Hyperbolic Cosine

Author

Eric W. Weisstein
August 23, 2008

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

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

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

Plots

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

Real Axis

Plot[Cosh[x],{x,-5,5},AxesLabel->TraditionalForm/@{x,Cosh[x]},
PlotRange->{Automatic,{0,50}},PlotStyle->Red]

Complex Plane

ComplexPlot3D[Cosh[z],{z,-5,5}]//Timing

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

Special Values

Cosh[0]

FullSimplify[Cosh[Log[GoldenRatio]]]

Series

Series[Cosh[x],{x,0,8}]

Plus@@Table[x^(2n)/(2n)!,{n,0,4}]

V5

<<DiscreteMath`RSolve`

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

V7

SeriesCoefficient[Cosh[x],{x,0,n}]//InputForm

GeneratingFunction[%,n,x]//InputForm

Sum[% x^n,{n,0,∞}]

Sum

(∑_(n=0)^∞)(x^n) ((Piecewise[{{0, Mod[n,2]≠0||n<0}, {(1/(n!)), True}}]))==Cosh[x]

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

Sum[% x^n,{n,0,∞}]

(∑_(n=0)^∞)((x^(2 n))/((2 n)!))==cosh(x)

Sum[x^(2n)/(2n)!,{n,0,∞}]

(∑_(n=0)^∞)((x^n)/((2 n)!))==cosh((Sqrt[x]))

Sum[x^(n)/(2n)!,{n,0,∞}]

Derivative

D[Cosh[z],z]

Integral

Integrate[Cosh[z],z]