It is defined by
Some related formulas don't quite work as definitions. For example, for real x, . (See inverse trigonometric functions.)
The following identities hold:
The inverse Gudermannian function, which is defined on the interval −π/2 < x < π/2, is given by
(See inverse hyperbolic functions.)
The derivatives of the Gudermannian and its inverse are
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 cyklisch-hyperbolischen functionen (1833), a book which expounded sinh and cosh to a wide audience (under the guises of and ).
and then derives "the definition" of the transcendent:
observing immediately that it is a real function of u.
- Hyperbolic secant distribution
- Mercator projection
- Tangent half-angle formula
- Trigonometric identity
- George F. Becker, C. E. Van Orstrand. Hyperbolic functions. Read Books, 1931. Page xlix.
- John S. Robertson, "Gudermann and the Simple Pendulum", The College Mathematics Journal 28:4:271–276 (September 1997) at JSTOR