Gudermannian function [SEP] named after
Gudermannian function
The Gudermannian function, named after Christoph Gudermann (1798–1852), relates the circular functions and hyperbolic functions without explicitly using complex numbers.
It is defined for all "x" by
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
of the tangent half-angle substitution, which is simpler in the case of this particular integral, published in 2013, is as follows:
Hyperbolic forms.
Let
Therefore,
Gudermannian and lambertian.
The integral of the secant function defines the inverse of the Gudermannian function:
The lambertian function (lam) is a notation for the inverse of the gudermannian which is encountered in the theory of map projections. In particular the Mercator projection may be written as
See also.
- Integral of secant