Gudermannian function

In mathematics, the Gudermannian function relates a hyperbolic angle measure ${\textstyle \psi }$ to a circular angle measure ${\textstyle \phi }$ called the gudermannian of ${\textstyle \psi }$ and denoted ${\textstyle \operatorname {gd} \psi }$.^[1] 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.^[2] The gudermannian is sometimes called the hyperbolic amplitude as a limiting case of the Jacobi elliptic amplitude ${\textstyle \operatorname {am} (\psi ,m)}$ when parameter ${\textstyle m=1.}$

The real Gudermannian function is typically defined for ${\textstyle -\infty <\psi <\infty }$ to be the integral of the hyperbolic secant^[3]

${\displaystyle \phi =\operatorname {gd} \psi \equiv \int _{0}^{\psi }\operatorname {sech} t\,\mathrm {d} t=\operatorname {arctan} (\sinh \psi ).}$

The real inverse Gudermannian function can be defined for ${\textstyle -{\tfrac {1}{2}}\pi <\phi <{\tfrac {1}{2}}\pi }$ as the integral of the (circular) secant

${\displaystyle \psi =\operatorname {gd} ^{-1}\phi =\int _{0}^{\phi }\operatorname {sec} t\,\mathrm {d} t=\operatorname {arsinh} (\tan \phi ).}$

The hyperbolic angle measure ${\displaystyle \psi =\operatorname {gd} ^{-1}\phi }$ is called the anti-gudermannian of ${\displaystyle \phi }$ or sometimes the lambertian of ${\displaystyle \phi }$, denoted ${\displaystyle \psi =\operatorname {lam} \phi .}$^[4] In the context of geodesy and navigation for latitude ${\textstyle \phi }$, ${\displaystyle k\operatorname {gd} ^{-1}\phi }$ (scaled by arbitrary constant ${\textstyle k}$) was historically called the meridional part of ${\displaystyle \phi }$ (French: latitude croissante). It is the vertical coordinate of the Mercator projection.

The two angle measures ${\textstyle \phi }$ and ${\textstyle \psi }$ are related by a common stereographic projection

${\displaystyle s=\tan {\tfrac {1}{2}}\phi =\tanh {\tfrac {1}{2}}\psi ,}$

and this identity can serve as an alternative definition for ${\textstyle \operatorname {gd} }$ and ${\textstyle \operatorname {gd} ^{-1}}$ valid throughout the complex plane:

${\displaystyle {\begin{aligned}\operatorname {gd} \psi &={2\arctan }{\bigl (}\tanh {\tfrac {1}{2}}\psi \,{\bigr )},\\[5mu]\operatorname {gd} ^{-1}\phi &={2\operatorname {artanh} }{\bigl (}\tan {\tfrac {1}{2}}\phi \,{\bigr )}.\end{aligned}}}$