically, { t l J >.(~t)
~
0.
Since the Bessel function does not vanish
2
[ dt ~ 0 and hence ~ = (~ - l) (~ l) = 0,
r
2
+
which implies
~
= l or
~
=
-~.
Hence tis either real or pure imaginary. For real}.. > -1 the Bessel
function Jx(z) has only real or pure imaginary zeros.
In order to investigate the pure imaginary zeros of the Bessel
functions, we consider the power series expansion
00
L: · - .
Jx(z)
(-1r (z)
-· = -1
z>.
2>- n-o n!
2
The substitution z = ai, a real
J>i.(Z)
-zA =
1
2>-
00
~
(a)
~o 2
2
n
1
---------···-.
r(n + A + 1)
0, yields
2
1
n
n!r(n
+ A + 1) •
+ }.. +
Since }.. is real, n
1 is positive for all n with a finite number of
exceptions, and since the gamma function takes on positive values
for positive arguments, all the coefficients of the power series are
positive except perhaps for a finite number at the beginning of the
series. For large I a I the higher powers prevail, and since (a/2) 2n > 0
for a ~ 0, we have J>.(z)/l > 0 for all sufficiently large I a j. Thus
zeros of the function Jx(z)/l can occur only on a finite piece of the
imaginary axis; hence, as an integral transcendental function,
Jx(z)/l can have only a finite number of pure imaginary zeros.
Jx(z)/l has no pure imaginary zeros for A > -1, since in this case
for all n
+}.. + 1 > 0,
r(n + X + 1) > o.
n
Thus all coefficients of the series are positive, and the value of the
series itself is positive. In particular for}.. = 0, 1, 2, · · · there are no
pure imaginary zeros.
We have the result: For real A > -1, Jx(z) has only real zeros.
For any real value of A, Jx(z) has only a finite number of pure imaginary
zeros. Actually it can be proved that, for real A, the total number of
zeros not on the real axis is always finite.
The function J,.,(z) has infinitely many real zeros; for every real
BESSEL FUNCTIONS
495
positive integer A this is clear from the discussion of the preceding
chapter, since the zeros of J n(z) furnish the system of the eigenvalues
of a differential equation.
We add a few remarks about the position of the real zeros of
Bessel functions.
Suppose A is real and set
JA(z)
-z>.- == v,
then we have (17)
zv"
+ (2A + l)v' + zv = 0.
If ~ is a positive zero of v', for z =
the form
~
the differential equation takes
and hence
v"(l;)
+ v(l;)
=
0.
It follows that at the point ~ the second derivative v" (~) cannot
also vanish. For, if it did, we would have vet) = 0, from which,
together with v'(t) = 0, the identical vanishing of the solution v(z)
of (17) would follow. We therefore conclude that v(~) and v"(l;) are
of opposite sign.
Now let t1 and ~2(>~1) be two adjacent positive zeros of v'(z), so
that v'(z) -:F 0 for t1 < z < t2. By Rolle's theorem there must then
be an odd number of zeros of v" between t1 and ~2 ; consequently
v"(~1) and v"(~), and hence also v(~1) and v(t2), are of opposite sign.
Therefore an odd number of zeros of v must lie between ~1 and ~,
hence at least one; but by Rolle's theorem there can be only one since
an odd number of zeros of v' lies between two adjacent zeros of v,
and by hypothesis v' has no zeros between ~1 and ~ . Accordingly,
v has precisely one zero between ti and ~2 • In other words, between
two adjacent positive zeros of v' there is one and only one zero of v.
The positive zeros of v and v' separate each other, the same is true of the
negative zeros.
In subsection 7 we derived relation (23)
d J>.(Z)
dz~
J>.+i(z)
--r-
496
VII. SPECIAL FUNCTIONS
DEFINED BY EIGENVALUE PROBLEMS
or
I
J>.+i(Z)
v = - -z->.-.
Since the zeros of v and v' separate each other, and since moreover
J>.(Z)
I
J>.+1(z)
v=-v=--z>. '
z>. '
and hence all positive and negative zeros of v and v' are also zeros of
J>.(z) and J>.+1(z), we find in addition: The zeros of J>.(z) and J>.+i(z)
separate each other.
For A = -!, A = ! we found that
J_t(z) =
~ /2 cos z,
v;z
The zeros of these functions are given by
1r
±2,
±371'"
2 '
±571'"
±
2 '
(2n
+ l)?r
2
'
...
and
o,
±n7r,.
±27r,
±1r,
respectively; thus they do, in fact, separate each other.
In this respect, too, the Bessel functions reveal a relationship to
trigonometric functions.
9. Neumann Functions. If A is not an integer, relations (5)
h.(z) = !(H~(z)
+ H~(z))
and (7)
J_>.(z) = !(ei>.rH~(z)
may be solved for H~(z) and H~(z).
(38)
Hi(z) = -
..
1
i Sill
(38')
and hence
(39)
+ e-i>.rH~(z))
We obtain
X (J>.(z)e-'"" - J_>.(z)),
1r
H~(z) = .. 1 X (J>.(z)ei>.r - J _>.(Z) ),
ism r
497
BESSEL FUNCTION S
This represen tation of the Neuman n functions by means of J x
and J _x fails, however, if A. is an integer. For, considered as functions
of A., both the numerato r J x cos A7r . . :. J -x and the denomin ator
sin A.r have simple zeros in this case. But since for z r6 0 both
numerato r and denomin ator are regular analytic functions of A., we
may differentiate them both in order to determine the value of the
function for integral A.. Passing to the limit, we find from the
quotient
aJx(z)
---ax cos A7r
aJ _x(z)
( ) .
- J x z 71" sm A7r - ~
7r cos A7r
that fol' integral A
Nx(z) = !(aJx(z) - (-1? aJ_x(z))
aJ
aJ
71"
(40)
holds. It is easy to verify directly that the expression just obtained
represents a solution of the differential equation for integral A.. In
fact, if we differentiate Bessel's eqution
2
d Jx(z)
dz 2
+ 1 ~x(z) +
dz
z
(i _z
2
A.
)
2
Jx(z)
=
0
'
which represents an identity in X, with respect to X, we obtain
2
d aJx(z)
dz 2
iJA.
+!i
aJx(z)
z dz
+
aA.
aJx(z)
(1 - ~)
z
2
=
cJA.
2A. Jx(z).
z2
Similarly, for - X we find
!_2 aJ_).(z) + ! !!:... aJ_x(z) +
dz
iJA.
z dz
cJA.
(i - ~) aJ_x(z) = 2A. J _x(z).
z2
cJA.
z2
We multiply the second equation by (-1? and subtract the resulting
expression from the first equation ; from the relation J x(z) =
( -l)x J _x(z) we see that the right side of the resulting equation
vanishes for integral X, and we obtain, as a further solution of Bessel's
equation , the function
aJx(z) - (-1? aJ_x(z) = rNx(z)
a>..
iJA.
(integral A.).
The relations just derived between the Neuman n function Nx(z)
498
VII. SPECIAL FUNCTIONS DEFINED BY EIGENVALUE PROBLEMS
and the functions J,. (z) and J _,_ (z) enable us to find corresponding
representations for N,,.(z) from the representations for the Bessel
functions. Thus, for example, one finds from the integral representation (9) for 'A ~ n,
(41)
N,.(z) = - 2
1r
and for 'A
~
sm 1r
X {
= n,
f re
N n (z) = - 2i
(42)
-izsinr
1r
1r
L
cos 1rX - e-i>.t} dt,
cos
L
f
N n (z) -_ 21
(42')
e-iz1sint { e'"r
JL
nr dr
(n even),
r e-izsint sm
•
nlr drl
(n odd).
l
Making use of the integral formula (20) of subsection 5, it follows,
e.g. for No(z), in virtue of
No(z)
= ~ (aJ")
ax
1r
>-o
that
(43) 7rNo(z) = 2(C
+ log 2)Jo(z) + ~ 1. cos (z cos S-) log (z sin
1r
2
t) ds,
0
where C is the well-known Euler constant.
Similarly, series expansions for Nx(z) may be obtained from those
for Jx(z) and J_x(z). We shall consider in greater detail the case of
integral 'A. We have (21)
J)t.(Z)
z)>..
= (2
Eo nr (z)2
00
(-1)"
2
"
r(n
1
+ x + 1)
and, since we may differentiate with respect to 'A under the summation sign,
aJ;~z)
(44)
=
aJ_x(z)
ax
log~ J,(z) + (~)'
=
t. (~:)" m
. (~t r~t))
t-•+41'
-log ~ J _)t.(z) 2
-1)" (z ) (d 1 )
(2z)->. ?;O n
l 2
dt r(t) e-n-A+l
00
(
2
"
499
BESSEL FUNCTION S
First we determine the values of the derivatives d[l/r(t)] /dt for
positive integers t. The gamma function satisfies the functional
equation
fort ~ 0, -1, -2, · · · .
r (t + 1) = tr(t)
By logarithmic differentiation we find
r'(t
r(t
+
1)
+ 1)
=! +
r'(t)
t
r(t)
and, repeating k times,
1
1
+ k + 1)
r + k + 1) = t + k + t + k r'tt
(t
1
+ ... + t +
l
r'(t)
r(t)
(k = 0, 1, 2, ... ).
We have
d
1
r'(t)
1
r'(t) .
iit r (t) = - r 2(t) = - r(t) r(t) '
if we sett = 1, k = n - 1 we obtain the above formula.
r'(n
r(n
+ 1) = ! + _1_ + ... + 1 + r'(l)
r(l)
+ 1) n n - 1
= ! + _l_ + ... + 1 - c
n n- 1
for n = 1, 2, 3, · · · . Knowing the value of r'(t)/r(t) for the
positive integers, we also know the desired value of the derivativ e
d[l/r(t)]/ dt at these points:
(11 r~oL
~ r~t) =
-(t
=
c;
~ 1)1{1~1 + t ~ 2 + ··· + 1 for t
c}
= 2, 3, · · ·.
In order to determine in addition the value of the derivativ e for
negative integral values t, we solve the equation
r'(t+
k
+
1)
1
1
1
Rt + k + 1) = t + k +. t + k
- 1
+ ... + - +
r'(t)
r(t)
500
VII. SPECIAL FUNCTIONS DEFINED BY EIGENVALUE PROBLEMS
for r'(t)/r(t).
Multiplying by - 1/r(t) we obtain
1 r'(t)
1 {1
- r(t) r(t) = r(t)
1
1
t + t + 1 + ··· + t + k
1 }
+t+ k
1 r'(t + k + 1)
r(t) r(t + k + 1) ·
- 1
-
If we let t converge to -k (k = 0, 1, 2, ···),the left side, and hence
also the right side, converges to the derivative (d[l/r(t)]/dt)e=-k. Now
as t ~ -k the expression 1/r(t) tends to zero, and since
l/t + l/(t + 1) + .. · + I/(t + k - 1)
and
r'(t
+ k + 1)/r(t + k + 1)
remain finite, all that is left on the right is the term 1/r(t)(t + k).
If we multiply top and bottom by t(t + 1) · · · (t + k - 1) and use
the functional equation of the gamma function, we see that the
denominator becomes equal to r(t + k + 1) and converges to
r(l) = 1 as t ~ -k; since the numerator tends to (-l)kk!, we
obtain
(~t r~t)),__. =
(-l)°kl.
If we substitute the value of the derivative d[l/r(t)]/dt at integral
values oft into the series (44), it follows for X = 1, 2, · · · that
rN,.(z) = aJ,.(z) - (-1)>.CJJ_>.(Z)
ax
ax
= 2J,.(z) (log~+
2
(45)
a)- (~)->.I:
2
n-o
- (~)' :, H+
z)>.
oo
- (-2 I:
n-1
1
- -1 - +
{ n+X
n+X-1
>.
(-1)"
n!(n
(~) "
2
(X - n - l)!
nl
2
~ 1 + · · · + 1}
(2z)2" .
+ X)l
+ .. · + 1 +n1- +n-1
-1- + · · · + 1}
LEGENDRE FUNCTIONS
and for ).
=
501
0 that
irNo(z) = 2Jo(z) (log;
+ C)
2
)
1
"° ( 1t ( ~) {1-+--+···+
1.
-2L--=n - 1
n
2
(n !)
"
n-1
2
From the last expansion we get an idea of the singularities which
can occur in connection with the solutions of Bessel's equation.
With the exception of the point z = oo, which is an essential
singularity for all nonidentically vanishing solutions, the origin is
the only point at which the solutions of Bessel's ·equation can be
singular. If ). is not an integer, the most general solution can be
represented with the aid of the functions Jx(z) and J _x(z) and thus
at the origin can exhibit singularities only of the form i and z->.,
respectively. If). = n is an integer, besides a pole of order n at the
origin the solutions can have only a logarithmic singularity of the
form z" log z. For, every solution may be expressed as a linear combination of the functions Jn(z) and Nn(z), and these have rio other
singularities.
The Bessel functions J n(z) with integral index n, in particular,
are just those solutions which remain regular at the origin.
§3. Legendre Functions
The Legendre functions and the higher spherical functions ob1
tained from them by differentiation have already been investigated
as functions of a real variable and many of their properties have been
derived. Now we shall develop integral representations for these
functions by going over to complex variables z = x + iy; at the
same time we shall try to find the remaining solutions of Legendre's
equation. In the process it will become clear that the parameter n
in the Legendre function P .,h) need not be restricted to positive
integral values.2
1. Schliifii's Integral. From the representation (Ch. II, §8)
p n(z) = _1_ er (z2 - l)n
2"n! dz"
In Ch. II, §8 and Ch. V, §10, 2.
In connection with this paragraph, see in particular E. T. Whittaker and
G. N. Watson. A Course of Modern Analysis, pp. 302-336, 4th ed., Cambridge
University Press, Cambridge, 1927.
1
2
502
VII. SPECIAL FUNCTIONS DEFINED BY
~IGENVALUE
PROBLEMS
of the nth Legendre polynomial, one immediately obtains the expression
f
(f - 1)" dr
-Pn(z) = -12ri c 2"(r - z)n+1
(46)
for arbitrary complex z, using Cauchy's integral formula. Here the
path of integration c in the plane of the complex variable r = ~ + i.,,
is taken in the positive sense about the point t = z. From this
expression, which is due to Schlafli, important consequences and
generalizations can be derived. First we observe that Legendre's
equation
!, ((1 - z')d:.·) + n(n + l)P. = 0
is a direct consequence of the integral representation (46).
differentiation under the integral sign gives the expression
l (r--.:::((n +
~~! l (r--.:::
~..~2!
=
2)(1 - z') -
(2(n
= n
+
Indeed,
2z(t- z) + n(t - z)') dt
l)l"(I - z) - (n
+ 2)(f -
1)) dl"
+ 1J !!:_ ((f - 1r+i) ds
2ri2"
c dt
(t - z)n+2
for the left side of the differential equation. Because the path of
1
2
2
integration is closed and (t - 1r+ /(r - zr+ is single-valued, this
expression vanishes identically in r. We can use this direct verification of Legendre's equation in order to extend the definition of
P n(z) to arbitrary values of n which are not positive integers. Eviden.tly Schlafti's integral (46) must represent a solution of Legendre's
2
1r+1/(r - zr+ returns
equation for arbitrary n whenever
to its starting value as the path of integration is traversed once;
for example, when the path is closed on the Riemann surface of the
integrand. In this case, however, the function Pn(z) will not in
general be an integral rational-not even a single-valued analyticfunction of z. Paths of this kind can be obtained in the following
way: We cut the r-plane along the real axis from -1 to - oo and
along an arbitrary path from the point 1 to the point z; the z-plane
is cut analogously. For C we choose a closed curve, oriented in the
er -
503
LEGENDRE FUNCTIONS
positive sense, which encloses the points r
excludes the point f = -1. The function
(47)
P.,(z)
=
1
2'1ri
= z
and
r
+1 but
J2 (f c
11
1)"
(t - z)*1 dt
so defined, which is single-valued in the cut z-plane, is also called
the Legendre function of index JI. It satisfies Legendre's differential
equation
((1 - i)u')'
(48)
+ J1(J1 + l)u = 0
and may be characterized as that uniquely determined solution which
1
for z = 1 is finite and has the value
P,(1)
=
1.
This property immediately becomes evident from the integral representation when z is allowed to approach 1. Since the above differential equation is transformed into itself by the substitution of
1 for J1, we have the identity
- JI P.(z) = P _,_1(z),
which is not easy to verify by calculation.
As can be seen from the representation, P,,(z) satisfies the recursion
formulas
+ l)P.,(z),
z(2J1 + l)P.,(z) + J1P.,_i(z)
P~+i(z) - zP~(z) = (J1
(49)
(11
+ l)P.,+1(z)
-
=
0,
the second of which was derived in Ch. II, §8, 3 for integral J1.
2. Integral Repr.esentations of Laplace. If the real part of z is
positive and z ~ 1, as we shall now assume, we can take for C the
circle of radius I yz 2 - 1 I about the point z. This circle has the
2
properties required above, as we see from the inequality I z - 1 1 <
I z + 1 II z - 1 I which holds for ffi-e (z) > 0, z ~ 1. If we introduce
z2 - 1 ei"',
the real variable of integration 'P by setting r = z
i 'P I ~ '11", Schlafli's integral yields at once the first integral repre-
+v
1 In fact, the second integral Q. , which will be defined in subsection 3,
becomes logarithmically infinite at z = 1. Hence the same is true of any
integral which is linearly independent of P • .
504
VII. SPECIAL FUNCTIONS DEFINED BY EIGENVALUE PROBLEMS
sentation of Laplace, valid also for z = 1:
11ll" (z + VZ2=1. cos 'Pr d<p,
in which the multiple-valued function (z + vZ2- 1 cos
be determined in such a way that
(50)
Ph)
= 7r-
0
'P )" is to
for 'P = 7r/2 it is equal to z"; here
z" denotes the principal value of z", in particular, a real value for
positive z and real v.
The formula P,, = P -11-1 immediately yields the second Laplace
representation
11r
(51)
P,,(z) = -
7r
o
d<p
(z
+ VZ2=1 cos <p) v+1 •
It should be noted that for those z for which the expression
z + z2 - 1 cos 'P vanishes on the path of integration the first representation fails to be valid only for .,, ~ -1, and the second only for
.,, ~ 0. Therefore, at least one representation defines a singlevalued branch of P,, in the plane (except on a cut along the real axis
from -1 to oo •)
3. Legendre Functions of the Second Kind. The differential equation (48) must also have a second solution which is linearly independent of P,,(z). This solution can be obtained easily from Schlafli's
integral by taking a path of integration different from that used
above. Such a path is given by the path ~ in the form of a figure
eight (see Figure 14, page 480), provided this path does not enclose
the point z. Legendre's equation is again satisfied by the analytic
function Q,,(z) which is defined by the integral
v
(52)
o
Q
=
-1
,, z
4i sin '117r
12"! <r !(
(z -
1),,
dr
r)v+l
where arg (t2 - 1) = 0 on the intersection of ~ with the positive
real axis at the right of r = 1. It is called a Legendre function of the
second kind, and is regular and single-valued in the z-plane which
has been cut along the real axis from + 1 to - oo. At first we explicitly assume that in this representation .,, is not an integer, since
otherwise the normalizing factor 1/sin v7r would become infinite.
In case the real part of .,, + 1 is positive, we can contract the path of
505
LEGENDRE FUNCTIONS
integration and write (compare the calculation on page 483)
(53)
=
Q,,(z)
1
2..+1
Ll
<1 -
rr
1 (z - r)*l dr.
This formula is now applicable for non-negative integral v also.
From representation (52) it is easily seen that Q,, becomes logarithmically infinite at z = 1 and z = -1, since the path must pass through
the lines joining z to the points + 1 and -1.
An integral representation analogous to the Laplace integrals for
P,,(z) holds also for the functions Q,,(z). In the above integral (53)
we set
r
=
e"vz+l - vz=-i
e"vz + 1 + vz - 1
and consider first real z
>
(54)
1
Q.,(z) =
1. After some calculation, we obtain
ao
d'{)
(v > -1),
0 (z +
2___1_c_o_sh_'P_)"--:-::+l
v--,-z
in which the value of the integrand is determined in the same way
as above.
4. Associated Legendre Functions. (Legendre functions of higher
order.) For the Legendre functions of higher order, which we define
by means of the equations
P.,,1a(z) = ( l Q,,,1a(z)
=
(1 -
z2)"'2
tz:
z2)M2
dd"1a Q,(z}
P,,(z),
.z
(see Ch. V, §10, 2, p. 327), we likewise obtain integral formulas by
differentiating Schlafli's integral representation (47) and then substituting r = z +
z~l ei"' (see subsection 2). Explicitly we
write
_ ( .)h{v
l){v + 2) · · · (v
h)
P ,,,,,, ( z) - i
v
+
(55)
+
r
· i'lr (z +~cos <P)" cos h<P d'{).
From this expression we see immediately, for example, that all the
Legendre functions P,,,1a(z) with h > 0 vanish for z = 1.
506
§4.
VII. SPECIAL FUNCTIONS DEFINED BY EIGENVALUE PROBLEMS
Application of the Method of Integral Transformation to
Legendre, TchebychejJ, Hermite, and Laguerre Equations
The theory of the Legendre equation, as well as that of the orthogonal functions discussed in Chapter II, can be developed with the
methods of the integral transformation described in §1. In the following we shall sketch briefly how this is done.
1. Legendre Functions. In the Legendre equation
(56)
L[u] = (1 - z2 )u" - 2zu' = -A(A + l)u
we are led, by means of the transformation
=
u(z}
l
K(z, t)v(t} dt,
to the condition
l{
2
(1 - z )Kz;e - 2zK,,
+ X(X + l)K}v(t} dt
= 0.
If we require the transformation kernel to satisfy the differential
equation
(57)
+ f(fK)rr = 0,
2
(1 - z )K•• - 2zK,,
of which the function K = 1/ yl - 2zf + f 2 is a solution, and
replace L[KJ by -f(fK)rr in the integral, we can transform the
resulting integral using integration by parts; we obtain for v(f) the
differential equation
+
f(vf)" - A(A
which has the solutions v
tegrals
1
Px(z} = 2ri
=
l)v = 0,
e' and·v = r->--1.
J yl -
r-x-1
C1
(58)
1
Q>.(Z) = 4i sin rX
2zf
1 v1 -
+f
2
r->--1
C2
2zf
This leads to the in-
df,
+ f 2df
where C1 and C2 (see Figures 16 and 17) denote the curves on the
Riemann surface of the integrand.
By means of the transformation
r = z + vz
2
-
1 cos <p
APPLICATION OF METHOD OF INTEGRAL TRANSFORMATION
507
and a suitable deformation of the path of integration, we at once obtain the Laplace integrals (51) and (54)
PA(z)
=!
'Ir
lr
(z
£
00
QA(z) =
+ yz2 -
1 cos
cp)-'>--l
dcp,
1 cosh
cp)->.-l
dcp
0
(z
+ v z2 -
(A
<-
1).
0
Figure 17
Figure 16
The kernel selected above,
K(z,
r)
=
1
Vl -
2zf
+r
2'
as well as any other which satisfies equation (57), is a generating function of the Legendre equation. For, the coefficient un(z) in the series
expansion
00
K(z, r)
=
E Un(z)r
0
of such a kernel is an integral of the above form,
1
Un(Z ) = 27r'l
f
K(z, t)
~ df;
since the path of integration is closed, un(z) is a solution of equation
(56) for X = n.
2. Tchebychefl Functions. In the case of Tchebycheff's equation
(59)
2
L[u] = (1 - z2)u" - zu' = -X u,
we take K to be a solution of the differential equation
(60)
for example, we take K(z, f) = 1 -
r /(1 2
2zt
+
f
2
),
which leads to
508
VII. SPECIAL FUNCTIONS DEFINED BY EIGENVALUE PROBLEMS
solutions of the form
P()-~f __!___~
z - 211'"i
>.
(61)
Q( ) . i
"z =
2;i
1 - 2zr
Ct
f
+rr
-H
1 - r2
1 - 2zr +-r r
c2
->.-1
dr,
d
r.
Here C1 and C2 are closed curves on the Riemann surface of the
integrand and enclose the zeros of the denominator
(see Figure 18).
Application of Cauchy's integral
theorem yields
(62)
P,.(z)
=
Q,.(z)
(z
+ yz
vz
(z -
2 -
1)\
2
lf'.
-
f:::'\Ct
~
0
Gl
Figure 18
The sum
T,.(z) = ;,. (P,.(z)
+ Q,.(z))
= (z
+ Vz2=1/ ~
(z -
vZ2=1)",
which is represented also by the integral
T ( )
"z
1 1
=
f
1 -
2" 27ri c 1 - 2zr
f
_,._1 d
r
+r r
2
in which C now encloses both points S-1 and S-2, goes over into the
n-th Tchebycheff polynomial for A = n.
3. Hermite Functions. In the case of the Hermite equation
(63)
L[u] = u" - 2zu' = -2Au
we require K to satisfy equation
(64)
Ku - 2zK. + 2tKr = 0,
2
which has the function e2zr-r as a solution. If for C we take one of
the curves represented in Figure 19, we obtain the solutions
P,.(z)
(65)
1
= -.
11'"1.
1
Q>.(Z) = -.
1rt
j
j
C1
e-r2+2•r
,.>-+1 dr,
l
e-r2+2sr
,.HI
Ct
l
dt.
509
APPLICATION OF METHOD OF INTEGRAL TRANSFORMATIO N
Their arithmetic mean,
HA(z) = !(PA(z)
in other words, the integral
=
HA(z)
+ QA(z)),
J rH•
e-r2+211r
1
dt,
27ri c
where C is the loop in Figure 21, becomes the Hermite polynomial
H.,h) for X = n.
If 9le ().) < 0, we can contract the path of integration to the
origin and obtain as solutions-up to factors independent of zthe integrals
e-rsH•r
(66)
c,
f'AH df
0
100
E
C2
and
oo
(67)
~z+tsr
loo e
fHl
L[u]
=
0
;
Figure 19
df.
Correspondingly, in the Laguerre equation
4. Laguerre Functions.
(68)
"
zu"
+ (1
- z)u'
=
-Xu
we impose on K(z, t) the partial differential equation
(69)
zK..
+
(1 - z)K1:
+ tKr
= 0
and arrive at integrals of the form
(70)
In the choice of the path of integration C, it must be noted that the
integrand has an essential singularity at the point r = 1. In particular if we take for C the curve shown in Figure 20, the integral
-11r1c1-r>
1
r->.-l df
e .
~ (z} = - .
(71)
1 1-r
2ri c
represents solutions which are essentially identical with the Laguerre
polynomials in the case X = n.
By means of the transformation
U -
r
- 1-
r'
510
VII. SPECIAL FUNCTIONS DEFINED BY EIGENVALUE PROBLEMS
(71) acquires the form
(72)
LA(z) =
J
1
e-uz
-.
A+i (1
2ri cu
+ u) A du;
now C denotes the path in Figure 21.
Furthermore, we note that, as in the case of Legendre's equation,
every solution of the partial differential equation associated with
each equation discussed here can be considered as a generating function of a family of solutions of the original differential equation. In
Figure 20
Figure 21
particular, the special. kernels we have used define, by their power
series expansions, the Tchebycheff, Hermite, and Laguerre polynomials.
§5.
Laplace Spherical Harmonics
The Laplace spherical harmonics Yn(8, 'P) were introduced in Ch.
V, §8, p. 314 as the everywhere-regular eigenfunctions, corresponding
to the eigenvalues A = n(n
1), of the differential equation
+
(73)
A*Y
+ AY
=
1
l
-.-2- Y.,., -1- --:--- (sin 8Ys)s
sm 8
sm 8
+ AY =
0
on the sphere. We see that the functions rnYn = Un are then
polynomials in the rectangular coordinates x, y, z which satisfy the
differential equation AU = 0 and are homogeneous of degree n.
Conversely, every integral rational solution Un of the differential
equation AU = 0 which is homogeneous of degree n becomes a
Laplace spherical harmonic when divided by rn. Since a polynomial which is homogeneous of degree n has (n + l)(n -:- 2)/2
coefficients, and since the condition AUn
0 determines (n - l)n/2
linear homogeneous relations between these coefficients-fo r, AUn is
homogeneous of (n - 2)-nd degree-, the Un have at least
=
(n
+ l)(n + 2)
2
(n - 1)n
2
=
2n
+1
511
LAPLA CE SPHER ICAL HARMONICS
2n + 1
indepe ndent coefficients; therefo re there must be at least
order.
n-th
of
nics
harmo
cal
linearl y indepe ndent spheri
condiIn this paragr aph it will be shown that the aforem ention ed
1
2n
ly
precise
exist
there
that
so
ndent,
tions are mutua lly indepe
be
will
It
order.
n-th
of
nics
harmo
cal
spheri
linearl y indepe ndent
eigenfuncshown also that these functio ns Y n really repres ent all the
A =
values
the
that
hence
and
m,
proble
alue
tions of our eigenv
ns
1) repres ent all the eigenvalues. · Finally , these functio
n(n
funcdre
will be expressed explic itly in terms of associ ated Legen
p. 327.
2,
§10,
V,
Ch.
and
§3
from
r
familia
are
we
tions, with which
We begin with the last point.
1 Spheri cal Harmo nics of n-th Order.
1. Determ ination of 2n
familia r
We again obtain special spheri cal harmo nics by means of the
tution
substi
trial substi tution Y(8, <P) = p(<P)q(8). We make this
with
n
ntiatio
in equati on (73) for A = n(n + 1), and denote differe
a
by
8
to
t
respec
respec t to 'P by a prime, and differe ntiatio n with
dot; equati on (73) then becomes
+
+
+
(
(sin 6q}' sin 6
p"(<P)
- n n
q
P(<P) == -
in which p must be a consta nt.
(sin Oq)'
. 8
+ l) sm
=
2
-p,
Thus we obtain for q the equati on
+ ( n(n + 1) sin 6 - s~ 8)q =
0,
that a
in which the param eter p must be determ ined in such a way
tuting
Substi
11".
=
8
and
0
=
6
for
solutio n exists which is regula r
prime,
a
by
z
to
t
respec
with
n
z = cos 8, and denoti ng differe ntiatio
this equati on immed iately becomes
+ (n(n + 1)
!:._ z2) q = 0
1
and z =
with the bound ary condit ion of regula rity for z = + 1
from
form,
nt
differe
hat
somew
a
in
us,
to
r
familia
-1 ; this proble m is
2
p = h and
ns
solutio
the
know
y
alread
We
327.
Ch. V, §10, 2, p.
h-th order, and
q = Pn,11.(z), where Pn,11.(z) is the Legen dre functio n of
2
0, 1, 2, · · · , n.
values
the
e
assum
can
h
n(z);
/dl)P
(d"
P n,h = (1 - z2)hl
2
hep; since
Since p"(<P) + h p(cp) = 0, pis given as a,,, cos hep + b,,, sin
Y = pq, \Ve at once obtain
Y(O, <P) = (a,,, cos hep + b,,, sin hcp)Pn,,,,(cos 8)
((1 - i)q')'
-
512
VII. SPECIAL FUNCTIONS DEFINED BY EIGENVALU E PROBLEMS
as a solution of (73).
Therefore the function
+
t
+ bn,h sin hq;)Pn,h (cos 8)
is a spherical harmonic of dimension n depending upon 2n + 1 ar(74)
Y n = a;.o P n (cos 8)
(an,h cos hq;
bitrary linear parameter s. It will soon be apparent that this function is the most general possible. The functions cos hq; P n,h(cos 8),
sin hq; P n,h( cos 8) are all linearly independe nt, since they are mutually
orthogona l; they will be called symmetric spherical harmonics of
order n.
2. Completeness of the System of Functions. From earlier theorems we see at once that the 2n
1 functions cos hq; Pn,h(cos 8),
sin hq; P n,h( cos 8) form a complete orthogona l system of functions on
the sphere. The functions sin hq;, cos hq; form a complete system in
the variable <p, and the functions P n,h(z) for every h form a complete
system in z because the system of all eigenfunctions of an eigenvalue
problem is always complete (see Ch. VI, §3, 1). In order to prove
that the system of functions is complete, we need only recall the
theorem of Ch. II, §I, 6, which contains a general rule for constructing a complete system of functions in two variables from two complete systems in one variable.
From this result it follows immediate ly that equation (73) has
no eigenfunctions other than those indicated, and thus can have no
eigenvalues except the values n(n
1). All the questions which
were raised above have thus been answered. (Note that this constitutes a transcende ntal proof of the algebraic fact that there exist
precisely 2n
1 linearly independe nt functions y n .)
This fact may, of course, be proved easily from the standpoin t of
algebra alone. Consider any polynomial u of x, y, z which is homogeneous of degree n: u =
arat xryazt (r + 8 + t = n). Then
every coefficient arst may be represented, up to a constant factor, as
a derivative of the form anu/a£ays az'. If .1u = 0, every derivative
of the form aum/axaayfJaz-r can be written in such a way that differentiation with respect to x occurs no more than once; we use the
differential equation Uzz = - Uw - Uu to eliminate all derivatives
of u with respect to x of order higher than one (for example,
3
iJ u/ax 2ay = - iJ 3u/iJy3 - iJ 3u/iJz2iJy). Thus, if .1u = 0, all coefficients of u are linear combinations of the 2n + 1 coefficients
ao,o,n, ao,l,n-1 , • · • , ao,n,o; a1,o,n-l , a1,1,n-2, " • · , a1,n-l,O, Which may
be chosen arbitrarily .
+
+
+
L
513
LAPLACE SPHERICAL HARMONICS
3. Expansion Theorem. Since the functions (74) represent all the
eigenfunctions of our problem, our earlier theorems (see e.g. Ch. V,
§14, 5) imply that every function g(8, 'P) which is continuous together
with its derivatives up to the second order on the sphere may be
expanded in an absolutely and uniformly convergent series in terms
of the spherical harmonics
g(8, 'P) =
t
n-0
[an,o P n (cos 8) + ±(an,h cos h'P + bn,h sin h'P)Pn,dcos8)]
h-1
where, in view of the formulas of Ch. V, §10, p. 327, the coefficients
an,0' an,h' bn,h are determined from the relations
'
(75)
Lr fr g(8, 'P)Pn (cos 8) sm. 8 d8 d'P,
an,o
+1
4 ; - rJo
= 2n
an,h
+
~ (n + h) !
= 2n
Lr1
1rl
.
1 (n - h) !
+
0
1 ((n - h)
bn,h = 2n
h) !I
- -
27'"
n+
r
r
-11"
r
o
•
g(8, 'P)P n,h (cos 8) cos h'Psm 8 d8 d'P,
(
. 8 d8 d'P·
. h'P sm
)Pn,h (COS 8) Sm
g 8, 'P
Extension of this result to more general functions g(8, 'P) need not
concern us here.
4. The Poisson Integral. We can now write the solution of the
boundary value problem of potential theory for a sphere of radius
1 with boundary values g(8, 'P) explicitly in the form
u
=
t
n-0
rn [an,oP n (cos 8)
t (an,1 cos h'P + bn,h sin h'P)P
+ h=l
n,h
(cos 8)].
If we introduce the integral representations (75), we can interchange
summation and integration since the convergence is uniform for
r ~ ro < 1. The summation may then be carried out in closed
form; in doing this it is easiest to assume at first 8 = 0, 'P = 0, and then
to note that the result must hold for an arbitrary point 8, 'P, since
any point on the sphere may be chosen as the north pole.
Since Pn(l) = 1, Pn,1(1) = 0 (h = 1, 2, · · · , n), we obtain
4-A-u(r, 0, 0) = { { {
f,
(2n
+ l)r"P. (cos 8)} g(8, <P) sin 8 a8 d<P;
here the sum may be obtained in closed form by using the relation
1 - h2
00
L (2n + l)hnPn(Z) =
2 3 2
n=O
(1 - 2hz
+h
) 1 '
514
VII., SPECIAL FUNCTIO NS DEFINED BY EIGENVA LUE PROBLEM
S
which can be easily derived from the equatio n of definition
co
L
hnPn(z)
n=O
=
(1 - 2hz
+h
2
)-
112
with the aid of the recursion formula (49). After carrying out the
summat ion and again thinking of the pole of the sphere as displaced,
we can write the result with complete general ity in the form
=
411"u(r 8, '{),)
(76)
2
(l - r)
Lrl
r
0
r
1
g(8', '{) ) sin 8' dfJ' d'P'
{r2-2r[ cosfJco s8'+sin 8sin8'c os('P- 'P')] +1} 312 •
This so-called Poisson integral expresses the potentia l function
at interior points in terms of its bounda ry values; it no longer contains any explicit reference to the spherical harmonics. In Volume
II we shall return to this integral in the framework of potentia l
theory.
5. The Maxwell-Sylvester Represe ntation of Spherical Harmon ics.
An entirely different represen tation for the spherical harmonics,
connected with the physical meaning of a potentia l, was given by
Maxwell. 1 In this section we shall investig ate the properti es of
spherical functions in connection with Maxwell's basic ideas and a
supplem entary remark of Sylvester, and thus arrive at a new developm ent of our theory.
We begin with the potentia l 1/r = 1/v' x 2 + y 2 + z2 corresponding
to a unit mass concent rated at the origin, and note that every derivative v = anu/axCla'/az"t (n = a + {j + -y) of a potentia l function
u is again a solution of the potentia l equatio n Av = 0, since from
Au = 0 we obtain by di:fferentiat ion
a
au
0 =-Au = A-,
ax
etc.
ax
Hence, if a, b, c are constan ts, the function
1 A Treatise
on Electrici ty and Magnetis m, Vol. 1, pp. 179-214, 2nd ed.,
Clarendo n Press, Oxford, 1881.
515
LAPLACE SPHERICAL HARMONICS
Using the symbolic linear form
1s also a potential function.
a
a
a
L=a-+b-+c-,
az
ay
ax
we can write it in the form L ! or in the form
r
a i);'
where a ·= v' a 2 + b2 + c2 and a/ av stands for differentiation in that
1
direction v whose direction cosines are proportional to a, b, c. Physically, this potential corresponds to a dipole of moment a and direction v. More generally, in the expression
a"!
(77)
u
= C av1av2 . .r . av. = CLi'Li • · • L.
!r
we obtain a potential which corresponds to a "multipole" with the
axes v1 , v2, · · · , vn. The Li denote linear forms in the operators
a/ax, a/ay, a/az, and their coefficientsai' bi' c, define the axis directions v 1 • It is easily seen that the potential u has the form
(78)
where Un is a homogeneous polynomial of degree n in x, y, z. The
function un itself satisfies the potential equation ~.u n = 0, as is
seen from the following general theorem: If u(x, y, z) is a solution o.f
2
2
2
the potential equation, then (I/r)u(x/r , y/r , z/r ) is also a solution. 2
For r = 1, such functions Un(x, y, z) according to our earlier definitions(Ch. V, §9, 1) are spherical harmonics of order n.
Since each of the n directions occurring in (77) is determined by
two parameters, and an additional arbitrary constant factor appears
in the potential u, there are 2n + 1 arbitrary constants in all. It
may be conjectured, therefore, that all the spherical harmonics of
order n can in fact be represented in the form (77). We shall prove
1 If complex values are admitted for the a, b, c, necessary precautions must
c2 = 0.
b2
naturally be taken in·the case of those triples for which a 2
2 The proof of this theorem is obtained immediately from the polar coordinate form of the potential equation (see Ch. IV, §8, 2).
+ +
.
I
516
VII. SPECIAL FUNCTIONS DEFINED BY EIGENVALUE PROBLEMS
this rigorously by first representing the 2n + 1 linearly independent
symmetric spherical harmonics Pn,h(cos 8) sin ]up, Pn,h(cos 8) cos ]up
in terms of multipole potentials. Subsequently it will follow that
every n-th order spherical harmonic is given by a sum of multipole
potentials. Finally, we shall show that every sum of sevetitl such
multipole potentials is equal to the potential of a single multipole,
which can be obtained by means of a simple geometric construction.
The 2n + 1 symmetric spherical harmonics of subsection 1 are
obtained easily by considering symmetric multipoles. Let n axes
with the directions 111 , 112 , • • • , Pn be arranged symmetrically in the
x, y-plane in such a way that any two consecutive axes form the
angle 211'/ n with each other. If we set
an!
- - -r - - = Un =
av1iJJ12 ••• avn
(79)
unT-2n-1
and note that the left side is invariant with respect to rotations of
the sphere about the z-axis through the angle 211'/n, we see at once
that the n-th order spherical harmonic
UnTn+l = Unr-n = Yn(8, ~),
which certainly does not vanish identically,1 has the period 211'/n
when considered as a function of ~· By subsection 3 every spherical
harmonic of order n permits the representation
n
L: (an,h cos h~ +
h=O
It follows that Yn(8,
~)
bn,h sin h~)Pn,h(cos 8).
has the form
Yn(8, ~) = [an,n cos~+ bn,n sin n~] Pn,n(cos 8) + an,o Pn,o(cos 8)
(80)
= a cos n(~ -
~o)Pn,n(cos
8) + f3Pn,o(cos 8}:
The two terms in (80) are the only ones which are periodic in~ with
period 211'/n. The term Pn,o(cos 8) can be expressed separately in
terms of a derivative of r-1• We have
(80')
1
519.
That no multipole potential can vanish identically will be proved on page
LAPLACE SPHERICAL HARMONICS
517
The fact that the right side in (80') is a multiple of r-n-I Pn,o(cos 0)
follows immediately from the remarks made above. The calculation
of the constant factor can be carried out by induction, using the
recursion formulas (49).
In order to obtain a multipole representation for the remaining
spherical harmonics of n-th order, we note that because of (80) the
potential Un can be decomposed as follows:
y)~~)r-•-'
u. = f(:t,
where f(x, y) = a cos n(<p - <po), f(O, 0) = 0. We replace n by h in
this expression and then differentiate n - h times with respect to z.
The resulting potential function Un,h again has the form
Un,h
= f(x,
y)g(~)r-n-I,
from which we conclude that then-th order spherical harmonic
must have the form a cos h(<p - <po) w(O).
l it necessarily has the form
(81)
Therefore by subsection
const. X cos h(<p - <po)P n,h(cos 0).
Conversely, since one of the axes can be chosen arbitrarily, every
function of this family is obtained by our procedure.
Since according to subsection 2 every spherical harmonic of order
n may be expressed as a sum of 2n + 1 spherical harmonics of the
form (81), it follows at once that we obtain every n-th order spherical
harmonic by forming sums of multipole potentials
(82)
u
:L:
i+k+l=n
= ·
a"~
aikl
.
:
ax•ay az
z•
That, conversely, every such sum is a spherical harmonic of order
In fact, any particular spherical
harmonic is represented infinitely often when we let the coefficients
aikl run through all possible values.
We shall now elaborate this
point.
We first prove that every sum of the above form is the potential
n is obvious according to page 510.
518
VII. SPECIAL FUNCTIONS DEFINED BY EIGENVALUE PROBLEMS
of a single multipole with suitable axes. To do this we make use of a
symbolic notation, considering the homogeneous polynomial of n-th
degree
and writing our potential in the form H/r, in which we replace the
r in H by the differentiation symbols a/ax, a/ay, a/az.
Since with this meaning of t, 11, r the function (f + 11 2 + t 2)/r is
identically zero, we have H/r = H1/r provided the difference H - H1,
as a polynomial in the indeterminates t, 71, r, can be expressed in the
form Q . (t2 + 71 2 + r2), where Q denotes a homogeneous polynomial
of (n - 2)-nd degree in t, 71, r.
At this point we need a simple theorem employed by Sylvester :1
Given any homogeneous polynomial of n-th degree H(t, 11, t), we can
determine n linear forms L1 , L2 , · · · , Ln and .a polynomial Q(t, 11, t)
of degree n - 2 in such a way that a relation of the form
variables~' 11,
H
= C • L1L2 · · · Ln
+ Q · (f + 11 + t
2
2
)
holds. If H is real, the linear forms L1 , L2 , · · · , Ln are determined
uniquely, up to constant factors, by the condition that their coefficients be
real. The proof of this theorem, as well as the geometric characterization of the forms Li , will be postponed to the end of the section
so as not to interrupt the train of thought~ Our assertion concerning the representation of the potential (82) by a single multipole
follows at once from Sylvester's theorem. For, if Pi denotes the
axial direction perpendicular to the plane Li = 0, we obtain
which furnishes the desired representation.
Our theory is now outlined in its essential points. We shall give
our discussion a somewhat different turn which avoids dependence
on the results of subsections 1 and 2 and puts more stress on the
1
J. J. Sylvester, Note on Spherical Harmonics, Phil. Mag., Vol. 2m, 1876,
pp. 291-307 and 400. Collected Mathematical Papers, Vol. 3, pp. 37-51, Cambridge University Press, Cambridge, 1909.
519
LAPLACE SPHERICAL HARMONICS
purely algebraic. nature of our theorems, although at the sanie time
it relaxes the connection with the explicit representations. First we
remark that two functions H/r and Hi/rare identical if and only if
the difference H*(~, 71, t) = H(~, .,,, t) - H1(~, .,,, t) is divisible by
2
2
2
~ +11 + r • As already pointed out, the first part of the assertion
is obvious. In order to prove the second part, we must show that
a relation H* /r = 0 implies that the homogeneous polynomial
2
2 1
NowbySylvester'stheorem
H*(~, 11, t) is divisible by f + 11 .+ r •
2
2
2
t ),
Q* · (~
H* = C·LtL: · · · L!
(83)
+ ., +
+
where Lf, L:, · · · , L ! denote linear forms which, in the case of
real H*, may be supposed real. If one of the linear forms L~ vanishes identically, our assertion is clear. However, if none of the
lin-ear forms vanishes identically, we have
H*
!
r
= C ·Li L: · · · L! ! = C
r
o" !
* *
0111 0112
r
*,
• • • 011 n
and the multipole potential on the right side, because of the singularity at the origin, can vanish in the whole space only if C = 0. Otherwise, for suitable m, 0 ~ m < n, we would have
am!
r
0111 • • • 011m
=
Vm ~
0,
OVm
=
O;
011m+l
hence Vm would have to have constant values on every parallel to
the axis 11m+i ' which is impossible because of the singularity at the
origin. We thus have
H*(~, .,,, t) = Q*(~, 11, t). (~2
+
.,,2
+
t2),
which is what we wished to prove.
Clearly, every homogeneous function H(~, 11, t) of degree n can
be expressed in one and only one way in the form
Here G,. denotes a homogeneous function of degree n in 11, t alone,
Gn-1 a homogeneous function of degree (n - 1), and Q a homogeneous
1
See the work by A. Ostrowski referred to on p. 521, footnote 2.
520
VII. SPECIAL FUNCTIONS DEFINED BY EIGENVALUE PROBLEMS
function of degree (n - 2). The difference between two functions
H(~, 11, s) and H(t, ,,,, s) of degree n is divisible by
+ 112 + r2 if
and only if the associated functions Gn , Gn-1, Gn, Gn-1 satisfy the
relations
t
identically. Since we have 2n + 1 ooefficients at our disposal in
the functions Gn and Gn-1, the lemma just proved shows that we
have precisely 2n + 1 linearly independent potentials in the form
H / r. Hence we obtain every spherical harmonic of degree n as a
sum of potentials of multipoles. In order actually to obtain the
representation of the spherical functions in this form one must use,
in addition to this pure existence proof, an argument analogous to
that carried out above.
Finally we shall prove Sylvester's theorem using a simple idea
from algebraic geometry. By Bezout's theorem, the n-th degree
cone H(t, 11, s) = 0 in t, 11, s-space intersects the absolute cone
2
t2
r2 = 0 in precisely 2n edges; all multiple intersections are
correctly weighted. We connect the 2n edges by means of n planes
given by the equations
+ ,, +
Li(t, ,,,, s) = ait
+ bi,,, + CiS
= 0 (i = 1, 2, ... , n)
in such a way that each plane contains two of the edges and every
edge is accounted for once. Multiple edges appear according to their
multiplicity. 1 We now consider the bundle of cones of n-th order
'AH
+ µL1L2
· · · Ln = 0
1
We can make the meaning of this rule more precise without referring to the
difficult general algebraic elimination theory: We uniformize the form ~2 +
11 2 + r 2 = 0 by setting
(*)
1 -
t2
~ = 1 + t2 '
r=i =
v-1.
A homogeneous function H(~, 11, t) of degree n is then transformed by (*)into
a rational function H*(t) of degree 2n, whose zeros determine the common
edges of the cones H(~, 11, t) = 0 and ~2 + 11 2 + f' 2 = 0. We shall say that a
common edge of these cones counts k-fold if on it H*(t), as a function of t,
has a precisely k-fold zero. The linear forms Li, /J2, · · · , Ln are now to be
chosen in such a way that every k-fold curve of intersection of the cone H =
0 with the absolute cone is also a k-fold edge of intersection of the family of
planes L1 , L2 , · · · , Ln = 0. That this rule may be realized in every case
is easily seen.
521
LAPLACE SPHERICAL HARMONICS
containing the two parameters X andµ. Every cone of this bundle
intersects the absolute cone in the 2n given fixed edges. Selecting
any one edge of the absolute conic not one of the above fixed 2n
edges), we determine the ratio X/ µ in such a way that the n-th degree
cone
'AH
+ µ.L1L2 · · · Ln = 0
passes also through this edge, which is certainly possible and which
lf'ads to a value for X/ µ. different from zero and infinity. The new
cone of degree n then has more than 2n lines of intersection in common with a cone of second degree, which is impossible unless it
contains .the cone of second degree completely. This case arises if
and only if the left side of the equation contains the expression
2
2
2
~ + 11 + r as a factor,1 i.e. if
XH
+
µL1L2 · · · Ln = Q ·
(f +
11
2
+ r
2
).
2
The proof of Sylvester's theorem is now complete. At the same
time a simple geometric interpretation is given for the axes of the
multipole associated with a spherical harmonic.
Concerning reality considerations, one must note that although
all lines of intersection are imaginary for real H, they are pairwise
complex conjugates; thus there is just one way of projecting them
onto n real planes.
1 The first part of the assertion is obvious. The second is proved most
simply by writing the given form, in accordance with (84), as
+ 11 + t )Q(~, 11, f).
Now if 11, r is any pair of values for which ,,, + t r!: 0, the two equations
0 == Gn(11, t) + V -(11' + r )Gn-1(11, !")
Gn(11, t)
+
~Gn-1(11,
t)
+
(~ 2
2
2
2
2
2
and
0 == Gn(11, !') -
V ~ (11 2 + r2 )Gn-1(11,
!')
hold simultaneously. We conclude at once
Gn(11, t) == Gn-1(11, t) == 0.
Thus Gn and Gn-1 vanish for all pairs of values ,,,, r with 11 2 + r2 ~ O; hence
they vanish identically in ,,, and r.
2 This algebraic theorem was used by Sylvester, loc. cit., without proof.
Th~ necessity of obtaining a proof for it was indicated by A. Ostrowski. See
Ostrowski, Mathematische Miszellen, I, Die Maxwellsche Erzeugung der
Kugelfunktionen, Deutsch. Math.-Ver. Jahresber., Vol. 33, 1925, pp. 245-251.
;)22
VII. SPECIAL FUNCTIONS DEFINED BY EIGENVALUE PROBLEMS
§6. Asymptotic Expansions
Asymptotic expressions for our functions for large values of the
arguments or parameters are often useful. In the preceding chapter
we considered the asymptotic behavior of Sturm-Liouville and Bessel
functions, restricting the variables to the real domain. In the present section we shall develop· methods of obtaining representations
which depend essentially on the use of complex variables and complex
integration.
1. Stirling's Formula. As a first example of an asymptotic expansion we consider Stirling's formula. It will be derived by a method
which will be used frequently in the future; here, however, complex
integration does not occur.· Fors > 0 we have
r(s
+ 1)
= ["
t8e-t dt
(t = ST)
sa+le- 8
[GO e-B(T-1-logr)
dr
0
1111
sa+le-a
e-s/(1') dr
[
(f(r) =
r -
1 - log r);
0
the integrand has the value 1 for r = 1, for all other values it tends
to zero with increasings. Hence we may expect that only the immediate neighborhood of r = 1 will contribute essentially to the
value of the integral when s is sufficiently large. Accordingly, we
replace this integral by the integral over the interval 1 - E to 1 + E
(0 < E <!),and begin by estimating the error incurred in neglecting
the integrals from 0 to 1 - E and from 1
E to oo •
For ! ~ r ~ 1
we have
+
T -
1 - log
T
= { ( ~ - 1) du 2 { (1 - u) du
Hr - 1) 2 ~ i(r - 1) 2,
and for 1
~ r
~
4
r - 1 - log r = [
(
1-
~)du ~
l [ (u - 1) du
=
Hr -
1)
2
•
523
ASYMPTOTIC EXPANSIONS
In the integrals
1-e
1
e-s/(T) dT 1
0
we replace the integrand by its largest value, assumed at the points
218
We thus obtain
•
E, and this in turn by the upper bound e-u
1
+
1 + 1•
1-E
~
4e-se21s.
l+e
0
However, for r ~ 4 we have r - 1 - log r ~
for s
>
3T
4
-
log
T
> 4T ; hence
4
and setting
=
E
s-215 ,
In order to find an approximation for the integral on the right, we
make use of the relation
=
f(r)
(r
~
2
l)
+ (r
- 1) 8,Y(r),
where ,Y( T) is a function, regular in the interval ! ~ ,, ~ -!, whose
absolute value in this interval does not exceed a finite bound M.
From this relation, for 1 - E ~ r ~ 1 + E, we conclude
-•(,.-1)2/2
e
-M•-1/&
e
-s/(t') < -s(f'-1)2/2 M•-1/;,
<
e
_e
_e
and in addition
From this follows
1
(1
+ O(s-1/5)) [~E e-•u212 du
(I
+
O(s- 115 ))
1
-2 [+E"t;/2
2
e--c dv
_
s - t../•12
-
The notation O(g(s)) has here the same meaning as in Ch. V, §11.
524
VII. SPECIAL FUNCTIONS DEFINED BY EIGENVALUE PROBLEMS
or in other words
(85)
r(s
+ 1)
hence we have
(86)
2. Asymptotic Calculation of Hankel and Bessel Functions for Large
Values of the Arguments. In a similar manner, we can obtain an
asymptotic estimate for the Hankel functions Hi(z) for large I z I
inside an angle - 7r /2
o < arg z < 7r/2 - o by considering the
integral
+
H >.1 ( z) =
r(! - X)(z/2)>.
rir(U
f . "(
e
r
2 _
l)>.-112 d
r
(see §2, 5); here the path of integration is given by the right path
in Fig. 13, z must satisfy -7r/2 < arg z < 7r/2, and log (r 2 - 1) is
assumed real for r > 1. Without changing the value of the integral, we can rotate the cuts in the r-plane, together with the path
of integration which encloses one of them, so that it has the direction
7r/2 - a,rg z. If we then make the substitution
r-l=i
11
z
the u-plane is slit by means of two cuts running horizontally from
0 to 2iz, respectively, out to infinity, and the new path of integration
surrounds the cut which runs along the positive real axis, going from
right to left in the upper half-plane, in the lower half-plane from left
to right. If by u>.-it 2 we understand the uniquely determined branch
in the cut plane which is positive on the lower edge of the positive
real axis, and by (1
iu/2z)'"'- 112 the branch which assumes the
value 1 for u = 0, we find
+
'\)
.
(
H~(z) = r7r~~7r~
ei(z+rX/2-r/4)
1
J e-uu'A-l/2 ( + ~~. )>.-112 du.
1
525
ASYMPTOTIC EXPANSIONS
Now if 9le (A - !) > - 1, we can contract the loop around u = 0
and thus replace the integral over this loop by the integral taken
over the lower edge of the positive real axis from 0 to oo , minus the
integral over the upper edge from oo to 0. But the latter equals the
first integral times e-2ri(Hlt 2>. Hence, using the supplementary formula for the gamma function, we have, after simple manipulation,
(87)
["°
)1/2 i(z-b/2-r/4)
e
-2
( 7rZ
j) o
r(A
H~(z) =
+
(1 + ui
•);\-1/2
e-"u;\-l/ 2
du.
2z
Taylor's theorem with Cauchy's remainder term (denoted by R) gives
us an expression for the last factor:
(88)
(l + ui);\2z
+
112
p
=
I:•==o (A -.,, !) (ui)"
2z
r1 (1 (A -p !) (ui)P
2z lo
t)P-1(1
+ tui);\-112-p dt·
2z
'
observe that we thus obtain a useful estimate for the remainder.
Assuming that ffie (A - ! - p) < 0, which is certainly true for
sufficiently large p, we have for positive u
11 + 2tuiz I > sm. o,
·)'A-1/2-p
1 + t~;
I(
<
erld#n(;\) I
(sin
rl1
o)V\.e(;\-1/2-p)
= Ap'
where AP is independent of z and t. Inserting (88) in (87) and integrating term by term, we obtain
1
_
H;\(z) -
(
2 )l/2ei(.-b/2-r/4)
U
r(A
1fZ
+
(89)
so that
IH, I~A, [v(>-; l)(a{f.' (1 -
1
t)•- dt
Rp = 0(1 z ,-P).
f .-· 1u'- •»1
11
du}I.
526
VII. SPECIAL FUNCTIONS DEFINED BY EIGENVALUE PROBLEMS
Again making the substitution
H~(z)
(90)
2 )1/2
= (wz
T
+
1 = iu/ z, we obtain
e-i(a-b/2-r/4)
r(X
+ !)
. [~(>' :.t)r<>.+ •+t>(-~)' +s.J.
s,, ==
0(1 z 1-P),
and it follows that
J,.(z) = !(H~(z)
1
+
r(>.
+ H~(z))
!)
( 2)
112
;z
~ (X
11-0
r(X + + !)
- !) --c2ZV--
.((-1)'' cos (• - >-; - ~)
(91)
• (
( - l)c..+1>12 Sill
l
JI
JI
w)
A11"
z-2-4
+ O(I z 1-,,_112),
where the upper expression inside the braces refers to even
lower to odd J1.
The first term of the expansion yields
(92)
h(z)
=
V! cos(• - >-; - ~) +
= 1!2
J1,
the
0(1 •I-"'),
which determines the limiting values of Ch. V, §11, 2:
a ao
11" '
3. The Saddle Point Method. In a great many cases a more general
method, called the saddle point method, can be used to determine
asymptotic values of integrals. Consider an integral
L
l/(T)
dr
over a path of integration C on which the real part of f (T) tends to
- oo as we approach either end point. For large positive values of
z the distant portions of the path of integration, in other words the
parts corresponding to large negative real part 9le f (T), furnish a
contribution which becomes smaller as z becomes larger. The path
527
ASYMPTOTIC EXPANSIONS
of integration in the complex plane will now be deformed so that
the part of it which, for large z, contributes significantly to the integral will be contracted to a neighborhood of a point. Thus we
must choose a path of integration for which ffi.e f( T) falls off from a
maximum value as fast as possible on each side. If we set T =
u + iv and think of the real part f.ke f (T) as a surface lying over the
u, v-plane-the surface has negative curvature at every point-our
object will be attained provided it is possible to take the path over a
saddle point or col in such a way that on both sides of the saddle
point it falls off to large negative values of ffi.e f (T) as steeply as
possible. In this case, for large positive values of z only the immediate neighborhood of the saddle point will be of significance.
The curves of steepest descent are given by the orthogonal trajectories of the level curves ffi.e f( T) = const., hence by the curves
9m j(T) = const. At a saddle point the derivatives of the functions
ffi.e f (T) and dni f (T) taken along the curve dni f (T) vanish; therefore
the derivative f' (T) of the function f (T) also vanishes and the saddle
points will be found among the roots of the equation
f'(T)
=
0.
The derivation of Stirling's formula is an example of this method;
for, the real axis is the path that falls off most steeply from the saddle
point T = 1.
4. Application of the Saddle Point Method to the Calculation of Hankel and Bessel Functions for Large Parameter and Large Argument.
We shall use this method to evaluate the function (see formula (3),
page 468)
H~(a>..)
= -
!
11"
J,
i<-ia•int+i.,)
dT
L1
asymptotically for the case of real a and large positive ~- We separate the factor of~ in the exponent into its real and imaginary parts:
- ia sin T + iT
= f(T)
=a cos u sinh v - v + i(u - a sin u cosh v).
The saddle points are roots of the equation a cos T = 1 through which
the curves u - a sin u cosh v = const. should pass; we shall now
try to combine them in such a way that suitable paths of integration
are formed.
(1) If a < 1, say a= 1/cosh a (a> 0), we have saddle points given
528
VII. SPECIAL FUNCTIONS DEFINED BY EIGENVALUE PROBLEMS
by r = ± ai and the corresponding curves are u - a sin u cosh v = 0.
They consist of the imaginary axis u = 0 and any branch through
r = ± ai which approaches the lines u = ± 7r from above and from
below, respectively. This is shown in Figure 22, where the direction of
increasing real part off( r) is indicated by arrows. Th~ curve made up
of the curves !Jni(f) = 0 traversed in an upward direction evidently
yields HL for, it may be deformed into L1
I
/I
except for a part beginning arbitrarily high
.
I
I
/1
which lies inside a strip - 7r ~ u ~ - 7r E
I I
I
and hence furnishes an arbitrarily small conI
I . )I.
:
I
ict.v
I
tribution to the value of the integral. The
I
I
real
part of -ia sin r
ir has its maximum
I
I
I
a
tanh
a
for
r
=
ai.
Again we replace
Q------~L
-Xr-----I
(see
page
522)
the
path
L1
by the straight1
I
I
line
-rxi
segment
I
L'
from
(
a
- E)i to ( - a
I
,,,,.. . . ,
I
E)i
With
E
=
A-~.
If
We
then separate
I /
·~. l
the
remaining
path
of
integration
into two
I l
\ I
I I
' I
adjacent finite parts and two parts leading
I .
\l to infinity, we find an estimate which corIf
I/
\1
responds exactly to that derived in subsecFigure 22
tion 1, given by
,
+
+
+
i
(-a-e)i
ioo
eMfr> dr
+
-r+ioo
1
(-a+e)i
e>./CT> dr=e>.<a-tanha>o( e-c 1>.e2) ,
where c1 (as well as c2, C3, etc. which will occur ·below) denotes a
positive constant independent of A (hence also of E). That is, on the
two finite intervals the absolute value of the integrand is at most
equal to the values which it has at the points ( - a ± E)i, and for
these values the indicated approximation holds. On the infinite
portions one easily finds a bound of the form e-cMB+c'> for the absolute value of the integrand, where s is the arc length on the path
of integration and c and c' are positive constants independent of E
and A. For the contributions of these parts to the total integral
this yields an estimate O(e-c 1""). On the portion L' itself, however,
I f(r)
hence
-
+ !f"(- ai)(r +
f"(-ai) = tanh a;
(a - tanh a
ai)
2
)
I<
3
C2E ,
529
ASYMPTOTIC EXPANSIONS
f
(-cttE)i
1(-a+E)i
dr =
e"·f(-r}
i'tanha((-r+ai) 2/2)
eMa-tanha)
dr(l
+ O(>..-1/5))
(-a-E)i
(-a-E)i
!:!
:
!I
Ii
J•\
I
I
·I\
I
/
l
!'II
1•
!:
I:\
I
'.a.i
·..er- ----r-- -.X-U
-
:
/
I
'I
I
I
I
1.
I'
I'
I
I
I
.I
I
II
i'·1
:~
II
ii
jl
I
I
\
II
If
•I
11
II
I
I
I
I
j1
1•
11
-q-
I
-JU--- ·~------~-lO
' , -a-t.e lf'- II
i
I
\I
•
-«+le
I
\I
/
I,
11
I•
11ci.
I
I
I
I
Ii
I
\
/:
I
Figure 24
Figure 23
'
Thus we have
H~(aX) =
(93)
(2) If a
points
T
-i
~ l<a-tanha)(l + O(X-1/5)).
> 1, say a =
1/cos a
(O
< a < r/2), we have the saddle
= ±a and the curves
u - a sin u cosh v = ±(a - a sin a),
cosh v
u
+ (a
- tan a)
= - - - - .- - -
a sm u
which are reproduced in Figure 23; the solid path represents H~(x).
In the neighborhood of the saddle point, we replace this path by a
line-segment making an angle - r / 4 with the real axis, and by connecting segments of bounded length on which !J~ f (r) does not assume values larger than its value at the points r = - a ± Eeari/ 4
(see Figure 24). Again we set E = x.- 215 , and obtain as before
530
VII. SPECIAL FUNCTIONS DEFINED BY EIGENVALUE PROBLEMS
II
I
I
j
'I
-~r----I
I
I
I
Ii
·
I
/
ii''
•I
I
I I
io/ /
I
:I
I
"'I
ee-r:
·:----1-Z
'·
\
.
\
lI 'i
1/
!o___ _
II
I
\
/
I
I
-et
\\ :I
\I
Figure 25
Figure 26
(3) If a = 1, /"( r) also vanishes at the saddle point r = 0 and the
curve 9- f( r) = u - sin u cosh v = 9- f(O) = 0 has three branches
through r = 0 (Figure 25), one of which is the imaginary axis. Again
we replace the curved part of the path L 1 (Figure 25) which is close
to r = 0 by a straight segment of length E = }..-it• inclined at the
angle 51r/6 with the real axis (see Figure 26) and obtain for all r of
the path of integration between - Ei and Ee5rite
lf(r) - i~' I~
c1l'.
531
ASYMPTOTIC EXPANSIONS
1
u6Ti/6
1
-ei
;ifra/6
dr =
u~"'i/6
o
-
1-
Ei
:;-
'V ~ (e6ri/6 + i) lo
=
E...v>:/6
/6
e_"'a du;
in the last transformation we use the substitution r = ~6/"A e6ri16u
in the first integral and r = - ~6/"A.iu in the second. The right
side of the last equation equals
ti~ (e6ri/6 +
i)
1"° e_"a du(l + O(e-cae3>.)),
if lA remains greater than a positive bound.
have
f
f e-'r
0
0
e_"'a du =
l
and, therefore, finally
(95)
H~(X)
=
On the other hand we
-L
r(l)(e5ri/ 6 + i)
213
dt = !r(l)
V1
+ ocx-114)) ...
(1
To find asymptotic formulas for J-,,,(a>.) in the cases a ~ 1 we use
the formulas derived above for H~(a>.) and the following three formulas which are found in a similar way:
(96)
H~(a"A)
=
(96')
H~(aX)
=
I 2 i<a-tanha)(l + ocx-1/6)) (a < 1),
'V 7rA tanh a
-e-3ri/4 .. I
2
e-•Mtana-a) (1 + ocx-1/6))
'V ?rA tan a
i ..
(a> 1),
(96")
H~(x) = _ _!_r(l)(e-6ri/ 637r
i) ..3 /~ (1
'V
"A
+·ocx-114))
(a
= 1).
We combine them by means of
J>.(X) = !(H~(x)
+ H~(x));
the principal terin turns out to be zero only in the case a < 1. In
this case we can take the path shown in Figure 27 for J-,,, also, ob-
532
VII. SPECIAL FUNCTIONS DEFINED BY EIGENVALUE PROBLEMS
taining by the same method
Jx(a'A)
2
= ------e>.(tanha-cr)(l
V27rA tanh a
+ oc~-1/6)).
5. General Remarks on the Saddle Point Method. Here the saddle
point method has been used only to derive asymptotic formulas for
the first terms of asymptotic series found as originally indicated.
The reader is referred to the detailed presentation of these series 1.n
G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, 1922,
and to the literature, in particular P. Debye,
I
,
I
Naherungsformeln fiir die Zylinderfunktionen
I
I, .
I
I
I fiir grosse W erte des Arguments und unbeI
at.
schrankt veranderliche W erte des Index, Math.
I
\
I Ann., Vol. 67, 1909, pp. 535-558.
I
I.
I
In applications of the saddle point method
-.11:1- - - -~----44'
it
is not necessary to take the path of inteI
I
I
I gration precisely in the manner described. It
is enough if eventually, i.e. for large values
II
I
/
·,
I
.
•
I of the parameter in terms of which the funcI
~
.
'\
I /
t' .\ I tions are expanded, the path comes suffiI ,
I I
I
\ I ciently close to the one described. In this
way Faber1 obtains a number of asymptotic
Figure 27
series, e.g. for Hermite and Laguerre polynomials.
6. The Darboux Method. A different method for deriving asymptotic formulas is due to Darboux. 2 Let the quantities a" in question
be given as the coefficients of a power series, hence by a generating
function K(t) = L~=o a.f. If we know the singularities of this
function on the circle of convergence--say JJ I = 1, t = ei"'-and
if by subtracting known functions fn(t) = L~=o an"t" we can insure
that the remainder K - fn converges uniformly, as we approach the
circle of convergence, to an n-times continuously differentiable func-
'\+;'
I
I
I
r
_ta:t
!
1 G. Faber, Abschatzung von
Funktionen grosser Zahlen, S.-Ber. Akad.
Miinchen (math.-phys. Kl.), 1922, pp. 285-304.
2 G. Darboux, Memoire sur !'approximation des fonctions de tres-grands
nombres, et sur une classe etendue de developpements en serie, J. de math.
pures et appl., Ser. 3, Vol. 4, 1878, pp. 5-56 and 377-416. See also A. Haar,
'Ober asymptotische Entwicklungen von Funktionen, M~th. Ann., Vol. 96,
1926, pp. 69-107.
ASYMPTOTIC EXPANSIONS
tion of
<p,
533
then the coefficients a,, - an,, of the power series
00
K(f) - /n(f)
=L
•-o
(a,, - an.)(
are the Fourier coefficients of an n-times continuously differentiable
(i.e. for n = 0, just continuous) function of <p; hence, by Ch. II,
§5, 3, they satisfy the condition
1
fun
,,_ .,,"-
I a.
- an. I
= o.
Thus if 11 is large, the approximation of a,, by an• becomes better as
n increases.
7. Application of the Darboux Method to the Asymptotic Expansion
of Legendre Polynomials. We apply the method to the Legendre
polynomials P.(x) which are given by means of the generating function
•
00
1
= :E P.(z)r.
(97)
K(z, t) = v'
2
1 - 2zf
+s
P=-0
2
Assume -1 < z < 1, z = cos 'P, 0 < 'P < 7r. Then 1 - 2zf + f =
(t - e"'i) (t - e-"i) ; the circle of convergence has radius 1, and on it
lie the singular points r = e%.tpi. In order to derive the series expansion of K according to powers of r - e±tpi, we make the convention that
v'r -
e±t>•
= e±Htp+r>f2y'1
- te=F'Pi,
where the square root on the right denotes the branch presented by
the binomial series. 1 We obtain
1 Thus if a is a positive number, for r = ev>• - a the root~ is positive imaginary, while for r = e-tpi - a the root v'r - e-.,. is negative imaginary.
The above convention is in agreement with the requirement, which follows
2zr + r2 should take the value
from formula (97), that for r = o the root
+L
vi -
534
VIL SPECIAL FUNCTIONS DEFINED BY EIGENVALUE PROBLEMS
If we set
1
-~--
f n(z, t) =
1)
n (
,L
-
2
v'2 sm cp v-0
11
37ri/4
. {e
(~r - e.pi)v-1/'2
(e!'i -
e-"'i)"
+
-3ri/4
e
(r
-.pi)v-1/~~
e
. ~
(e-<Fi - e"i)• J '
~ -
then K - fn is n-times continuously differentiable on the circle of
convergence. Hence if we expand fn in powers of t and, for convenience, write (37r/4)
(cp
7r)(11 - !) = w, we obtain
+ +
z
f nC , t)
= __ __!_ v2 sin
cp
n
~
(-!){ei"'(!_- te-.piy-11~
cp
v=O
with
p
- _1_
nµ. - y2 Sin cp
[exp{i[7r/4
+ (,, -
(98)
p
nµ.
=
2
µ
t (-!) (" - !)
v=O
1
C(Jr •
(2 sin
µ
11
zxp{-i[11"/4
-~-----
V2 sin
L.J
µ=O
!)cp - (µ - v/2)7r]}
µ -
+
11
e-iw(l - re<Pir~-~2}
(e-.pi - e<Pi)v
.!) ~ (" - .!) sµ .
= _____
L.J
1
~ (- 2
V2 sin
+
(e<Pi - e-.pi)v
,,
cp
+
(v -
t (-!) (" - !)
v=O
µ
JI
cos
!)cp - (µ - v/2)11"]}],
µ -
(-1)"
(2 sin
cpf
(i (1 + 2v) -
(v - µ. -
u~),
we obtain
P,,(z) = Pn,µ.(z)
+
O(µ -n)
uniformly in every interval of the form -1
(0 < E < 1). In view of the fact that Pn+i,µ
follows that
Pµ.(z) = Pn,µ.(z)
+
+
-
0(µ-n-1).
e
<
Pn,µ =
z < 1 - "
O(µ - n -1), it
535
ASYMPTOTIC EXPANSIONS
The first term of this asymptotic expansion may be written as
P,,.(z)
(99)
2
= v2 sin cp
1 ·3 · · · (2µ. - 1)
2·4 · · · 2µ.
cos
+
(i-<,, + !)q>) + 0 G).
If z is not real between -1 and 1, then since the singular points
has a modulus I I·< 1,
= 1, one of them, say
satisfy
the other a modulus I I > 1. Only the singular point lies on the
circle of convergence I I = I 1 I; hence we need consider only this
singularity. Accordingly, if we transform the first n terms of the
into a power series
expansion of K(z, t) in terms of powers of
in t, the coefficients provide asymptotic expressions for P11(z); however, now only
ri, r2
r1r2
r2
r
ri,
r1
r
r1
r - ri
holds.
Appendix to Chapter VII
§7. Trans/ormation of Spherical Harmonics 1
1.
Introduction and Notation
Let x, y, z be Cartesian coordinates and let r, fJ,
polar coordinates defined by
(100)
x = r cos cp sin 0,
y
= r sin cp sin 0,
cp
be spherical
z = r cos 0.
Let A denote Laplace's operator
a2
(101)
a2
a2
A=-+-+az2
iJy 2
ax2
= a + ~ !__ + _! a + cot o ~ + _1_ .!._
2
ar
2
2
r ar
r ao
2
2
r ao
2
r2 sin2 0 iJcp2 •
A function u which satisfies Au = 0 is called harmonic. If Pn(O, cp)
is a one-valued function of x, y, z such that rnpn is a harmonic func1 The author is indebted to W. Magnus for this appendix, which is based on
unpublished notes by the late G. Herglotz.
536
VII. SPECIAL FUNCTIONS DEFINED BY EIGENVALUE PROBLEMS
tion, Pn is called a spherical harmonic of order n. It was observed
in §5 that rnpn is a homogeneous harmonic polynomial of degree n in
x, y, z and vice versa.
In several problems of mathematical physics the question arises
how any set of linearly independent spherical harmonics is transformed if the coordinate system is rotated but the origin r = 0 kept
fixed. To answer this question, we shall characterize the orthogonal
transformation s in terms of independent parameters q.,, (v = 1, 2, 3, 4).
Coordinate rotation in a system of 2n
1 linearly independent
spherical harmonics of order n is expressed by a linear substitution
whose coefficients form a complete system of orthogonal hyperspherical harmonics of order 2n in the four-dimensional space of the
parameters q.,,.
There exists no analogue to this result in the case of more than
three variables.
+
2. Orthogonal Transformations
Let 0 be an orthogonal matrix of th:ree rows and three columns;
0 is characterized by
(102)
00' =I
wh~re 0' is the transposed matrix of 0 and I denotes the identity
or unit matrix. We shall consider only "proper" orthogonal matrices; i.e., we assume that the determinant I 0 I of 0 equals + 1. Let
(103)
be a skew-symmetri c matrix, i.e. one for which A' = -A.
q4 be a fourth parameter and let
Let
Then we have
For any real values of the q.,, (v = 1, 2, 3, 4) for
0, the matrix
CAYLEY's THEOREM:
which v
>
APPENDIX: TRANSFORMATION OF SPHERICAL HARMONICS
537
or
q!
(loo)
o
=
~
+ q~ -
2q1q2 (
2q1q3
+
qi- q:, 2q1q2
+
2qaq4 , q! +qi- q~ - qL 2q2qa
2q2q4' 2q2q3
-
2q2q4)
2qsq4 , 2q1q3
+
2q1q4 ,
2q1q4, q!+q:- q~ -qi
is orthogonal; q1/w, q2/w, q3/w are the direction cosines of the axis
of rotation with respect to the x-, y-, and z-axes. If w denotes the angle
of rotation, then
(107)
q4
w
cos-= 2
v'
We normalize the parameters q,, in such a way that
(108)
holds; then any two sets q,, and q,,* of values of the parameters define the
same 0 if and only if q,, = q,,* orq,, = -q,,* for v = 1, 2, 3, 4. If w = 0,
every direction in the space remains fixed and 0 becomes the identity I.
Proof: We say that two matrices Band C commute if BC = CB.
We may then perform all algebraic operations as if B and C were
numbers provided we do not divide by any matrix unless its determi1
nant is different from zero; for instance, if I B I ¢ 0, B- and C also
1
commute. Using this fact and the well-known relation (B'f'" =
1
(B- )' (where the prime denotes a transposed matrix) we see at once:
If q4 I - A has an inverse and if, as in (105), we set
(109)
then
(110)
0' = (q4/' - A')-1(q4J'
+ A')
= (q.I - A)(q.I
+ A)-
1
= 0-1.
Therefore 0 is orthogonal. Similarly, we find for any orthogonal 0
for which I I + 0 I ¢ 0 that
(111)
A = q4(I -
0)(1
+
0)-1
is a skew-symmetric matrix. But (ll 1) is exactly the expression
for A which we obtain if we "solve" (109) with respect to A. This
538
VII. SPECIAL FUNCTIONS DEFINED BY EIGENVALUE PROBLEMS
proves the first part of Cayley's theorem (i.e. (105) and (106)) unless
I I+ 0 I == 0. Even in this case, the right side in (109) has a meaning if we write it in a different form. Since every matrix satisfies
its characteristic equation, we have
(112)
which (for v
(113)
>
0) leads to
q4l +A = v-2(v2I
+ 2q~ + 2A 2)(q4/ -
A).
We may define the right side in (109) as v-2 (v2/ + 2q~ + 2A 2 );
then, however, we have to show that this is an orthogonal matrix
even if I q4I - A I = 0, Le. if q4 is an eigenvalue of A. Since A is
skew-symmetric, its eigenvalues are pure imaginary. Hence q4 = 0
and ·
(114)
v-2(v 2I
+
2q~
+
2A 2) = I
+
2w-2A 2 ;
because of (112) we have
(115)
(I+ 2w-2A 2) 2 = I+ 4w- 2A 2
+
4w-4A 4 = I.
Therefore, for q4 = 0, I + 2w-2A 2 is a symmetric orthogonal matrix.
It can be verified that all orthogonal matrices of this type can be
written in the form I+ 2w-2A 2 • On the other hand, any orthogonal
matrix for which I 0 I = +1 and I I+ 0 I = 0 is necessarily symmetric because such an 0 must have the eigenvalues + 1, -1, -1;
its square has the eigenvalues 1, 1, 1, and is therefore the identity.
But 0 2 = 00' = I implies that 0 is symmetric.
In order to interpret the q,. geometrically, we observe first that
the vector (q1, q2, qa) is an eigenvector of A belonging to the eigenvalue zero. It follows from (105) that (q1, q2, q3) is an eigenvector
of 0 belonging to the eigenvalue 1 and therefore defining the axis of
rotation. Since the eigenvalues of 0 are exp{iw}, exp{ -iw}, and 1,
and since the sum of these is the trace of 0, (107) follows from (106)
except for the sign of cos (w/2), which need not be determined; for
if we change w/2 by ?r, then w changes by 2?r.
Apparently, Cayley's theorem can be extended to more than three
dimensions, but then the explicit formulas (105) and (106) become
more complicated. In three dimensions, (106) can be stated in a
APPENDIX: TRANSFORMATION OF SPHERICAL HARMOXICS
539
simple manner by using quaternions. 1 Let the vectors x and x'
be defined by x = (x, y, z) and x' = Ox. Let~' ~', a be quaternions
defined by
(116)
~ =
xi
+
yj
+
zk,
e=
x'i
+
y'j
+
z' k
and
(116')
then
(117).
1
a-1a = 1 anda- = v- 2 ( - q4 - qii - q2j- q3k).
whereaa- 1
It may be observed that there exists a one-to-two correspondence
between the elements of the orthogonal group and the points on a
four-dimensional hypersphere. There also exists a group G such
that the orthogonal group is a quotient group of G and the elements
of Gare in a one-to-one correspondence with the points of the hypersphere. For a definition of this group see §4; for its significance consult, e.g., B. L. van der Waerden, Die gruppentheoretische Methode in
der Quantenmechanik, §16, J. Springer, Berlin, 1932.
3. A Generating Function for Spherical Harmonics
Let P n.z (x) denote the functions defined in Ch. VII, §5, 1 for l =
0, 1, 2, · · · , n and let
(118)
Pn,-i(x)
=
Then we know that the 2n
)
z (n - Z)!
l) ! Pn,z(X .
(-1) (n
+
+
1 functions
(119)
(l =
0, ±1, ±2, · · · , ±n)
1 Quaternions are "hypercomplex'' numbers with three "imaginary" units
i, j, k which satisfy the relations
= j 1 = k 1 = -1,
ki = j,
jk == i,
ij = k,
ik + ki = o,
ij + ji = 0,
ii
jk
+ kj = 0.
The general quaternion is a = a+ bi + cj + dk where a, b, c, dare real. All
laws of algebra with the exception of the commutative law of multiplication
are satisfied by quaternions.
540
VII. SPECIAL FUNCTIONS DEFINED BY EIGENVALUE PROBLEMS
form a complete set of linearly independent spherical harmonics of
order n. We assert that:
The homogeneous polynomials hn,l of degree n in x, y, z which are defined by
+n
(120)
{x - iy - 2zt - (x
+ iy)t2}n = tn ·L
hn,i(x, y, z)tl
l--n
form a complete system of linearly independent harmonic polynomials of
degree n. If spherical polar coordinates are introduced by equation (100),
we have
(121)
where
n 2n n! p (
) ilrp
Pn,z (8, cp). = ( - l) (n + l) ! n,z cos 8 e .
(122)
If x, y, z are connected by a relation
x2
(123)
+ y2 + z2
=
0,
then hn,l can be expressed in terms of two parameters
( 124)
hn,l (X, y, Z ) = ( - l)
n(n 2n+ l)
W1
W1, W2
by
n+l
n-l
W2
where
+ +
2
2
Proof:If a, b, careanythreenumbe rs which satisfy a 2
b
c = 0,
then clearly ~(ax
by
czr = 0 holds; substituting a = 1 - t 2,
2
b = -i(l
t ), c = -2t, we obtain (120). In order to prove equations (121) and (122) we substitute e-1 for t in (120) and find
+
(126)
+
+
hn,-l = ( -1) 1 hn,l
where the bar denotes a conjugate complex quantity. Therefore it
suffices to prove (122) for l ~ 0. Introducing t = e-i"s in (120), we
find
+n
(127)
{(1 - s2) sin 8 - 2s cos fJ}n = Sn
L Pn.1(8, cp)e-il"s 1
l=...,.n
APPENDIX: TRANSFORMATION OF SPHERICAL HARMONICS
541
and see that
Pn,le -ilrp =
f n,l (8)
From (126) we obtain
is a real function of 8 only.
fn,-z(fJ) = ( -1)' f n,1(8).
(128)
If we multiply (127) by s-n, substitute exp {i(w
use (128), we obtain
(129) (cos 8
+ i cos w sin 8)1'
= ( -2)-n{fn,o
+ 7r/2)}
for s, and
+ 2 tZ=l i 1fn,l cos "lw}.
On the other hand, the integral representations for P n,1(x) with
l = 0, 1, 2, · · · , n (see §4, 4) show that
az
=
i'
(n
+ l) ! Pn,z(x)
is the Fourier coefficient in the expansion of
n
(130)
(x
+ iyl
- x 2 cos
wr = ao + 2 L: az cos l,w.
Z=l
Setting x = cos 0 and comparing equations (130) and (129), we see
that (122) holds.
Finally, (124) becomes a trivial application of the binomial theorem
if in (120) we substitute the expressions for x, y, z given by (125).
It should be emphasized that equation (123) is not only a consequence of (125), but that any three numbers x, y, z satisfying (123)
define exactly two pairs of numbers W1 , w2 such that equations
(125) hold. Since x + iy and x - iy are algebraically independent,
the polynomials in w 1 , w2 on the right side of (124) are linearly
independent. Therefore, (125) maps the harmonic polynomials hn,z
on the linearly independent polynomials (of even degree) in W1 and
w2 . This also shows that every harmonic polynomial can be characterized by exactly one polynomial in three variables the sum of the
squares of which is zero. Earlier in this chapter, the same fact was
established by showing that the harmonic polynomials can be obtained if we apply a polynomial in the differential operators &/ox,
2
112
2
2
1
iJ/iJy, iJ/iJz to r- = (x + y + z ) - , where the sum of the squares
1
of these operators (if applied to r- ) is evidently zero.
542
VII. SPECIAL FUNCTIONS DEFINED BY EIGENVALUE PROBLEMS
4. Transformation Formula
The significance of the formulas (120) to (125) will become clear
when we have shown that any orthogonal transformation of x, y, z
can be obtained from a linear substitution of w1 and w2 • This
expresses the well-known fact that the real orthogonal group in three
variables has a representation by (complex) two-by-two matrices.
Lemma: Let W1, W2, x, y, z be connected by (125) and assume that
(131)
w~ =
aw1
+
iy' =
+
/jw2,
and
(132)
x'
w?,
-x'
+
iy' = w~ 2 ,
z'
= w~w~.
Then (125), (131), (132) define a linear substitution M by which the
vector x
(x, y, z) is mapped on x' = (x', y', z') = Mx. If we
choose
(133)
a
= (q4 - ·iqa)/v,
then Mis the matrix 0 defined by (106).
The proof is trivial. If we replace a, /j by - a, - /j we obtain
the same matrix M = 0 as before. This shows that there are (at
least) two different substitutions (131) which lead to the same orthogonal substitution 0. As a matter of fact, there are exactly two
such substitutions (132), and the group of these is the group G mentioned at the end of subsection 2 of this Appendix.
We may use this lemma to prove the following theorem which in
its present form is due to G. Herglotz:
Let x = (x, y, z) be trans!armed into x' = (x', y', z') by the orthogonal
transformation 0 defined by (105), (106). Then the spherical harmonics
hn,l (x', y', z') can be expressed in terms of the hn,z (x, y, z) (which are
defined by (120)) by
(134)
hn,l (X , ,y /,z')
=
~ (z ~ n)
-2nn<n+l,n+r>( )h (
£- ( - - - - ) V
2n
qµ n,r X,
r--n
2n
)
y, Z
r+n
where the H~:+z.n+r> are homogeneous polynomials of degree 2n in
the variables q" (µ = 1, 2, 3, 4) which satisfy
(135)
APPENDIX: TRANSFORMATION OF SPHERICAL HARMONICS
543
and can be defined in terms of the generating function
(136)
Gin(qu
j S,
t)
= {iqa(l -
st)
+ iq1(s + t) + q2(s -
t)
+ q4(1 + st)} 2"
by
~
G2n(qis; s, t) = £.J
(137)
j.k=O
(2n) H u.'" (qis)t.s; •
.
J
k
2n
0
Explicit expressions for the H~:· and a modified form of equation
(135) (in polar coordinates) will be given in subsection 5.
Equation (135) can be deduced from (136) in the same way that ,
the statement ll.hn,z = 0 was deduced from (120). To prove equations (134) and (136) we use (124) and consider the effect of the
linear transformation (131) upon the hn,i. From equation (124)
and equations (133) and (131} we find
hn, z(X /, Y1,
(138)
(139)
hn.h', y', z',
{ -q2
where s = wi/w2.
(136) is
{[q4
Z
')
=
_
-
(-
l)n ( n 2n l)
+
ln+l ln-l
,
W2
W1
(-1r(: )2\n 2~ z)
2
+ iq1 + (q4-
iq3)s}n+l {q4
+ iqa + (q2 + iq1)s}"-z
On the other hand, the right side in equation
+ iqa + (iq1 + q2)s] + [iq1 -
q2
+ (q4 -
2
iq3)s]t} ";
2
therefore, the coefficient of ( -1r(w2/v) n in (139) is precisely the
coefficient of tn+z in (137) withs = wi/w2. We obtain
k 2n-k ,
~ n<n+l,k) W1W2
( ' ' ') = (-l)n ( 2n
(140) hn,zx,y,z
£..J 2n
+l) V-2n k=O
n
which together with (124) proves equation (134).
5. Expressions in Terms of Angular Coordinates
In polar coordinates the standard spherical harmonics express
themselves in terms of r, 8, 'fJ as products of functions of a single
variable; however, the transformation formulas become simple in Cartesian coordinates. Indeed, if we set
(141)
x'
=
r' cos 'fJ1 sin 8',
y1 = r' sin 'fJ' sin 8',
z'
= cos 8',
544
VII. SPEClAL FUNCTIONS DEFINED BY EIGENVALUE PROBLEMS
we haver' = r, but O', <p' cannot be expressed in a simple manner in
terms of 8 and 'P· Nevertheless, the spheric~l harmonics
I
(r)
-n
I
I
I
n 2nn!
I
iltp'
hn,z(x,y,z) - (-1) (n
l)!Pn,z(cosO)e
+
can be expressed in terms of the P n. z (cos 8) exp {il'P} if we use ( 134).
Only the problem of finding simple explicit expressions for the
H~:+z,n+r>(q,,) remains. If we were to introduce four-dimensional
(hyperspherical) polar coordinates in the space of the variables q",
we would still obtain rather complicated formulas; however, there
exists a particular coordinate system in which the expressions for
the H~:+z,n+r> become fairly simple. We may summarize the results as follows:
Let v ~ 0, 0 S p S 2T, 0 S <T S 27r, 0 S T S 7r/2 and
qi = -v sin
(142)
qa = v sin
Then v,
p,
(143)
dq~+ dq~
p
<T
sin
cos
T,
T,
q2
= v cos
q4
=
<T
sin
T
V COS p COS T.
<T, Tare orthogonal coordinates; the line element is
+ dqi + dq! =
dv
2
+ v {cos Td/ + sin Td<T -+ dT l
2
and the surface element of the hypersphere
(144)
dn
=
2
2
2
n (defined by v =
2
1) is
cos r sin T dp du dT.
In these coordinates
v-2nH~:+l,n+r)(q,.) = s~~r)(p, u,
(145)
1
(n
-i(l+r)p -i(r-l)v(
+ r.) I e
e
T)
• T )r-lp(l,r)(
cos T )l+r(sin
2n
crn;2 r )
where
F(l,r)(t) =
(146)
2n
~+r
dtn+r
t"-l(t _ l)n+z.
The S~~r> and their conjugate complex functions are biorthogonal
on the unit-hypersphere n:
if
0
(147)
L
JJ s~~·r·»-s~~r) du
=
l'
~
l
Ol'
r' ~
I
2
27r
(- 2n ) / (
2n )
2n
1 r
n
l
n
+
+
if
+
l'
=
l,
and
r' = r •
APPENDIX: TRANSFORMATION OF SPHERICAL HARMONICS
545
The trans.formation formulas of the spherical harmonics are
(148)
8) ir.,,.
- r)! s<l,r>p (
~ (n
P n,l ( COS 8') eil<F' = £..J
e
n,r COS
( _ l) I 2n
r-n
.
n
Proof: If in (136) and (137) we substitute the expressions (142) for
the q. and introduce s* and t* by
s* = se-•<.. +P>,
we find, for the s~~r) in (145),
{(1
+ s*t*) cos T + (s*
(149)
--
- t*) sin T} 2"
~ ( 2n ) s<2nl,r) ei(r+l)p ei(r-l)..t*"+'8 *"+r
£..J
l,r-n
n
+l
which proves that s~ ~r) exp {i[ (r + l) p + (r - l)u]} = u~ ~r) depends
only on T. Comparing the coefficients of t*"+i on both sides of
(149), we find
+n
L
r-n
(150)
+ s* sin T)"- 1(-sin T + s* cos T)"+'
(cos T) 1-"(sin T)-"-'[cos2 + s* cos sin T]"-'
· [cos T - 1 + s* cos T sin rl"+I.
U2 ~l,r>s•"+'
=
(cos T
T
T
2
Application of Taylor's formula to the last line in (150) yields equations (145) and (146). From these formulas and from (144) we can
derive (147) by repeated integration by parts if we observe that
s~~r) =
( - l)'+r s~-;l,-r).
We mention without proof that, according to a formula which was
discovered by Jacobi, the u~~r) can also be expressed in terms of
hypergeometric series (see Chapter II, equation (24) on page 90).
These series are, for l + r ~ 0,
+ l)c cos r )-l-r(sm.
- r
( l)"+'(n
u<l.r)
n
= 2"
T
· tF1 ( - n - r, n
and, for r
+
)l-r
+1-
2
r; 1 - l - r; cos T),
l 2::, 0,
u~~r) = ( -l)n+r(n
+
l)(cos r)z+r(sin rr-l
n-r
·'lF1(r - n, n
+ r + 1; 1 + l + r; cos r).
2
Additional Bibliography
CHAPTER I
Birkhoff, G., and MacLane, S. A survey of modern algebra. Macmillan, New
York, 1941.
Frazer, R. A., Duncan, W. J., and Collar, A. R. Elementary matrices. Cam·
bridge, England, the University Press, 1938.
Halmos, P.R. Finite dimensional vector spaces. Princeton, Princeton University Press, 1942 (Annals of mathematics studies no. 7).
Hamburger, H. L., and Gremshaw, M. E. Linear transformations in n-dimensional vector space. Cambridge, Eng., the University Press, 1951.
MacDuffee, C. C. Theory of matrices. J. Springer, Berlin, 1933 (Ergebnisse
der Mathematik und ihrer Grenzgebiete, bd. 2, bd. 5).
Schreier, 0., and Sperner, E. Einfuehrung in die analytische Geometrie und
Algebra. B. G. Teubner, Leipzig und Berlin, 1935 (Hamburger mathematische Einzelschriften, Heft 10, Heft 19).
Turnbull, H. W., and Aitken, A. C. An introduction to the theory of canonical
matrices. Blackie and Son, Ltd., London and Glasgow, 1932.
Wedderburn, J. H. M. Lectures on matrices. American Mathematical
Society, New York, 1934 (Colloquium publications, vol. 17).
CHAPTER II
Bochner, S. Vorlesungen ueber Fouriersche Integrale. Akademische Verlagsgesellschaft, Leipzig, 1932.
Bochner, S., and Chandrasekharan, K. Fourier transforms. Princeton University Press, Princeton, 1949 (Annals of mathematics studies, no. 19).
Carleman, T. L'lntegrale de Fourier et questions qui s'y rattachent. Almquist and Wiksells boktr., Upsala, 1944.
Hardy, G. H., and Rogosinski, W.W. Fourier series. Cambridge, Eng., the
University Press, 1950 (Cambridge tracts in mathematics and physics, no.
38).
Jackson, D. Fourier series and orthogonal polynomials. Oberlin, Ohio, 1941
(Carus mathematical monographs, no. 8).
Jeffreys, H., and Jeffreys, B. S. Methods of mathematical physics. Cambridge, Eng., the University Press, 1946.
Kaczmarz, S., and Steinhaus, H. Theorie der Orthogonalreihen. Z subwencji Funduszu kultury narodowej (Monografje matematyczne, vol. 6)
Warsawa-Lwow, 1935.
Paley, R. E. A. C., and Wiener, N. Fourier transforms in the complex domain. American Mathematical Society, New York, 1934 (Colloquium publications, vol. 19).
Rogosinski, W. Fouriersche Reihen. W. de Gruyter, Berlin, 1930.
Sneddon, I. N. Fourier transforms. McGraw-Hill, New York, 1951.
ADDITIONAL BIBLIOGRAPHY
547
Szego, G. Orthogonal polynomials. American Mathematical Society, New
York, 1939 (Colloquium publications, vol. 23).
Titchmarsh, E. C. Introduction to the theory of Fourier integrals. The
Clarendon Press, Oxford, 1937.
Wiener, N. The Fourier integral and certain of its applications. Cambridge,
Eng., the University Press, 1933.
Zygmund, A. Trigonometrical series. Z subwencji Funduszu kultury narodowej (Monografje matematyczne, vol. 5), Warsawa-Lwow, 1935.
CHAPTER III
Banach, S. Theorie des operations lineaires. Z subwencji Funduszu kultury
narodowej (Monografje matematyczne, vol. 1), Warsawa, 1932.
Hamel, G. Integralgleichungen. J. Springer, Berlin, 1937.
Lov~tt, S. V. Linear integral equations. McGraw-;Hill, New York, 1924.
Riesz, F. Les systemes d'equations lineaires a une infinite d'inconnues.
Gauthier-Villars, Paris, 1913.
Stone, M. H. Linear transformations in Hilbert space and their application
to analysis. American Mathematical Society, New York, 1932 (Colloquium
publications, vol. 15).
CHAPTER IV
Bliss, G. A. Lectures on the calculus of variations. University of Chicago
Press, Chicago, 1946.
Caratheodory, C. Variationsrechnung · und partielle Differentialgleichungen
erster Ordnung. B. G. Teubner, Leipzig and Berlin, 1935.
Courant, R. Dirichlet's principle, conformal mapping, and minimal surfaces.
Interscience, New York, 1950.
Forsyth, A. R. Calculus of variations. Cambridge, Eng., the University
Press, 1927.
Fox, C. An introduction to the calculus of variations. Oxford University
Press, London, 1950.
Lanczos, C. The variational principles of mechanics. University of Toronto
Press, Toronto, 1949.
Lusternik, L., and Schnirelmann, L. Methodes topologiques dans les problemes
variationnels. Hermann et Cie., Paris, 1934.
Lusternik, L., and Schnirelmann, L. Functional topology and abstract variational theory. Paris, Hermann and Cie., 1938.
Morrey, C. B. Multiple integral problems in the calculus of variations and
related topics. University of California Press, Berkeley and Los Angeles,
1943.
Morse, M. Tne calculus of variations in the large. Americaa Mathematical
Society, New York, 1934 (A.M.S. Colloquium publications v. 18).
Seifert, H., and Threlfall, W. Variationsrechnung im Grossen. B. G. Teubner, Leipzig and Berlin, 1938. (Hamburger mathematische Einzelschriften
Heft 24)
548
ADDITIONAL BIBLIOGRAPHY
Polya, G., and Szego, G. Isoperimetric inequalities in mathematical physics.
Princeton University Press, 1951 (Annals of mathematics studies, 27).
CHAPTER IV (Papers)
Courant, R. Variational methods for the solution of problems of equilibrium
and vibrations. American Mathematical Society, Bulletin, vol. 49, pp. 1-23,
1943.
CHAPTERS V AND VI
Churchill, R. V. Fourier series and boundary value problems. McGrawHill, New York, 1941.
Collatz, L. Eigenwertprobleme und ihre numerische Behandlung. Becker
und Erler. Leipzig, 1945.
Hohenemser, K., Die Methoden zur angenaherten Losung von Eigenwertproblemen in der Elastokinetik. Chelsea, New York, 1949.
Ince, E. L., Ordinary differential equations. Dover, New York, 1944.
Rellich, F ., Storungstheorie der Spektralzerlegung. J. Springer, Berlin,
1936-42.
Titchmarsh, E. C. Eigenfunction expansions associated with second-order
differential equations. Oxford, the Clarendon Press, 1946.
Weyl, H. Ramifications, old and new, of the eigenvalue problem. Bulletin of
the American Mathematical Society, Volume 56, 1950, pp. 115-139.
Weinstein, A. Etude des spectres des equations aux derivees partielles de la
theorie des plaques elastiques. Memorial des Sciences Mathematiques,
Volume 88, 1937.
Proceedings of the Symposium on Spectral Theory and Differential Problems.
Oklahoma A. & M., Stillwater, Oklahoma, 1951.
CHAPTER VII
Gray, A., Matthews, G. B., and MacRobert, T. M. A treatise on Bessel functions and their applications to physics. Macmillan, London, 1922.
Hobson, E. W. The theory of spherical and ellipsoidal harmonics. Cambridge, England, the University Press, 1931.
Jahnke, E., and Emde, F. Tables of functions with formulae and curves.
Dover, New York, 1943.
MacRobert, T. M. Spherical harmonics: an elementary treatise on harmonic
functions. London, Methuen and Co., 1927.
Madelung, E. Die mathematischen Hilfsmittel des Physikers. Berlin, J.
Springer, 1936 (Die Grundlehren der mathematischen Wissenschaften in
Einzeldarstellungen, Bd. 4).
l\fagnus, W., and Oberhettinger, F. Formeln und Satze fiir die speziellen
Funktionen der mathematischen Physik. Berlin, J. Springer, 1948 (Die
Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen,
Bd. 52).
Mclachlan, N. W. Bessel functions for engineers. Oxford, the Clarendon
Press, 1934.
ADDITIONAL BIBLIOGRAPH Y
549
Relton, F. E. Applied Bessel functions. London, Blackie and Son Ltd., 1946.
Watson, G. N. A treatise on the theory of Bessel Functions. Cambridge,
England, the University Press, 1944; New York, Macmillan, 1944.
Weyrich, R. Die Zylinderfunkt ionen und ihre Anwendungen . Leipzig und
Berlin, B. G. Teubner, 1937.
Papers
Copson, E. T. The asymptotic expansion of a Function defined by a definite
integral or contour integral, Admiralty Computing Service.
Van der Waerden, B. L. On The method of saddle points, Applied Scientific
Research B (The Hague), vol. II, No. 1, 1951, pp. 33-45.
INDEX
A
Abel's integral equation, 158
Accumulation principle for functions
'
57
Adjoint differential expression, 279,
280
Adjoint orthogonal systems of unsymmetric kernels, 159
Alternative theorem for integral
equations, 115
for linear equations, 6
fo! ordinary differential equations,
359
Analytic continuation, 395
Approximation in the mean, 52, 54
Approximation theorem of Weierstrass, 65
Argument function, 168
Arzela's theorem, 59
Associated Legendre functions, 327,
505
Asymptotic behavior
of eigenvalues, in Sturm-Liouville
problem, 414
of Bessel's equation, 415
of vibrating plate, 460
of solutions of the Sturm-Liouville
problem, 331-339
Asymptotic dimension of a sequence
of functions, 63
Asymptotic distribution of eigenvalues of a differential equation,
429-445' 460
Asymptotic expansions, 522-535
for Bessel and Hankel functions
'
524, 532
B
Beats, 390
Bessel's equation, 303, 467
Bessel's equation (continued)
asymptotic behavior of eigenvalues of, 415
singularities of solutions of, 501
Bessel functions 303-306 324-325 340
'
372,390-391, 402-403, 467-soi
asymptotic behavior of, 332-4
asymptotic representations of, 524532
functional relation for, 489
integral representations for 474-82
integral theorem for, 340-34i, 490
orthogonality relations for, 305
power series for, 482-484
relations between, 485
singularities of, 501
zeros of, 451, 492-496
Bessel's inequality
for functions, 51, 425
for vectors, 5
Bessel transformation, 490-492
Beta function, 483
Bezout's theorem, 520
Biharmonic equation, 370
Bilinear forms, 11-14
reciprocal, 12
resolvents of, 18, 29
Bilinear formula, for iterated kernels
'
137
Bilinear integral form, 123
Bilinear relation, 360
Biorthogonality relations, 404
du Bois-Reymond, theorem of, 200
Boundary, free, 208
Boundary conditions
containing a parameter, 392-393,
461
homogeneous and inhomogeneous,
277
natural, 208
551
INDEX
Boundary conditions (continued)
for the rod, 295
for the string, 291
Brachistochro ne, 170, 207
Broken extremals, 259
Buckling, variational problem of, 272
c
Calculus of perturbations , 344
Calculus of variations, 164-274
applied to eigenvalue problems
397-465
fundamental lemma of, 185
Canonical differential equations, 239
Canonical form of variational problems, 238
Castigliano's principle, 270
Catenary, 172, 218
Cayley's theorem, 22, 536-539
Characteristic values of a matrix, 17
Closed systems of functions, 110
Complete systems of functions, 52, 54
Completeness
of eigenfunction s of a differential
equation, 359-360, 369, 424
of Hermite polynomials, 96-97
of Laguerre polynomials, 96
of Legendre polynomial~, 82
of powers of x, 65, 101-102
of spherical harmonics, S12-513
of a system of functions, 110
of systems of functions of several
variables, 56
of systems of vectors, 4
of trigonometric functions, 68
Completeness relation
for vectors, 5
for functions, 51
Components of a function, 51
Conduction of heat, 311
Confocal rectangular parallelepiped ,
319
Conformal mapping, 377
Conjugate potential fo.nctions, 241
Conservation of momentum, 265
Constrained systems, eigenvalues of,
408
Constraints
on quadratic forms, 33, 44
on variational problems, 233
Continued fraction for the cotangent,
489
Continuous dependence of eigenvalues and eigenfunction s on the
kernel, 151
Continuous spectrum, 100, 339-343
Convergence in the mean, 98, 110
Convergence theorems of Lebesgue,
109-111
Cotangent, continued fraction for,
489
Cylinder functions, see Bessel,
Hankel, Mathieu, Neumann functions
Cylindrical domain, 319
D
Darboux method, 532-535
Definite kernels, 123, 162
Degeneracy of eigenvalues, 129
Degenerate kernels, 114-118, 129
Degenerate quadratic forms, 27
Dense function systems, 100-101
Derivative, variational, 186
Determinant inequality of Hadamard, 36
Dido's problem, 258
Differential equations
canonical, 239
equivalence with integral equations, 358-363
integral transformatio ns applied
to, 466-468
Dimension number, asymptotic, 63
Dini's theorem, 57
Dipole, 514-15
Direct solution of variational problems, 174-183
Dirichlet's discontinuous factor, 81
Dirichlet's integral formula, 78
Dirichlet's problem, 240
for the circle, 179
Discontinuou s factor of Dirichlet, 81
Discontinuou s kernels, 152
552
INDEX
Discrete spectrum, 130
Divergence expressions , 195
Du Bois-Reymo nd, theorem of, 200
E
Eigenfreque ncy, 283
Eigenfuncti ons
continuous dependence on kernel,
151
of a differential equation, nodes of,
451-455
completene ss of, 424
Kellogg's method for, 156
of a hrnel, 113
minin;mm properties of, 161
nodes of, 300, 302, 304, 395, 451-455,
458, 463
for rectangular membrane, 301
of symmetric kernels, 126-132, 148-,150
zeros of, 451-455
Eigenvalues
of Bessel's equation, asymptotic
behavior of, 415
of a constrained system, 408
continuous dependence on kernel,
151
degenerate, 129
of a differential equation, asymptotic distribution of, 429-445,
460
continuity of, 418-424
dependence on boundary conditions, 410-411
dependence on coefficients, 411
extremum properties of, 398-407
infinite growth of, 412
maximum-m inimum property of,
405
minimum properties of, 459
for a subdomain, 408-409
of functions of a kernel, 157
of functions of a matrix, 22
of infinite multiplicity , 154, 395
of a kernel, 113, 132, 137
of a matrix, 17, 26
for the membrane, 298
minimum-m aximum property, 31,
132-134, 405
multiple, 113, 129
negative, 294, 416
for the plate, asymptotic behavior
of, 460
of a positive definite kernel, 132
reciprocal, 17
for the rod, 296
for the string, 287
of Sturm-Liou ville problem, asymptotic behavior of, 414
of a symmetric kernel, 148-150
first, 122-125
maximum-m inimum property of,
132-134
totality of, 126-132
of a unitary matrix, 46
Eigenvalue problems
for closed surfaces, 461
definition of, 309
of mathematic al physics, 275-396
with two parameters, 460
of Schrodinger type, 445-450
variational methods for, 397-465
Eigenvector s, 24
use in solving linear equations, 30
Eigenvibrat ions, 283, 310
of homogeneou s string, 287
Elastic bodies, 268
Electrostati cs, 267
Elementary divisors of a matrix, 45
Ellipsoidal coordinates , 226
Elliptic coordinates , 319, 391-392
Elliptic functions, 227
Energy
kinetic, 242
potential, 242
Energy integral, 265
Enskog's method for symmetric integral equations, 156
Equicontinu ous functions, 58, 114,
119, 125
Equilibrium conditions of a mechanical system, 243
Equilibrium problems, 308
Erdmann vertex condition, 260
553
INDEX
Euler equations, 183-207
homogeneous form of, 196
integration of, 206
invariance of, 222
for isoperimetric problem, 207
Euler·expression, identical vanishing
of, 193
Euler integral, 483
Euler transformation, 467
Expansions, asymptotic, 522-535
Expansion theorem, 310
for definite kernels, 138-140
for differential equations, 360-361,
427-428
for iterated kernels, 137
limits of validity of, 395--396
for spherical harmonics, 513
for symmetric integral equations,
for arbitrary kernels, 118-122, 150
for degenerate kernels, 115-118
Schmidt's derivation of, 155
Free boundaries, 208
Free vibrations, 282-283
Friedrichs transformation, 234
Function space, 57
gradient in, 222
Functions of a kernel, eigenvalues of,
157
Functions of a matrix, eigenvalues of,
22
Functionals, 167-169
Fundamental domain, 48, 112
Fundamental lemma of the calculus
of variations, 185
Fundamental solution, 370
of an ordinary differential equation,
353
134-136
Extrema, Weierstrass's theorem on,
Fundamental tone, 286
20, 164
Extremals, 185
broken, 259
Extremum properties of eigenvalues,
398-407
F
Fejer's summation theorem, 102
Fermat's principle, 165
Finite differences, 176
First variation, 184, 208-14
Fischer-Riesz theorem, 110
Forced motion
of the membrane, 300
of the string, 289, 294
Forced vibrations, 285, 389-390
Forms, see under particular type of
form; e.g., bilinear, quadratic,
Hermitian, etc.
Form factor, 286
Fourier coefficients, 51, 424
order of magnitude of, 74
Fourier integral, 77--82, 98-100
Fourier integral theorem, 78
Fourier series, 69-77
Fredholm formulas, 142-147
Fredholm's theorems
G
Generalized Green's function, 356357
Generating function
for Hermite and Laguerre polynomials, 510
for spherical harmonics, 539-541
Geodesic curves, 170, 189, 213
Gibbs' phenomenon, 105--107
Gradient in function space, 222
Gram determinant
of a system of functions, 62, 107108
of a system of vectors, 34-36
orthogonalization
Gram-Schmidt
process, 4, 50
Green's formula, 278, 280
Green's function, 311, 351-388
for the biharmonic equation, 370
for a circular ring, 386-388
obtained by conformal mapping,
377
construction by reflection, 378-379,
384-385
examples of, 371-388
in the generalized sense, 356-357
INDEX
for Hermite's equation, 373-375
as kernel of an integral equation,
358
for Laguerre's equation, 375-376
for ordinary differential equations,
351-363
construction of, 354-355
definition of, 352-353
of higher order, 362-363
for the potential equation, 363370, 377
in the plane, 384-386
in space, 378-384
symmetry of, 354
Green's tensors for systems of differential equations, 393-394
Group, orthogonal, 539
H
Haar, theorem of, 203
Hadamard's inequality for determinants,. 36
Hamilton's variational principle, 242
Hammerstein's theorem, 162
Hankel functions, 468-473
asymptotic representations for,
524-532
integral representation of, 476
singularities of, 501
Harmonic functions, see potential
equation
Heat conduction, equation of, 311
Herglotz, theorem of, 542
Hermite equation, 508
Green's function for, 373-375
Hermite functions, 508-509
Hermite orthogonal functions, 154
Hermite polynomials, 91-93, 328
completeness of, 96-97
generating function for, 510
Hermitian forms, 13, 28
Hilbert space, 55
orthogonal transformations in, 55
Holonomic conditions, 221
Hurwitz's solution of the isoperimetric problem, 97-98
I
Independence, measure of, 61
lndicatrix, 258
Inertial theorem for quadratic forms,
28
Infinitely many variables, 168, 177
applied to integral equations, 160
Infinitesimal linear transformations,
41-42
Influence function, see Green's function
Inner product
of functions, 49
of vectors, 1
Integral equation of Abel, 158
Integral equations (linear), 112-163
alternative theorem for, 115
with discontinuous kernels, 152
Enskog's method for, 156
equivalence with differential equations, 358-363
Fredholm theorems for, 115-122,
150
for functions of several variables,
152
homogeneous, 112
·inhomogeneous, solution of, 136
of the first kind, 159
of the second kind, 112
of the third kind, 161
polar, 161
reciprocal, 141
singular, 154
symmetric, 156
transposed, 116
Volterra, 158, 337-339
Integral formula of Dirichlet, 78
Integral representations
of Bessel functions, 474-482
of Hankel functions, 476
of Hermite function, 508-509
of Laguerre functions, 509-510
of Legendre functions, 502-505
of Neumann functions, 498
of Tchebycheff functions, 507-508
555
INDEX
Integral theorem
for Bessel functions, 340-341, 490
of Fourier, 78
Integral transforms, 113
Integral transformat ions
applied to differential equations,
466-468
definition of, 112
of Mellin, 104
Integro-diff erential equations, 404
Invariance of Euler equations, 222
Invariant variational problems, 262
Involutory transformat ion of variational problems, 233
Isoperimetr ic problem(s), 171-172,
216
Euler equation of, 207
Hurwitz solution of, 97
for plane curves, 174
for polygons, 166
reciprocity for, 258
Iterated bilinear form, 12
Iterated kernels, 137
J
Jacobi's method of multiplicati ve
variation, 458
Jacobi polynomials , 90-91, 327-328
Jump conditions, 404
K
Kellogg's method for eigenfuncti ons,
156
Kernels
definite, 123, 162
definition of, 112
degenerate, 114-118, 129
discontinuo us, 152
eigenfunctio ns of, 113
eigenvalues of, 113, 132, 137
expansion theorem for, 134-140
functions of, 157
iterated, 137
positive definite, 132
possessing no eigenvalues , 157, 158
reciprocal, 141
resolvents of, 141-147, 162
symbolic functions of, 157
symmetric, see Symmetric kernels
symmetriza ble, 161
unsymmetri c, adjoint orthogonal
systems of, 159
Kinetic energy, 242
L
La.grange's equations of motion, 243
Lagrange multipliers, 165
Laguerre's equation, Green's function for, 375-376
Laguerre functions, 509-510
Laguerre orthogonal functions, 154
Laguerre polynomials , 93-97, 329-330,
342-343
completenes s of, 95
generating function for, 510
Lame equation, 320
Lame functions, 320
Lame problem, 319-324
Laplace's equation, see potential
equation
Laplace integral, 82
Laplace's integral representat ion for
Legendre functions, 503-504
Laplace spherical harmonics, 316,
510-521
Laplace transformat ion, 467
Laplacian, transformat ion to polar
coordinates , 224
Least time, principle of, 165
Lebesgue convergence theorem, 109111
Lebesgue integral, 108-111
Legendre condition, 186, 200, 202, 214
strong form of, 216
Legendre functions, 325-327, 501-507
associated, 327, 505
integral representati ons for, 502505
of the second kind, 504-505
Legendre polynomials , 82-87, 402
asymptotic expansion of, 533-535
556
INDEX
completeness of, 82
differential equation for, 86
generating function for, 85
of higher order, 325, 505
minimum property. of, 86
recursion formula for, 86
as special spherical harmonics, 318
Light rays, 165, 170, 189, 213, 258
Linear dependence
of functions, 50
of vectors., 2
Linear differential operator, 275
Linear equations, 5-6
perturbations of, 42-44
solution by means of eigenvectors,
30
Linear transformations, 1-47
infinitesimal, 41-42
M
Mathematical physics, problems of,
275-396
Mathieu functions, 392
Matrices, 7-11
characteristic values of, 17
eigenvalues of, 17, 26
elementary divisors of, 45
functions of, 22
spectra of, 17
Maximizing sequences, 175
Maximum-minimum property
of eigenvalues of a differential
·
equation, 405
of nodal points, 463
Maxwell-Sylvester representation for
spherical harmonics, 514-521
Mean, approximation in the, 52, 54
Mean convergence, 98, 110
Mean square error, 52
Measure of independence, 61
Measure of a set, 108
Mellin transformation, 103-105
Membrane,246, 249, 297-307,463
circular, 302-306
eigenvalues for, 298
forced motion of, 300
non-homogeneous,306
rectangular, 300--302
Mercer's theorem, 138-140
application to differential equations, 360, 363, 369
Minimal surfaces, 193, 258
Minimizing sequences, 175
Minimum property
of eigenfunctions, 161
of eigenvalues, derived from completeness, 456
of Legendre polynomials, 86
Minimum-maximum principle, for
quadratic forms, 31
for symmetric kernels, 132-134
Momentum, conservation of, 265
Multiple Fourier series, 73
Multiplicative variation, Jacobi's
method of, 458
Multiplicity of an eigenvalue, 113,
129
Multipliers, Lagrange, 165
Multiplier rule for variational problems, 222
Multipole, 515
Mi.intz's theorem, 101-102
N
Natural boundary conditions, 208
Neighborhood of a function, 169
Neumann functions, 471-473, 496-501
singularities of, 501
Neumann series, of bilinear forms, 18
for integral equations, 140-142
of matrices, 10
Nodal lines, 300, 302, 304, 395
Nodal points for Sturm-Liouville
problem, 454, 463
Nodes,451-454,458
Noether's theorem, 262
Norm
of a function, 49
of a vector, 2
Normal coordinates, 282
Normal modes, 283
Null sets, 108
557
INDEX
0
Orthogonal function system, of
Hermite, 154
of Laguerre, 154
Orthogonal group, 539
Orthogonal systems
of functions, 49-50, 87
of vectors, 3-5
Orthogonal transformations, 14-16,
536
in Hilbert space, 55
Orthogonalization
of functions, 50
of vectors, 4
Orthonormal system
of functions, 49
of vectors, 3
Oscillation theorem, 454
Overtones, 286
p
Paraboloidal coordinates, 229
Parallelepiped, confocal, 319
Parameters in boundary conditions,
392-393, 461
Particle mechanics, integrals in, 265
Perturbations, 42
Perturbation theory, 343-350
Picard's theorem, 160
Piecewise continuous functions 48
Piecewise smooth functions, 48 '
Pitch, 283
Plate,250-251,307-308,403,463
asymptotic behavior of eigenvalues
for, 460
Plucked string, 388
Point force, 351
Poisson's equation, 368
Poisson's integral, 513-14
Poisson's summation formula, 76
Polar form of a quadratic form, 12
Polar integral equations, 161
Polynomials, completeness of, 65,
101-102
Potential energy, 242
Potential equation
Green'~ function for, 363-370, 377
in the plane, Green's function for,
384-6
in space, Green's function for, 378384
transformation to polar coordinates,
224
Potential functions, conjugate, 241
Potential theory, 178-182, 315-324,
363-371, 377-388
Principal axes transformations, of
quadratic forms, 23-31
of two quadratic forms simultaneously, 37-41
Principle of Castigliano, 270
Principle of Fermat, 165
Propagation of light, 170, 189, 214
Pythagorean theorem for function
spaces, 253
Q
Quadratic forms, 12-16, 23-34
constraints on, 33, 44
degenerate, 27
inertial theorem for, 28
polar forms of, 12
simultaneous transformation to
principal axes, 37-41
transformation to principal axes,
23-31
Quadratic integral form, 122
Quaternions, 539
R
Rayleigh-Ritz method, 175
Reciprocal bilinear form, 12
Neumann series for, 19
Reciprocal eigenvalue, 17
Reciprocal kernel, 141
Reciprocal spectrum, 17
Reciprocal variational problems, 235,
252-257, 270
Reciprocity for isoperimetric problems, 258
Reciprocity formulas for Fourier
integrals, 80
558
INDEX
Rectangular parallelepiped, confocal,
319
Recursion formula for Legendre polynomials, 86
Reflection, law of, 165
Reflection method of constructing
Green's function, 378-379, 384385
Refraction, law of, 166
Regions with separate components,
405
Rellich, lemma of, 414, 450
Re.solvent
of a bilinear form, 18, 29
of a kernel, 141-147, 162
of a symmetric kernel, 142
Resonance, 285
Riesz-Fischer theorem, 110
Ritz, see Rayleigh-Ritz
Rod, 245, 295-297
boundary conditions for, 295
eigenvalues for, 296
s
Saddle point method, 526-533
Schli:fli's integral, 501-502
Schmidt method for derivation of
Fredholm theorem, 155
Schmidt orthogonalization process,
see Gram-Schmidt
Schwarz inequality
for functions, 49
for vectors, 2
Schrodinger eigenvalue problems,
445-450
Schrodinger's equation, 342-343
Second variation, 214
Self-adjoint differential expression,
279
Separated regions, 405
Separation of variables, 287, 313
Series
Fourier, 69--77
Neumann, 10, 18, 140--142
Series expansions of arbitrary functions, 48-111
Singular equations, eigenvalues of,
461
Singular integral equations, 154
Small vibrations, 244
Smooth sequences of functions, 63
Snell's law of refraction, 166
Spectral decomposition, 99-100
Spectral density, 100
Spectrum,339-340
continuous, 100, 339--343
discrete, 130
of a matrix, 17
reciprocal, 17
of a unitary matrix, 46
Spherical coordinates, 225
Spherical harmonics, 314, 316, 342343, 510--521
completeness of, 512-513
expansion theorem for, 513
generating function for, 539--541
Maxwell-Sylvester representation
for, 514-521
symmetric, 512
transformations of, 535-545
Spheroidal coordinates, 229
Stationary functions and curves, 186
Stationary values in variational problems, 186
Steepest descent, method of, 526--533
Steiner's problem, 166
Stirling's formula, 522-524
String, 244, 388-392, 463
boundary conditions for, 291
eigenvalues for, 287
free motion of, 287
forced vibrations of, 289, 389
impulsive excitation of, 389
non-homogeneous, 291
plucked, 388
Sturm-Liouville problem, 291-295
asymptotic behavior of eigenvalues
of, 414
asymptotic behavior of solutions of,
331-339
eigenvalue problem for, 402
singular case, 324
zeros of eigenfunctions of, 468
559
INDEX
Subsidiary conditions for variational
problems, 216
Summation formula of Poisson, 76
Superposition, principle of, 276
Sylvester, 518
Symmetric integral equations,
Enskog's method for, 156
Symmetric kernels
definition of, 115
eigenfunctions of, 126-132, 148-150
eigenvalues of, 122-134, 148-150
expansion theorem for, 134-136
resolvents of, 142
Symmetrizable kernels, 161
Synchronism, 309
Synchronous vibrations, 286
T
Tchebycheff functions, 507-508
Tchebycheff polynomials, 88-90, 328
Tensors, 7
Green's, 393-394
Thales, theorem of, 255
Theta function
applications of, 381-384, 388
functional equation for, 75
Thomson's principle, 267
Three-dimension al continua, vibrations of, 313
Tone, fundamental, 286
Transformation(s )
Bessel, 490-492
of Euler, 467
integral, 112
linear, 1-47
infinitesimal, 41-42
orthogonal, 14-16
of quadratic forms to principal
axes, 23-31, 37-41
unitary, 16
of variational problems, 231-242
Transforms, integral, 113
Transposed integral equation, 116
Transversality, 211
for multiple integrals, 266
u
Unitary matrix, 11
spectrum of a, 46
Unitary transformations, 16.
Unsymmetric kernels, adjoint orthogonal systems for, 159
v
Variation
first, 186
of a function, 184
second, 214
Variational derivative, 186
Variational principle of Hamilton,
242
Variational problems
canonical form of, 238
constraints on, 233
direct solution of, 174-183
invariant, 262
involutory transformation of, 233
multiplier rule for, 222
reciprocal, 235, 252-257, 270
with subsidiary conditions, 216
transformations of, 231-42
with variable domains, 260
without a solution, 173
Variations, calculus of, 164-274
Vectors, 1-5
Vertex condition of Weierstrass and
Erdmann, 260
Vibrating membrane, see Membrane
Vibrating plate, see Plate
Vibrating rod, see Rod
Vibrating string, see String
Vibration problems of mathematical
physics, 275-396
Vibrations
forced, 285, 389-390
free, 282-283
small, 244
Volterra integral equations, 158, 337339
560
INDEX
w
Wave front, 214
Weierst rass-Erd mann vertex condition, 260
Weierst rass's approxi mation theorem ,
65
Weierst rass's theorem on extrema , 20,
164
Weight function , 87
z
Zeros of Bessel function s, 451, 492-
496
Zeros of eigenfun ctions, 451-455