Absolute Square

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Min Max Re Im

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

(1)

where denotes the complex conjugate of and is the complex modulus.

If the complex number is written , with and real, then the absolute square can be written

(2)

If is a real number, then (1) simplifies to

(3)

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

An important identity involving the absolute square is given by

			(4)
			(5)
			(6)

If , then (6) becomes

			(7)
			(8)

If , and , then

(9)

Finally,

			(10)
			(11)
			(12)