Complex Argument (original) (raw)
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A complex number 
may be represented as
| 
| (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
 |
(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 functionArcTan[x,_y_], and is often (including by the Wolfram Language function Arg) restricted to the range 
. In the degenerate case when 
,
 |
(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
 |
(13) |
|---|
giving the special case
 |
(14) |
|---|
Note that all these identities will hold only modulo factors of 
if the argument is being restricted to ![theta in (-pi,pi]](https://mathworld.wolfram.com/images/equations/ComplexArgument/Inline43.svg).
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/
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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
Cite this as:
Weisstein, Eric W. "Complex Argument." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ComplexArgument.html