Inverse Hyperbolic Cosecant

Author

Eric W. Weisstein
August 24, 2008

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

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

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

Definition

TrigToExp[ArcCsch[z]]

FullSimplify[ArcCsch[z]-Log[((1+Sqrt[1+z^2])/z)],z>0]

FullSimplify[ArcCsch[z]-Log[((1-Sqrt[1+z^2])/z)],z<0]

Plots

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

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

Real Axis

PlotDiscontinuous[ArcCsch[x],{x,-10,0,10},
PlotStyle->Red,
PlotRange->{Automatic,{-5,5}},AxesLabel->TraditionalForm/@{x,ArcCsch[x]}]

Complex Plane

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

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

Special Values

ArcCsch[0]

Limit[ArcCsch[x],x->0,Direction->#]&/@{-1,1}

Derivative and Integral

Derivative

D[ArcCsch[z],z]

Integral

Integrate[ArcCsch[z],z]

Series

<<DiscreteMath`RSolve`

About 0

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

l=Rest[First/@Partition[CoefficientList[Normal[Series[ArcCsch[x],{x,0,20}]],x],2]]

Through[{Numerator,Denominator}[l]]

SeriesTerm[ArcCsch[z],{z,0,n}]

SeriesTerm[ArcCsch[z]+Log[z],{z,0,n}]

About ∞

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

Series[ArcCsch[x],{x,∞,10}]

SeriesTerm[ArcCsch[x],{x,∞,n}]

SeriesTerm[ArcCsch[1/x],{x,0,n}]

Sum[LegendreP[-1+n,0]/n (x^n),{n,10}]

l=Table[LegendreP[-1+n,0]/n,{n,1,20,2}]

Through[{Numerator,Denominator}[l]]

Sum[(((((-1))^(n+1))Pochhammer[1/2,n-1])/((n-1)!(2n-1)))(x^(-2n+1)),{n,10}]

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

FullSimplify[%]

Sums

(∑_n^∞)((((((-1))^(n+1)) (Subscript[((1/2)), n-1])) (x^(1-2 n)))/((2 n-1) (n-1)!))==(csch^(-1))(x)

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

FullSimplify[%]

(∑_k^∞)(((Subscript[P, k-1])(0) (x^(-k)))/k)==(csch^(-1))(x)

Sum[((LegendreP[k-1,0])/k)(x^(-k)),{k,∞}]

With[{x=1.1},{Sum[((LegendreP[k-1,0])/k)(x^(-k)),{k,50}],ArcCsch[x]}]

(∑_n^∞)((((((-1))^(n+1)) (2 n-1)!!) (x^(2 n)))/(2 n (2 n)!!))-ln(x)+ln(2)==(csch^(-1))(x)

Table[(((((-1))^(n+1))((2n-1)!!))/(2n(2n)!!))(x^(2n)),{n,10}]

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

FullSimplify[%]

FullSimplify[%-ArcCsch[x]]

FullSimplify[-Log[x]+Log[1+(Sqrt[1+(x^2)])]-ArcCsch[x],x>0]

Table[-ArcCsch[x]+Log[((1+(Sqrt[1+(x^2)]))/x)]/.x->RandomReal[{0,10}],{100}]//Chop//Union

{-Log[x]+Log[1+(Sqrt[1+(x^2)])],ArcCsch[x]}/.x->-1`20*^-20