Hyperbolic Secant

$\begin{figure}\begin{center}\BoxedEPSF{Sech.epsf}\end{center}\end{figure}$

$\begin{figure}\begin{center}\BoxedEPSF{SechReIm.epsf scaled 700}\end{center}\end{figure}$

The hyperbolic secant is defined as

$\begin{displaymath} \mathop{\rm sech}\nolimits x\equiv {1\over\cosh x} = {2\over e^x+e^{-x}}. \end{displaymath}$

It has a Maximum at

and inflection points at $x=\pm\mathop{\rm sech}\nolimits ^{-1}(1/\sqrt{2}\,)\approx 0.881374$ .

References

Abramowitz, M. and Stegun, C. A. (Eds.). ``Hyperbolic Functions.'' §4.5 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 83-86, 1972.

Spanier, J. and Oldham, K. B. ``The Hyperbolic Secant $\mathop{\rm sech}\nolimits (x)$ and Cosecant $\mathop{\rm csch}\nolimits (x)$ Functions.'' Ch. 29 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 273-278, 1987.