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Complex Modulus

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

|x+iy|=sqrt(x^2+y^2).
(1)

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

|re^(iphi)|=|r|.
(2)

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

The square |z|^2 of |z| is sometimes called the absolute square.

Let c_1=Ae^(iphi_1) and c_2=Be^(iphi_2) be two complex numbers. Then

|(c_1)/(c_2)|=|(Ae^(iphi_1))/(Be^(iphi_2))|=A/B|e^(i(phi_1-phi_2))|=A/B
(3)
(|c_1|)/(|c_2|)=(|Ae^(iphi_1)|)/(|Be^(iphi_2)|)=A/B(|e^(iphi_1)|)/(|e^(iphi_2)|)=A/B,
(4)

so

|(c_1)/(c_2)|=(|c_1|)/(|c_2|).
(5)

Also,

|c_1c_2|=|(Ae^(iphi_1))(Be^(iphi_2))|=AB|e^(i(phi_1+phi_2))|=AB
(6)
|c_1||c_2|=|Ae^(iphi_1)||Be^(iphi_2)|=AB|e^(iphi_1)||e^(iphi_2)|=AB,
(7)

so

|c_1c_2|=|c_1||c_2|
(8)

and, by extension,

|z^n|=|z|^n.
(9)

The only functions satisfying identities of the form

|f(x+iy)|=|f(x)+f(iy)|
(10)

are f(z)=Az, f(z)=Asin(bz), and f(z)=Asinh(bz) (Robinson 1957).

See also

Absolute Square, Absolute Value, Complex Argument, Complex Number, Imaginary Part, Maximum Modulus Principle, Minimum Modulus Principle, Real Part

Related Wolfram sites

http://functions.wolfram.com/ComplexComponents/Abs/

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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.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.

Referenced on Wolfram|Alpha

Complex Modulus

Cite this as:

Weisstein, Eric W. "Complex Modulus." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ComplexModulus.html

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