Tanhc Function

Author

Eric W. Weisstein
September 28, 2007

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

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

©2007 Wolfram Research, Inc. except for portions noted otherwise

Plots

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

Plot[Tanhc[x],{x,-10,10},PlotStyle->Red,Ticks->None,
AxesLabel->TraditionalForm/@{x,HoldForm[Tanhc[x]]}
]

ComplexPlot3D[Tanhc[z],{z,-10,10},PlotPoints->150,ComplexPlotLabel->HoldForm[Tanhc[x]]]//Timing

ComplexContourPlot[Tanhc[z],{z,-10,10},ComplexPlotLabel->HoldForm[Tanhc[x]]]//Timing

Special Values

Limit[Tanhc[x],x->0]

Derivative

D[Tanhc[x],x]

Integral

Integrate[Tanhc[x],x]

Integrate[#,x]&/@Apart[TrigToExp[Tanhc[x]]]

Critical Points

Maximum

Maximize[((Tanh[x])/x),x]

Maximize[((Tanh[x])/x),x,Reals]

Maximize[{((Tanh[x])/x),-2<x<2},x,Reals]

Maximize[Tanhc[x],x,Reals]

Inflection Points

Numerator[Together[D[Tanhc[x],{x,2}]]]

Plot[(x ((Sech[x])^2)-Tanh[x]+(x^2) ((Sech[x])^2) Tanh[x]),{x,-1,1}]

FindRoot[(x ((Sech[x])^2)-Tanh[x]+(x^2) ((Sech[x])^2) Tanh[x])==0,{x,.9}]

Reduce[(x ((Sech[x])^2)-Tanh[x]+(x^2) ((Sech[x])^2) Tanh[x])==0&&x>0,x]//FullSimplify

Fixed Points

Plot[{Tanhc[x],x},{x,-2,2},PlotStyle->{Red,Black},AxesLabel->{x,Tanhc[x]}]

FindRoot[Tanhc[x]==x,{x,1}]

Reduce[Tanhc[x]==x,x]

Reduce[Tanhc[x]==x&&x>0,x]