Inverse Hyperbolic Tangent

Author

Eric W. Weisstein
August 24, 2008

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

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

©2008 Wolfram Research, Inc. except for portions noted otherwise

Definition

TrigToExp[ArcTanh[x]]

FullSimplify[1/2Log[(1+x)/(1-x)]-ArcTanh[x],x<1]

Plot

<<Utilities`Plot`
<<Utilities`Typesetting`

Branch Cuts

BranchCuts[ArcTanh]//TraditionalForm

With[{f=ArcTanh},
BranchCutPlot[f,Epilog->Text[TraditionalForm[f[z]],{-2,2.3}],
PlotRange->2.7{{-1,1},{-1,1}}]
]

Real Axis

PlotDiscontinuous[ArcTanh[x],{x,-1,1},
AxesLabel->TraditionalForm/@{x,ArcTanh[x]},
VerticalAsymptotes->{-1,1},
PlotRange->{Automatic,{-5,5}}]

ComplexPlot3D[ArcTanh[z],{z,-5,5},PlotPoints->70,ClipFill->None,Mesh->False]

ComplexContourPlot[ArcTanh[z],{z,-5,5}]//Timing

Derivative and Integral

Derivative

D[ArcTanh[z],z]

Integral

Integrate[ArcTanh[z],z]

Special Values

Limit[ArcTanh[x],x->1]

ArcTanh[0]

ArcTanh[1]

ArcTanh[∞]

ArcTanh[I]

Series

About 0

Series[ArcTanh[x],{x,0,10}]

SeriesCoefficient[ArcTanh[x],{x,0,n}]

About ∞

Expansion about point on branch cut

Series[ArcTanh[1/x],{x,0,10}]

Series[ArcTanh[x],{x,Infinity,10}]

SeriesCoefficient[ArcTanh[x],{x,∞,n}]

Sums

(∑_n^∞)((x^(2 n-1))/(2 n-1))==(tanh^(-1))(x)

Sum[((x^(2n-1))/(2n-1)),{n,∞}]

(∑_n^∞)((x^(1-2 n))/(2 n-1))-((i π)/2)==(tanh^(-1))(x)

-I (π/2)+Sum[((x^(-2n+1))/(2n-1)),{n,∞}]

FullSimplify[%]

{ArcTanh[x],-((I π)/2)+ArcCoth[x]}/.x->.1