Inverse Cosecant

Author

Eric W. Weisstein
August 24, 2008

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

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

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

Definition

TrigToExp[ArcCsc[z]]

Plots

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

Branch Cut

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

Real Axis

PlotDiscontinuous[ArcCsc[x],{x,{-10,-1},{1,10}},
PlotRange->{Automatic,{-1.7,1.7}},
AxesLabel->TraditionalForm/@{x,ArcCsc[x]},
HorizontalAsymptotes->{0},
VerticalAsymptotes->{-1,1},
PlotStyle->Red,
Epilog->{PointSize[.025],Red,Point/@{{1,Pi/2},{-1,-Pi/2}}},
Ticks->{Automatic,{#,TraditionalForm[#]}&/@Range[-Pi/2,Pi/2,Pi/4]}]

Complex Plane

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

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

Special Values

ArcCsc/@{-1,1}

Derivative and Integral

Derivative

D[ArcCsc[z],z]

FullSimplify[%,z∈Reals]

FullSimplify[%,z>0]

Integral

Integrate[ArcCsc[z],z]

FullSimplify[%,z∈Reals]

FullSimplify[%,z>0]

Identities

With itself

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

FullSimplify[ArcSec[x/Sqrt[x^2-1]]-Pi,x<-1]

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

With other arc trigs

For all z

FullSimplify[ArcSin[1/z]]

FullSimplify[π/2-ArcSec[z]]

FullSimplify[-π/2+ArcSec[-z]]

For x>1

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

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

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

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

Series

About 0

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

Series[ArcCsc[x],{x,0,10},Assumptions->x∈Reals]//FullSimplify

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

About ∞

series=Series[ArcCsc[x],{x,∞,20}]

coef=CoefficientList[series,1/x]

Through[{Numerator,Denominator}[Last/@Partition[coef,2]]]//ColumnForm

V5

<<DiscreteMath`RSolve`

SeriesTerm[ArcCsc[x],{x,∞,n}]

Series[ArcCsc[x],{x,∞,10}]

term=SeriesTerm[ArcCsc[1/x],{x,0,n}]

FullSimplify[term,n>1]

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

V7

Series[ArcCsc[x],{x,∞,20}]

SeriesCoefficient[ArcCsc[x],{x,∞,n}]

Sums

(∑_n^∞)-(((i (i^n) (Subscript[P, n-1])(0)) (x^(-n)))/n)==(csc^(-1))(x)

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

With[{x=1.1},{Sum[-((I (I^n) LegendreP[-1+n,0])/n)(x^(-n)),{n,100}],ArcCsc[x]}]

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

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

FullSimplify[%]