Inverse Hyperbolic Sine

Author

Eric W. Weisstein
August 24, 2008

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

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

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

Definition

TrigToExp[ArcSinh[x]]

Plot

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

Branch Cuts

BranchCuts[ArcSinh]//TraditionalForm

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

Real Aixs

Plot[ArcSinh[x],{x,-10,10},
PlotStyle->Red,AxesLabel->TraditionalForm/@{x,ArcSinh[x]}]

Complex Plane

ComplexPlot3D[ArcSinh[z],{z,-5,5},PlotPoints->70,Mesh->False]//Timing

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

Derivative and Integral

Derivative

D[ArcSinh[z],z]

Integral

Integrate[ArcSinh[z],z]

Special Values

ArcSinh[0]

Limit[ArcSinh[x],x->∞]

Series

About 0

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

V5

<<DiscreteMath`RSolve`

SeriesTerm[ArcSinh[x],{x,0,n}]

Sum[LegendreP[n-1,0]/n (x^n),{n,15}]

V7

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

About ∞

Series[ArcSinh[x],{x,Infinity,10}]//Simplify

V7

SeriesCoefficient[ArcSinh[x]+Log[1/x],{x,∞,n}]

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

Sums

(∑_(n=0)^∞)((((((-1))^n) (2 n-1)!!) (x^(2 n+1)))/((2 n+1) (2 n)!!))==(sinh^(-1))(x)

Sum[(((((-1))^n)(2n-1)!!)/((2n+1)(2n)!!))(x^(2n+1)),{n,0,∞}]

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

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

FullSimplify[%]

FullSimplify[%-ArcSinh[x]]

-ArcSinh[x]+Log[1+(Sqrt[1+(1/(x^2))])]-Log[(1/x)]/.x->.9