Defining the hyperbolic tangent function

The hyperbolic tangent function is an old mathematical function. It was first used in the work by L'Abbe Sauri (1774).

This function is easily defined as the ratio between the hyperbolic sine and the cosine functions (or expanded, as the ratio of the half‐difference and half‐sum of two exponential functions in the points and ):

After comparison with the famous Euler formulas for the sine and cosine functions, and , it is easy to derive the following representation of the hyperbolic tangent through the circular tangent function:

This formula allows the derivation of all the properties and formulas for the hyperbolic tangent from the corresponding properties and formulas for the circular tangent.