Hyperbolic Cosecant

Author

Eric W. Weisstein
August 24, 2008

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

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

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

Plots

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

Real Axis

PlotDiscontinuous[Csch[x],{x,-10,0,10},
VerticalAsymptotes->{0},
PlotStyle->Red,
AxesLabel->TraditionalForm/@{x,Csch[x]},
PlotRange->{Automatic,{-5,5}}]

Complex Plane

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

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

Bifurcation Plot

BifurcationPlot[Csch[x],{x,.1},{1000,200},{.26,.32,2 10^-4},PlotRange->All]//Timing

Series

Series[Csch[z],{z,0,12}]

V5

<<DiscreteMath`RSolve`

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

Table[%,{n,-1,12}]

V7

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

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

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

Derivative and Integral

Derivative

D[Csch[z],z]

Integral

Integrate[Csch[z],z]

Sums

(∑_(n=-1)^∞)(((z^n) ((2^(n+1)) (Subscript[B, n+1])((1/2))))/((n+1)!))==csch(z)

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

Table[(((2^(1+n)) BernoulliB[1+n,(1/2)])/((1+n)!))(z^n),{n,-1,10}]

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

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

TimeConstrained[Sum[Piecewise[{{1,n==-1},{(-2*(-1+2^n)*BernoulliB[1+n])/(1+n)!,Mod[n,2]==1&&n≥0}},0]z^n,{n,-∞,∞}],60]//Timing

(∑_n^∞)-(((z^(2 n-1)) (2 ((2^(2 n-1))-1) (Subscript[B, 2 n])))/((2 n)!))+(1/z)==csch(z)

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

Table[-((2((2^(2n-1))-1)BernoulliB[2n])/((2n)!))(z^(2n-1)),{n,10}]

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

FullSimplify[%]

Subtract@@@Table[{Coth[(z/2)]-Coth[z],Csch[z]}/.z->RandomComplex[5(I+1){-1,1}],{100}]//Chop//Union

(∑_(k=1)^∞)csch(π k)

Sum[Csch[π k],{k,∞}]

N[%,30]

(∑_(k=1)^∞)(csch^2)(π k)

Sum[((Csch[π k])^2),{k,∞}]

N[%,20]

N[(1/6)-(1/(2π)),20]

(∑_(k=1)^∞)(csch^3)(π k)

Sum[((Csch[π k])^3),{k,∞}]

N[%,20]

(∑_(k=1)^∞)(csch^4)(π k)

Sum[((Csch[π k])^4),{k,∞}]

N[%,30]