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).$