hyperbolic functions

The hyperbolic functionsDlmfMathworldPlanetmath sinh (sinus hyperbolicus) and cosh (cosinus hyperbolicus) with arbitrary complex argument x are defined as follows:

sinhx := ex-e-x2,
coshx := ex+e-x2.

One can then also also define the functionsMathworldPlanetmath tanh (tangens hyperbolica) and coth (cotangens hyperbolica) in analogy to the definitions of tan and cot:

tanhx := sinhxcoshx=ex-e-xex+e-x,
cothx := coshxsinhx=ex+e-xex-e-x.

We further define the sech and csch:

sechx := 1coshx=2ex+e-x,
cschx := 1sinhx=2ex-e-x,

where coshx resp. sinhx is not 0.

Figure 1: Graphs of the hyperbolic functions.

The hyperbolic functions are named in that way because the hyperbola


can be written in parametrical form with the equations:


This is because of the equation


There are also addition formulasPlanetmathPlanetmath which are like the ones for trigonometric functionsDlmfMathworldPlanetmath:

sinh(x±y) = sinhxcoshy±coshxsinhy
cosh(x±y) = coshxcoshy±sinhxsinhy.

The Taylor seriesMathworldPlanetmath for the hyperbolic functions are:

sinhx = n=0x2n+1(2n+1)!
coshx = n=0x2n(2n)!.

There are the following between the hyperbolic and the trigonometric functions:

sinx = sinh(ix)i
cosx = cosh(ix).
Title hyperbolic functions
Canonical name HyperbolicFunctions
Date of creation 2013-03-22 12:38:27
Last modified on 2013-03-22 12:38:27
Owner mathwizard (128)
Last modified by mathwizard (128)
Numerical id 13
Author mathwizard (128)
Entry type Definition
Classification msc 26A09
Related topic UnitHyperbola
Related topic ComplexTangentAndCotangent
Related topic ParallelCurve
Related topic HyperbolicAngle
Related topic ExampleOfCauchyMultiplicationRule
Related topic DerivationOfFormulasForHyperbolicFunctionsFromDefinitionOfHyperbolicAngle
Related topic HeavisideFormula
Related topic Catenary
Related topic HyperbolicSineIntegral
Related topic InverseGudermannia
Defines sinh
Defines cosh
Defines tanh
Defines coth
Defines sech
Defines csch
Defines hyperbolic sine
Defines hyperbolic cosine
Defines hyperbolic tangent
Defines hyperbolic cotangent
Defines hyperbolic secant
Defines hyperbolic cosecant