Cosecant

Author

Eric W. Weisstein
March 5, 2012

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

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

©2012 Wolfram Research, Inc. except for portions noted otherwise

Definition

TrigToExp[Csc[z]]

Plots

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

Real Axis

V5

PlotDiscontinuous[Csc[x],{x,Sequence@@(Range[-2,2]Pi)},PlotRange->{Automatic,{-5,5}},
VerticalAsymptotes->Range[-2,2]Pi,
HorizontalAsymptotes->{-1,1},
Ticks->{{#,TraditionalForm[#]}&/@Range[-2Pi,2Pi,Pi],Automatic},AxesLabel->TraditionalForm/@{x,Csc[x]}]

V6

Plot[Csc[x],{x,-2π,2π},
PlotStyle->Red,
Exclusions->Range[-2,2]Pi,
AxesLabel->{x,Csc[x]},
Ticks->{{#,Text[#,Background->White]}&/@Range[-2Pi,2Pi,Pi/2],Automatic},
PlotRange->{Automatic,{-5,5}},
Prolog->{Blue,Dashing[.02],Line[{{-2π,#},{2π,#}}]&/@{-1,1},
Line[{{#,-5},{#,5}}]&/@(Range[-3,3]Pi)
}]

Complex Plane

ComplexPlot3D[Csc[z],{z,-2Pi,2Pi},PlotPoints->100,Mesh->False,ClipFill->None]//Timing

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

Series

Series[Csc[z],{z,0,10}]

V5

<<DiscreteMath`RSolve`

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

Through[{Numerator,Denominator}[Last/@Partition[CoefficientList[Series[Csc[z]-1/z,{z,0,15}],z],2]]]

V7

SeriesCoefficient[Csc[z],{z,0,n}]

SeriesCoefficient[Csc[z],{z,0,n}]//InputForm

TimeConstrained[GeneratingFunction[%,n,x],60]//Timing

Sums

(∑_(n=-1)^∞)(x^n) Piecewise[(({{1, n==-1}, {((2 (i^(n-1)) ((2^n)-1) (Subscript[B, n+1]))/((n+1)!)), n mod 2==1∧n≥0}})),0]==csc(x)

SeriesCoefficient[Csc[z],{z,0,n}]

SeriesCoefficient[Csc[z],{z,0,n}]//InputForm

Sum[Piecewise[{{1,n==-1},{(2*I^(-1+n)*(-1+2^n)*BernoulliB[1+n])/(1+n)!,Mod[n,2]==1&&n≥0}},0]x^n,{n,-1,∞}]

(∑_(n=0)^∞)(((x^(2 n-1)) (2 (((-1))^(n+1)) ((2^(2 n-1))-1) (Subscript[B, 2 n])))/((2 n)!))==csc(x)

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

Sum[(((((-1))^(n+1))2((2^(2n-1))-1)BernoulliB[2n])/((2n)!))(x^(2n-1)),{n,0,20}]/.x->1.3

Csc[1.3]

Derivative and Integral

Derivative

D[Csc[z],z]

Integral

int=Integrate[Csc[z],z]

Simplify[D[int,z]]

FullSimplify[int,-π<z<π]

FullSimplify[Subtract@@{int,Log[Csc[z]-Cot[z]]},-π<z<π]

FullSimplify[Subtract@@{int,Log[Csc[z]-Cot[z]]},-2π<z<-π]

InputForm[%82]

Table[int/.z->Random[Real,π{-2,1},50],{100}]//Chop//Union