hyperbolic functions

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

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

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

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

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

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

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

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:

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

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:

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

The Taylor series for the hyperbolic functions are:

 $\displaystyle\sinh x$ $\displaystyle=$ $\displaystyle\sum_{n=0}^{\infty}\frac{x^{2n+1}}{(2n+1)!}$ $\displaystyle\cosh x$ $\displaystyle=$ $\displaystyle\sum_{n=0}^{\infty}\frac{x^{2n}}{(2n)!}.$

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

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