Inverse Function

Author

Eric W. Weisstein
May 6, 2004

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Calculus/InverseFunction.nb.

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

©2005 Wolfram Research, Inc. except for portions noted otherwise

Classes of Inverse Functions

<<Utilities`Plot`

Linear (Reversible)

{3(z/3),(3z)/3}

One-way

Sqrt[z^2]

{Sqrt[z]^2,Sqrt[z^2]}

ComplexPlot3D[Sqrt[z^2],{z,-5,5},PlotPoints->25,Mesh->False]

ComplexContourPlot[Sqrt[z^2],{z,-5,5}]

ArcSin[Sin[z]]

{Sin[ArcSin[z]],ArcSin[Sin[z]]}

ComplexPlot3D[ArcSin[Sin[z]],{z,-4Pi,4Pi}]

ComplexContourPlot[ArcSin[Sin[z]],{z,-4Pi,4Pi}]

Log[Exp[z]]

{Exp[Log[z]],Log[Exp[z]]}

ComplexPlot3D[Log[Exp[z]],{z,-4Pi,4Pi}]

ComplexContourPlot[Log[Exp[z]],{z,-4Pi,4Pi}]

((((z^(5/3))))^((3/5)))

{(z^(3/5))^(5/3),(z^(5/3))^(3/5)}

{(z^(3/2))^(2/3),(z^(2/3))^(3/2)}

ComplexPlot3D[(z^(5/3))^(3/5),{z,-4Pi,4Pi},PlotPoints->20,Mesh->False]

ComplexContourPlot[(z^(5/3))^(3/5),{z,-4Pi,4Pi}]

Neither way

{(z^a)^(1/a),(z^(1/a))^a}

{(z^(2/3+3/5I))^(1/((2/3+3/5I))),
(z^(1/((2/3+3/5I))))^(2/3+3/5I)}

Mathematica

?InverseFunction

InverseFunction[#][x]&/@{Sin,Cos,Tan,Sinh,Cosh,Tanh,Exp,Log,Sqrt,#&,#^2&}

InverseFunction[#&][x]

InverseFunction[ArcSin][x]

InverseFunction[Sqrt]//InputForm

Evaluate[InverseFunction[Power,1,2][#,1/2]]&

InverseFunction[Power,1,2][x,2]

InverseFunction[Sqrt][x]

InverseFunction[Power,1,2][x,3]

InverseFunction[#^2&]

Counterintuitive Property

Simplify[(Sqrt[z])(Sqrt[(1/z)]),Im[z]≠0]

Simplify[(Sqrt[z])(Sqrt[(1/z)]),Im[z]==0&&Re[z]>0]

Simplify[(Sqrt[z])(Sqrt[(1/z)]),Im[z]==0&&Re[z]<0]

Plot

<<Graphics`Colors`

Plot[(Sqrt[z])(Sqrt[(1/z)]),{z,-2,2},PlotStyle->Red]

Plot3D[(Sqrt[x+I y])(Sqrt[(1/(x+I y))]),{x,-2,2},{y,-2,2},PlotPoints->100,Mesh->False]