Absolute Square

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The absolute square of a complex number z, also known as the squared norm, is defined as


where z^_ denotes the complex conjugate of z and |z| is the complex modulus.

If the complex number is written z=x+iy, with x and y real, then the absolute square can be written


If z=x+0i is a real number, then (1) simplifies to


An absolute square can be computed in terms of x and y using the Wolfram Language command ComplexExpand[Abs[z]^2, TargetFunctions -> {Conjugate}].

An important identity involving the absolute square is given by


If a=1, then (6) becomes


If a=1, and b=1, then




See also

Complex Argument, Complex Modulus, Complex Number, Imaginary Part, Real Part, Sign

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Cite this as:

Weisstein, Eric W. "Absolute Square." From MathWorld--A Wolfram Web Resource.

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