hyperbolic functions

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

$\sinh x$ $:=$ ${\displaystyle \frac{e^x-e^{-x}}{2}},$

$\cosh x$ $:=$ ${\displaystyle \frac{e^x+e^{-x}}{2}}.$

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

$\tanh x$ $:=$ ${\displaystyle \frac{\sinh x}{\cosh x}}={\displaystyle \frac{e^x-e^{-x}}{e^x+e^{-x}}},$

$\coth x$ $:=$ ${\displaystyle \frac{\cosh x}{\sinh x}}={\displaystyle \frac{e^x+e^{-x}}{e^x-e^{-x}}}.$

We further define the $\mathrm{sech}$ and $\mathrm{csch}$:

$\mathrm{sech}x$ $:=$ ${\displaystyle \frac{1}{\cosh x}}={\displaystyle \frac{2}{e^x+e^{-x}}},$

$\mathrm{csch}x$ $:=$ ${\displaystyle \frac{1}{\sinh x}}={\displaystyle \frac{2}{e^x-e^{-x}}},$

where $\cosh x$ resp. $\sinh x$ is not $0$.

Figure 1: Graphs of the hyperbolic functions.

The hyperbolic functions are named in that way because the hyperbola

$$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$$

can be written in parametrical form with the equations:

$$x=a\cosh t,y=b\sinh t.$$

This is because of the equation

$${\cosh }^{2}x-{\sinh }^{2}x=1.$$

There are also addition formulas which are like the ones for trigonometric functions:

$\sinh (x\pm y)$ $=$ $\sinh x\cosh y\pm \cosh x\sinh y$

$\cosh (x\pm y)$ $=$ $\cosh x\cosh y\pm \sinh x\sinh y.$

The Taylor series for the hyperbolic functions are:

$\sinh x$ $=$ ${\displaystyle \sum _{n=0}^{\mathrm{\infty }}}{\displaystyle \frac{x^{2n+1}}{(2n+1)!}}$

$\cosh x$ $=$ ${\displaystyle \sum _{n=0}^{\mathrm{\infty }}}{\displaystyle \frac{x^{2n}}{(2n)!}}.$

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

$\sin x$ $=$ ${\displaystyle \frac{\sinh (ix)}{i}}$

$\cos x$ $=$ $\cosh (ix).$

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