Complex Modulus
AbsoluteValue
AbsReIm
AbsContours
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)
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If z is expressed as a complex exponential (i.e., a phasor), then
| |re^(iphi)|=|r|. |
(2)
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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)| = [画像:|(Ae^(iphi_1))/(Be^(iphi_2))|=A/B|e^(i(phi_1-phi_2))|=A/B]
(3)
(|c_1|)/(|c_2|) = [画像:(|Ae^(iphi_1)|)/(|Be^(iphi_2)|)=A/B(|e^(iphi_1)|)/(|e^(iphi_2)|)=A/B,]
(4)
so
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)
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and, by extension,
| |z^n|=|z|^n. |
(9)
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The only functions satisfying identities of the form
| |f(x+iy)|=|f(x)+f(iy)| |
(10)
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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 PartRelated Wolfram sites
http://functions.wolfram.com/ComplexComponents/Abs/Explore with Wolfram|Alpha
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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 ModulusCite this as:
Weisstein, Eric W. "Complex Modulus." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ComplexModulus.html