PlanetMath: hyperbolic functions

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hyperbolic functions

(Definition)

The hyperbolic functions $\sinh$ (sinus hyperbolicus) and $\cosh$ (cosinus hyperbolicus) with arbitrary complex argument 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.

**Figure:** Graphs of the hyperbolic functions.

The hyperbolic functions are named in that way because the hyperbola

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

can be written in parametrical form with the equations:

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

This is because of the equation

$\displaystyle \cosh^2 x-\sinh^2 x=1.$

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 connections between the hyperbolic and the trigonometric functions:

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

"hyperbolic functions" is owned by mathwizard. [ full author list (2) | owner history (2) ]

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Also defines:

sinh, cosh, tanh, coth, sech, csch, hyperbolic sine, hyperbolic cosine, hyperbolic tangent, hyperbolic cotangent, hyperbolic secant, hyperbolic cosecant

Attachments:: area functions (Definition) by pahio
derivatives of hyperbolic functions (Derivation) by alozano
hyperbolic identities (Topic) by Wkbj79
addition and subtraction formulas for hyperbolic functions (Derivation) by Wkbj79

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Cross-references: Taylor series, trigonometric functions, addition formulas, equations, hyperbola, definitions, functions, argument, complex
There are 16 references to this entry.

This is version 10 of hyperbolic functions, born on 2002-05-15, modified 2008-03-18.
Object id is 2905, canonical name is HyperbolicFunctions.
Accessed 18428 times total.

Classification:

AMS MSC:

26A09 (Real functions :: Functions of one variable :: Elementary functions)

Pending Errata and Addenda

None.[ View all 9 ]

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