hyperbolic functions (original) (raw)

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 sech and csch:

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

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 between the hyperbolic and the trigonometric functions: