Defining the hyperbolic secant function

The hyperbolic secant function is an old mathematical function.

This function is easily defined as the ratio of one and hyperbolic cosine functions:

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

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