Hyperbolic Functions

hyperbolic functions [¦hī·pər¦bäl·ik ′fəŋk·shənz]

(mathematics)

The real or complex functions sinh (x), cosh (x), tanh (x), coth (x), sech (x), csch (x); they are related to the hyperbola in somewhat the same fashion as the trigonometric functions are related to the circle, and have properties analogous to those of the trigonometric functions.

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Hyperbolic Functions 

functions defined by the equations

sinh x = (hyperbolic sine of x)

cosh x = (hyperbolic cosine of x)

In some cases, the hyperbolic tangent is also considered:

tanh x =

(See Figure 1 for graphs of the hyperbolic functions.) Hyperbolic functions are connected by relations similar to those connecting the trigonometric functions:

cosh²x - sinh² = 1

tanh x = sinh x/cosh x

sinh (x±y) = sinh x cosh y ± cosh x sinh y

cosh (x±y) = cosh x cosh y ± sinh x sinh y

Figure 1

Figure 2

Hyperbolic functions may be expressed in terms of trigonometric functions:

Geometrically, the hyperbolic functions are obtained by analysis of the rectangular hyperbola x² - y² = 1, which may be defined in terms of the parametric equations x = cosh t and y = sinh t. The argument t represents twice the area of the hyperbolic sector OAC (see Figure 2). Inverse hyperbolic functions are defined by the equations

REFERENCE

Ianpol’skii, A. R. Giperbolicheskie funktsii. Moscow, 1960.