Complex Modulus (original) (raw)

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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

so

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

Also,

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

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.

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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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