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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

\tan x\equiv {\sin x\over \cos x},

the hyperbolic tangent is defined as

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

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

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

See also Bernoulli Number, Catenary, Correlation Coefficient--Gaussian Bivariate Distribution, Fibonacci Hyperbolic Tangent, Fisher's z'-Transformation, Hyperbolic Cotangent, Lorentz Group, Mercator Projection, Oblate Spheroidal Coordinates, Pseudosphere, Surface of Revolution, Tangent, Tractrix, Tractroid


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.

© 1996-9 Eric W. Weisstein