Gudermannian function


The Gudermannian function, named after Christoph Gudermann, relates the circular functions and hyperbolic functions without explicitly using complex numbers.
It is defined for all x by

Properties

Alternative definitions

Some identities

Inverse

Some identities

Derivatives

History

The function was introduced by Johann Heinrich Lambert in the 1760s at the same time as the hyperbolic functions. He called it the "transcendent angle," and it went by various names until 1862 when Arthur Cayley suggested it be given its current name as a tribute to Gudermann's work in the 1830s on the theory of special functions. Gudermann had published articles in Crelle's Journal that were collected in Theorie der potenzial- oder yklisch-hyperbolischen Funtionen, a book which expounded sinh and cosh to a wide audience.
The notation gd was introduced by Cayley where he starts by calling gd. u the inverse of the integral of the secant function:
and then derives "the definition" of the transcendent:
observing immediately that it is a real function of u.

Applications