Hyperbolic functions

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The functions given by the formulas: \begin{equation} \sinh x = \frac{e^x-e^{-x}}{2}, \end{equation} the hyperbolic sine; and \begin{equation} \cosh x = \frac{e^x+e^{-x}}{2}, \end{equation} the hyperbolic cosine. The hyperbolic tangent \begin{equation} \tanh x = \frac{\sinh x}{\cosh x}, \end{equation} is also sometimes considered. Other notations include: $\operatorname{sh} x$, $\operatorname{Sh} x$, $\operatorname{ch} x$, $\operatorname{Ch} x$, $\operatorname{tgh} x$, $\operatorname{th} x$, $\operatorname{Th} x$. The graphs of these functions are shown in Fig. a.

Figure: h048250a

The principal relations are:

Figure: h048250b

The geometrical interpretation of hyperbolic functions is similar to that of the trigonometric functions (Fig. b). The parametric equations of hyperbolas

make it possible to interpret the abscissa and the ordinate of a point on the equilateral hyperbola as the hyperbolic sine and cosine; the hyperbolic tangent is the segment . The parameter of the point equals twice the area of the sector , where is the arc of the hyperbola. The parameter is negative for a point (for ).

The inverse hyperbolic functions are defined by the formulas

The derivatives and basic integrals of the hyperbolic functions are:

The hyperbolic functions and may also be defined by the series

in the entire complex -plane, so that


Extensive tabulated values of hyperbolic functions are available. The values of the hyperbolic functions may also be obtained from tables giving and .


[1] E. Jahnke, F. Emde, F. Lösch, "Tafeln höheren Funktionen" , Teubner (1966)
[2] , Tables of circular and hyperbolic sines and cosines in radial angle measure , Moscow (1968) (In Russian)
[3] , Tables of and , Moscow (1955) (In Russian)


The right-hand sides of the defining relations (1), (2) allow analytic continuation to the whole complex plane. After this, using the Euler formulas one sees that (3) holds, from which the series expansions are readily derived.


[a1] A. Segun, M. Abramowitz, "Handbook of mathematical functions" , Appl. Math. Ser. , 55 , Nat. Bur. Standards (1970)
[a2] H.B. Dwight, "Tables of integrals and other mathematical data" , Macmillan (1963)
How to Cite This Entry:
Hyperbolic functions. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by V.I. Bityutskov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article