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Tanhc Function

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TanhcTanhcReImTanhcContours

By analogy with the tanc function, define the tanhc function by

tanhc(z)={(tanhz)/z for z!=0; 1 for z=0.
(1)

It has derivative

(dtanhc(z))/(dz)=(sech^2z)/z-(tanhz)/(z^2).
(2)

The indefinite integral can apparently not be done in closed form in terms of conventionally defined functions.

It has maximum at z=0, and positive inflection point at the solution to

xsech^2x^2-tanhx+x^2sech^2xtanhx=0,
(3)

which is 0.919937667... (OEIS A133919).

TanhcFunctionFixedPoint

It has a unique real fixed point at 0.82242927726... (OEIS A133918).

See also

Sinhc Function, Tanc Function

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References

Sloane, N. J. A. Sequences A133918 and A133919 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Tanhc Function

Cite this as:

Weisstein, Eric W. "Tanhc Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/TanhcFunction.html

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