Hyperbolic Tangent

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

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

By way of analogy with the usual Tangent

$\begin{displaymath} \tan x\equiv {\sin x\over \cos x}, \end{displaymath}$

the hyperbolic tangent is defined as

$\begin{displaymath} \tanh x\equiv{\sinh x\over\cosh x}={e^{x}-e^{-x}\over e^x+e^{-x}}={e^{2x}-1\over e^{2x}+1}, \end{displaymath}$

where $\sinh x$ is the Hyperbolic Sine and $\cosh x$ is the Hyperbolic Cosine. The hyperbolic tangent can be written using a Continued Fraction as

$\begin{displaymath} \tanh x = {x\over 1+{\strut\displaystyle x^2\over\strut\disp... ... {\strut\displaystyle x^2\over\strut\displaystyle 5+\ldots}}}. \end{displaymath}$

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 Tangent $\tanh(x)$ and Cotangent $\coth(x)$ Functions.'' Ch. 30 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 279-284, 1987.