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

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

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

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

The hyperbolic functions are named in that way because the hyperbola

can be written in parametrical form with the equations:

This is because of the equation

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

The Taylor series for the hyperbolic functions are:

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