The hyperbolic functions $\sinh$ (sinus hyperbolicus) and $\cosh$ (cosinus hyperbolicus) with arbitrary complex argument $x$ are defined as follows: \begin{eqnarray*} \sinh x&:=&\frac{e^x-e^{-x}}{2},\\ \cosh x&:=&\frac{e^x+e^{-x}}{2}. \end{eqnarray*}One can then also also define the functions $\tanh$ (tangens hyperbolica) and $\coth$ (cotangens hyperbolica) in analogy to the definitions of $\tan$ and $\cot$ : \begin{eqnarray*} \tanh x&:=&\frac{\sinh x}{\cosh x}=\frac{e^x-e^{-x}}{e^x+e^{-x}},\\ \coth x&:=&\frac{\cosh x}{\sinh x}=\frac{e^x+e^{-x}}{e^x-e^{-x}}. \end{eqnarray*}We further define the $\sech$ and $\csch$ : \begin{eqnarray*} \sech x&:=&\frac{1}{\cosh x}=\frac{2}{e^x+e^{-x}},\\ \csch x&:=&\frac{1}{\sinh x}=\frac{2}{e^x-e^{-x}}, \end{eqnarray*}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: \begin{eqnarray*} \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. \end{eqnarray*}The Taylor series for the hyperbolic functions are: \begin{eqnarray*} \sinh x&=&\sum_{n=0}^{\infty}\frac{x^{2n+1}}{(2n+1)!}\\ \cosh x&=&\sum_{n=0}^{\infty}\frac{x^{2n}}{(2n)!}. \end{eqnarray*}There are the following connections between the hyperbolic and the trigonometric functions: \begin{eqnarray*} \sin x&=&\frac{\sinh (ix)}{i}\\ \cos x&=&\cosh (ix). \end{eqnarray*}