Inverse Hyperbolic Cotangent

Author

Eric W. Weisstein
August 24, 2008

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

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

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

Definition

FactorTerms[TrigToExp[ArcCoth[z]]]

FullSimplify[ArcCoth[x]-Log[(x+1)/(x-1)]/2,#]&/@{x<0,x>1}

Plots

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

Branch Cuts

BranchCuts[ArcCoth]

With[{f=ArcCoth},
BranchCutPlot[f,Epilog->Text[TraditionalForm[f[z]],{-2,2.3}]]
]

Real Axis

PlotDiscontinuous[ArcCoth[x],{x,{-10,-1},{1,10}},PlotRange->{Automatic,{-4,4}},
VerticalAsymptotes->{-1,1},
HorizontalAsymptotes->{0},
PlotStyle->Red,
AxesLabel->TraditionalForm/@{x,ArcCoth[x]}]

Complex Plane

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

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

Derivative and Integral

Derivative

D[ArcCoth[z],z]

Integral

Integrate[ArcCoth[z],z]

Special Values

ArcCoth[0]

ArcCoth[1]

ArcCoth[∞]

ArcCoth[I]

ArcCoth/@{-1,1}

Limit[ArcCoth[x],x->#,Direction->-#]&/@{-1,1}

Series

About 0

Limit[ArcCoth[x],x->0,Direction->-1]

Limit[ArcCoth[x],x->0,Direction->1]

Series[ArcCoth[x],{x,0,7}]

Series[ArcCoth[x],{x,0,7}]

V7

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

About ∞

Series[ArcCoth[1/x],{x,0,7}]

Sums

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

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

FullSimplify[%]

{((I π)/2)+ArcTanh[x],ArcCoth[x]}/.x->1.1

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

Sum[((x^(-(2n-1)))/(2n-1)),{n,∞}]

FullSimplify[%]