# hyperbolic function

(redirected from*Arcsinh*)

## hyperbolic function

*n.*

Any of a set of six functions related, for a real or complex variable

*x,*to the hyperbola in a manner analogous to the relationship of the trigonometric functions to a circle, including:**a.**The hyperbolic sine, defined by the equation sinh

*x*=

^{1}/

_{2}(

*e*-

^{x}*e*

^{-}

*).*

^{x}**b.**The hyperbolic cosine, defined by the equation cosh

*x*=

^{1}/

_{2}(

*e*+

^{x}*e*

^{-}

*).*

^{x}**c.**The hyperbolic tangent, defined by the equation tanh

*x*= sinh

*x*/cosh

*x.*

**d.**The hyperbolic cotangent, defined by the equation coth

*x*= cosh

*x*/sinh

*x.*

**e.**The hyperbolic secant, defined by the equation sech

*x*= 1/cosh

*x.*

**f.**The hyperbolic cosecant, defined by the equation csch

*x*= 1/sinh

*x.*

## hyperbolic function

*n*

(Mathematics) any of a group of functions of an angle expressed as a relationship between the distances of a point on a hyperbola to the origin and to the coordinate axes. The group includes sinh (

**hyperbolic sine**), cosh (**hyperbolic cosine**), tanh (**hyperbolic tangent**), sech (**hyperbolic secant**), cosech (**hyperbolic cosecant**), and coth (**hyperbolic cotangent**)## hy′perbol′ic func′tion

*n.*

a function of an angle expressed as a relationship between the distances from a point on a hyperbola to the origin and to the coordinate axes, as hyperbolic sine or hyperbolic cosine.

[1885–90]

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