Complex Modulus

Min Max

Min   Max       Re       Im

The modulus of a complex number , also called the complex norm, is denoted  and defined by

(1)

If  is expressed as a complex exponential (i.e., a phasor), then

(2)

The complex modulus is implemented in the Wolfram Language as Abs[z], or as Norm[z].

The square  of  is sometimes called the absolute square.

Let  and  be two complex numbers. Then

			(3)
			(4)

so

(5)

Also,

			(6)
			(7)

so

(8)

and, by extension,

(9)

The only functions satisfying identities of the form

(10)

are , , and  (Robinson 1957).

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.

Krantz, S. G. "Modulus of a Complex Number." §1.1.4 n Handbook of Complex Variables. Boston, MA: Birkhäuser, pp. 2-3, 1999.

Robinson, R. M. "A Curious Mathematical Identity." Amer. Math. Monthly 64, 83-85, 1957.