Inverse Hyperbolic Secant

Author

Eric W. Weisstein
August 24, 2008

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

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

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

Definition

TrigToExp[ArcSech[x]]

FullSimplify[%,x>0]

FullSimplify[ArcSech[x]-Log[((1-(Sqrt[1-(x^2)]))/x)],x<-1]

Plot

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

Branch Cuts

BranchCuts[ArcSech]//TraditionalForm

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

Real Axis

PlotDiscontinuous[ArcSech[x],{x,0,1},AxesLabel->TraditionalForm/@{x,ArcSech[x]},
PlotStyle->Red,
VerticalAsymptotes->{0},
Epilog->{Red,PointSize[.02],Point[{1,0}]}]

Complex Plane

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

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

Derivative and Integral

Derivative

D[ArcSech[z],z]

Integral

Integrate[ArcSech[z],z]

Special Values

{Limit[ArcSech[x],x->0,Direction->-1],Limit[ArcSech[x],x->1]}

Series

About 0

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

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

About ∞

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

Series[ArcSech[x],{x,Infinity,8}]

TimeConstrained[SeriesCoefficient[ArcSech[x],{x,∞,n}],60]//Timing