Inverse Secant

Author

Eric W. Weisstein
August 24, 2008

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

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

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

Definition

TrigToExp[ArcSec[z]]

Plots

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

Branch Cuts

BranchCuts[ArcSec]//TraditionalForm

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

Real Axis

PlotDiscontinuous[ArcSec[x],{x,{-10,-1},{1,10}},
PlotRange->{Automatic,{0,Pi}},
AxesLabel->TraditionalForm/@{x,ArcSec[x]},
PlotStyle->Red,
VerticalAsymptotes->{-1,1},
HorizontalAsymptotes->{Pi/2},
Epilog->{Red,PointSize[.02],Point/@{{-1,Pi},{1,0}}},
Ticks->{Automatic,{#,TraditionalForm[#]}&/@Range[0,Pi,Pi/4]}]

Complex Plane

ComplexPlot3D[ArcSec[z],{z,-5,5},PlotPoints->100,Mesh->False,
ClipFill->None]//Timing

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

Derivative and Integral

Derivative

D[ArcSec[z],z]

FullSimplify[%,z∈Reals]

FullSimplify[%,z>0]

Integral

Integrate[ArcSec[z],z]

FullSimplify[%,z∈Reals]

FullSimplify[%,z>0]

Special Values

ArcSec/@{-∞,-1,0,1,∞}

FunctionExpand[ArcSec[#]]&/@{-I,I}

Indentites

In Terms of Itself

FullSimplify[Pi-ArcSec[-z]]

FullSimplify[Pi-ArcSec[-z]-ArcSec[z]]

FullSimplify[ArcCos[1/z]]

FullSimplify[Pi/2-ArcCsc[z]]

FullSimplify[Pi/2+ArcCsc[-z]]

In Terms of Other Arc Trigs

FullSimplify[ArcCsc[z/Sqrt[z^2-1]]-ArcSec[z],z>1]

Plot[Switch[z,_?(#<-1&),Pi+ArcCsc[z/Sqrt[z^2-1]]-ArcSec[z],
_?(-1<#<1&),0,
_?(#>1&),ArcCsc[z/Sqrt[z^2-1]]-ArcSec[z]
],{z,-5,5},PlotRange->All,PlotStyle->Red]

FullSimplify[ArcCot[1/Sqrt[x^2-1]],x>1]

Plot[Switch[x,
_?(#<-1&),Pi-ArcCot[1/Sqrt[x^2-1]]-ArcSec[x],
_?(-1<#<1&),0,
_?(#>1&),ArcCot[1/Sqrt[x^2-1]]-ArcSec[x]
],{x,-5,5},PlotStyle->Red,PlotRange->All]

FullSimplify[ArcSin[Sqrt[x^2-1]/x],x>0]

Plot[Switch[x,
_?(#<-1&),Pi+ArcSin[Sqrt[x^2-1]/x]-ArcSec[x],
_?(-1<#<1&),0,
_?(#>1&),ArcSin[Sqrt[x^2-1]/x]-ArcSec[x]
],{x,-5,5},PlotStyle->Red,PlotRange->All]

FullSimplify[ArcTan[Sqrt[x^2-1]],x>0]

Plot[Switch[x,
_?(#<-1&),Pi-ArcTan[Sqrt[x^2-1]]-ArcSec[x],
_?(-1<#<1&),0,
_?(#>1&),ArcTan[Sqrt[x^2-1]]-ArcSec[x]
],{x,-5,5},PlotStyle->Red,PlotRange->All]

Series

About 0

Series[ArcSec[x],{x,0,10},Assumptions->x∈Reals]

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

About ∞

series=Series[ArcSec[x],{x,∞,30}]

coefs=Last/@Partition[CoefficientList[Normal[series],1/x],2]

Through[{Numerator,Denominator}[coefs]]

V7

ser=SeriesCoefficient[ArcSec[x],{x,∞,n}]

ser//InputForm

Table[ser x^n,{n,-5,5}]

Series[ArcSec[x],{x,∞,5}]

Sums

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

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

FullSimplify[%]