Defining the hyperbolic sine function

The hyperbolic sine function is an old mathematical function. It was first used in the works of V. Riccati (1757), D. Foncenex (1759), and J. H. Lambert (1768).

The hyperbolic sine function is easily defined as the half difference of two exponential functions in the points and :

After comparison with the famous Euler formula for sine (), it is easy to derive the following representation of the hyperbolic sine through the circular sine:

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

The following formula can sometimes be used as an equivalent definition of the hyperbolic sine function:

This series converges for all finite numbers .