Defining the hyperbolic cosine function

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

This function is easily defined as the half‐sum of two exponential functions in the points and :

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

The previous formula allows the derivation of all properties and formulas for hyperbolic cosine from corresponding properties and formulas for the circular cosine.

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

This series converges for all finite numbers .