Complex Argument
A complex number z may be represented as
| z=x+iy=|z|e^(itheta), |
(1)
|
where |z| is a positive real number called the complex modulus of z, and theta (sometimes also denoted phi) 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 z is implemented in the Wolfram Language as Arg [z].
The complex argument can be computed as
| [画像: arg(x+iy)=tan^(-1)(y/x). ] |
(2)
|
Here, theta, sometimes also denoted phi, corresponds to the counterclockwise angle from the positive real axis, i.e., the value of theta such that x=costheta and y=sintheta. The special kind of inverse tangent used here takes into account the quadrant in which z 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 -pi<theta<=pi. In the degenerate case when x=0,
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), |
(13)
|
giving the special case
| arg(z^n)=narg(z). |
(14)
|
Note that all these identities will hold only modulo factors of 2pi if the argument is being restricted to theta in (-pi,pi].
See also
Affix, Argument, Complex Modulus, Complex Number, de Moivre's Identity, Euler Formula, Imaginary Part, Inverse Tangent, Phase, Phasor, Real PartRelated Wolfram sites
https://functions.wolfram.com/ComplexComponents/Arg/Explore with Wolfram|Alpha
More things to try:
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 ArgumentCite this as:
Weisstein, Eric W. "Complex Argument." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ComplexArgument.html