Gudermannian function

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In mathematics, the Gudermannian function relates a hyperbolic angle measure ψ {\textstyle \psi } to a circular angle measure ϕ {\textstyle \phi } called the gudermannian of ψ {\textstyle \psi } and denoted gd ⁡ ψ {\textstyle \operatorname {gd} \psi } . The Gudermannian function reveals a close relationship between the circular functions and hyperbolic functions. It was introduced in the 1760s by Johann Heinrich Lambert, and later named for Christoph Gudermann who also described the relationship between circular and hyperbolic functions in 1830. The Gudermannian function and its inverse were used historically to construct tables of hyperbolic functions or to compute hyperbolic functions given only a table of circular functions. The gudermannian is sometimes called the hyperbolic amplitude as a limiting...

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