: Mathematics: Gudermannian function

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mathematical analysis > functional analysis > special function > Gudermannian function

mathematical analysis > calculus > integral calculus > hyperbolic angle > Gudermannian function

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Gudermannian function

In mathematics, the Gudermannian function relates a hyperbolic angle measure ${\ extstyle \\psi }$ to a circular angle measure ${\ extstyle \\phi }$ called the gudermannian of ${\ extstyle \\psi }$ and denoted ${\ extstyle \\operatorname {gd} \\psi }$ . 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 is sometimes called the hyperbolic amplitude as a limiting case of the Jacobi elliptic amplitude ${\ extstyle \\operatorname {am} (\\psi ,m)}$ when parameter ${\ extstyle m=1.}$

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