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 by

Some related formula, such as ${\displaystyle \operatorname {arccot}(\operatorname {csch} x)}$, doesn't quite work as definition. (See inverse trigonometric functions.)

(See inverse hyperbolic functions.)

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 ${\displaystyle {\mathfrak {Sin}}}$ and ${\displaystyle {\mathfrak {Cos}}}$).

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