Hyperbolic Secant

Author

Eric W. Weisstein
August 23, 2008

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

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

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

Plots

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

Real Axis

Plot[Sech[x],{x,-5,5},AxesLabel->TraditionalForm/@{x,Sech[x]},PlotStyle->Red]

Complex  Plane

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

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

Derivative and Integral

Derivative

D[Sech[z],z]

Integral

Integrate[Sech[z],z]

Sums

(∑_(n=0)^∞)(((x^n) (Subscript[E, n]))/(n!))==sech(x)

Sum[((EulerE[n])/(n!))x^n,{n,0,∞}]

(∑_(n=0)^∞)(x^n) Piecewise[(({{((Subscript[E, n])/(n!)), n≥0}, {0, "PiecewiseDefault"[True]}}))]==sech(x)

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

Sum[x^n(Piecewise[{{((EulerE[n])/(n!)), n≥0}, {0, True}}]),{n,0,∞}]

(∑_(n=1)^∞)sech(π n)=(1/2) (((Sqrt[π])/((Γ((3/4)))^2))-1)

V6

Sum[Sech[Pi n],{n,Infinity}]

Sum[(2/((E^(-n π))+(E^(n π)))),{n,∞}]

Apart[(2/((E^(-n π))+(E^(n π))))]

Sum[((2 (E^(n π)))/(1+(E^(2 n π)))),{n,∞}]

N[Sum[Sech[Pi n],{n,Infinity}]-(Sqrt[Pi]/Gamma[3/4]^2-1)/2,200]

V7

Sum[Sech[Pi n],{n,Infinity}]

(∑_n^∞)(n^2) sech(π n)==??

Sum[n^2 Sech[n π],{n,∞}]

(∑_n^∞)(n^4) sech(π n)==((18 ((Γ((3/4)))^2) (∑_n^∞)(n^2) sech(π n))/(Sqrt[π]))

Sum[n^4 Sech[n π],{n,∞}]

Sum[n^2 Sech[π n],{n,∞}]

Series

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

V5

<<DiscreteMath`RSolve`

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

V7

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

GeneratingFunction[%,n,x]

Inflection Points

Factor[FullSimplify[Sech''[x]]]

Solve[-1+2 ((Sech[x])^2)==0,x]

N[%]

FullSimplify[ArcSech[(1/(Sqrt[2]))]]

RealDigits[ArcCosh[(Sqrt[2])],10,20]

Fixed Point

FindRoot[Sech[x]==x,{x,.7},WorkingPrecision->100]

RealDigits[FindRoot[Sech[x]==x,{x,.7},WorkingPrecision->100][[1,2]],10]