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Complex Argument


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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

 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,

 theta={-1/2pi   if y<0; undefined   if y=0; 1/2pi   if y>0.
(3)

Special values of the complex argument include

arg(1)=0
(4)
arg(1+i)=1/4pi
(5)
arg(i)=1/2pi
(6)
arg(-1)=pi
(7)
arg(-i)=-1/2pi.
(8)

From the definition of the argument, the complex argument of a product of two numbers is equal to the sum of their arguments,

arg(zw)=arg(|z|e^(itheta_z)|w|e^(itheta_w))
(9)
=arg(e^(itheta_z)e^(itheta_w))
(10)
=arg[e^(i(theta_z+theta_w))]
(11)
=arg(z)+arg(w).
(12)

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 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

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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