Complex Argument

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A complex number z may be represented as


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


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,

 theta={-1/2pi   if y<0; undefined   if y=0; 1/2pi   if y>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


giving the special case


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 Part

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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.

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