Absolute Square

Author

Eric W. Weisstein
August 18, 2007

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/AbsoluteSquare.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/AbsoluteSquare.html.

©2007 Wolfram Research, Inc. except for portions noted otherwise

Plots

<<Utilities`Plot`

Plot[Abs[x]^2,{x,-1.5,1.5},AspectRatio->Automatic,AxesLabel->{x,Abs[x]^2},ImageSize->200]

ComplexPlot3D[Abs[z]^2,{z,-2,2},ComplexPlotLabel->HoldForm[Abs[z]^2]]

ComplexContourPlot[Abs[z]^2,{z,-5,5},ComplexPlotLabel->HoldForm[Abs[z]^2]]

Expansion Identities

ComplexExpand[Abs[x+I y]^2]

ComplexExpand[Abs[x+I y]^2,TargetFunctions->{Conjugate}]

FullSimplify[%]

FullSimplify[ComplexExpand[Abs[a+b Exp[-I δ]]^2,TargetFunctions->{Conjugate}]]

FullSimplify[ComplexExpand[Abs[1+b Exp[-I δ]]^2,TargetFunctions->{Conjugate}]]

FullSimplify[ComplexExpand[Abs[1-Exp[-I δ]]^2,TargetFunctions->{Conjugate}]]

Simplify[TrigFactor[%]]

FullSimplify[ComplexExpand[Abs[Exp[I (Subscript[ϕ, 1])]+Exp[I (Subscript[ϕ, 2])]]^2,TargetFunctions->{Conjugate}]]

Simplify[TrigFactor[%]]