Difference between revisions of "Hyperbolic functions"
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The functions given by the formulas: | The functions given by the formulas: | ||
− | + | \begin{equation} | |
− | + | \sinh x = \frac{e^x-e^{-x}}{2}, | |
− | + | \end{equation} | |
the hyperbolic sine; and | the hyperbolic sine; and | ||
− | + | \begin{equation} | |
− | + | \cosh x = \frac{e^x+e^{-x}}{2}, | |
− | + | \end{equation} | |
the hyperbolic cosine. The hyperbolic tangent | the hyperbolic cosine. The hyperbolic tangent | ||
− | + | \begin{equation} | |
− | + | \tanh x = \frac{\sinh x}{\cosh x}, | |
− | + | \end{equation} | |
− | is also sometimes considered. Other notations include: | + | 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. |
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/h048250a.gif" /> | <img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/h048250a.gif" /> |
Latest revision as of 13:15, 9 December 2012
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:
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Figure: h048250b
The geometrical interpretation of hyperbolic functions is similar to that of the trigonometric functions (Fig. b). The parametric equations of hyperbolas
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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
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The derivatives and basic integrals of the hyperbolic functions are:
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The hyperbolic functions and
may also be defined by the series
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in the entire complex -plane, so that
![]() | (3) |
Extensive tabulated values of hyperbolic functions are available. The values of the hyperbolic functions may also be obtained from tables giving and
.
References
[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 ![]() ![]() |
Comments
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.
References
[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) |
Hyperbolic functions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hyperbolic_functions&oldid=14739