Editing Gudermannian function (section)

{{Short description|Mathematical function relating circular and hyperbolic functions}}
[[File:Gudermannian function.png|thumb|upright=1.5|The Gudermannian function relates the area of a [[circular sector]] to the area of a [[hyperbolic sector]], via a common [[stereographic projection]]. If twice the area of the blue hyperbolic sector is {{math|''ψ''}}, then twice the area of the red circular sector is {{math|''ϕ'' {{=}} gd ''ψ''}}. Twice the area of the purple triangle is the stereographic projection {{math|''s'' {{=}} tan {{sfrac|1|2}}''ϕ'' {{=}} tanh {{sfrac|1|2}}''ψ''.}} The blue point has coordinates {{math|(cosh ''ψ'', sinh ''ψ'')}}. The red point has coordinates {{math|(cos ''ϕ'', sin ''ϕ'').}} The purple point has coordinates {{math|(0, ''s'').}}  ]]
[[File:Gudermannian graph.png|thumb|right|upright=1.4|[[Graph of a function|Graph]] of the Gudermannian function.]]
[[File:Inverse Gudermannian graph.png|thumb|right|upright=1.2|Graph of the inverse Gudermannian function.]]

In mathematics, the '''Gudermannian function''' relates a [[hyperbolic angle]] measure <math display=inline>\psi</math> to a [[angle|circular angle]] measure <math display=inline>\phi</math> called the ''gudermannian'' of <math display=inline>\psi</math> and denoted <math display=inline>\operatorname{gd}\psi</math>.<ref>The symbols <math display=inline>\psi</math> and <math display=inline>\phi</math> were chosen for this article because they are commonly used in [[geodesy]] for the [[Latitude#Isometric latitude|isometric latitude]] (vertical coordinate of the [[Mercator projection]]) and [[Geodetic coordinates|geodetic latitude]], respectively, and geodesy/cartography was the original context for the study of the Gudermannian and inverse Gudermannian functions.</ref> The Gudermannian function reveals a close relationship between the [[circular function]]s and [[hyperbolic function]]s. 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.<ref>Gudermann published several papers about the trigonometric and hyperbolic functions in [[Crelle's Journal]] in 1830–1831. These were collected in a book, {{harvp|Gudermann|1833}}.</ref> The gudermannian is sometimes called the '''hyperbolic amplitude''' as a [[limiting case (mathematics)|limiting case]] of the [[Jacobi elliptic functions#am|Jacobi elliptic amplitude]] <math display=inline>\operatorname{am}(\psi, m)</math> when parameter <math display=inline>m=1.</math>

The [[real number|real]] Gudermannian function is typically defined for <math display=inline>-\infty < \psi < \infty</math> to be the integral of the hyperbolic secant<ref>{{harvp|Roy|Olver|2010}} [http://dlmf.nist.gov/4.23#viii §4.23(viii) "Gudermannian Function"]; {{harvp|Beyer|1987}}</ref>

:<math>
\phi = \operatorname{gd} \psi
\equiv \int_0^\psi \operatorname{sech} t \,\mathrm{d}t
= \operatorname{arctan} (\sinh \psi).</math>

The real inverse Gudermannian function can be defined for <math display=inline>-\tfrac12\pi < \phi < \tfrac12\pi</math> as the [[integral of the secant function|integral of the (circular) secant]]

:<math>
\psi = \operatorname{gd}^{-1} \phi
= \int_0^\phi \operatorname{sec} t \,\mathrm{d}t
= \operatorname{arsinh} (\tan \phi). </math>

The hyperbolic angle measure <math>\psi = \operatorname{gd}^{-1} \phi</math> is called the ''anti-gudermannian'' of <math>\phi</math> or sometimes the '''lambertian''' of <math>\phi</math>, denoted <math>\psi = \operatorname{lam} \phi.</math><ref>{{harvp|Kennelly|1929}}; {{harvp|Lee|1976}}</ref> In the context of [[geodesy]] and [[navigation]] for latitude <math display=inline>\phi</math>, <math>k \operatorname{gd}^{-1} \phi</math> (scaled by arbitrary constant <math display=inline>k</math>) was historically called the '''meridional part''' of <math>\phi</math> ([[French (language)|French]]: ''latitude croissante''). It is the vertical coordinate of the [[Mercator projection]].

The two angle measures <math display=inline>\phi</math> and <math display=inline>\psi</math> are related by a common [[stereographic projection]]

:<math>s = \tan \tfrac12 \phi = \tanh \tfrac12 \psi,</math>

and this identity can serve as an alternative definition for <math display=inline>\operatorname{gd}</math> and <math display=inline>\operatorname{gd}^{-1}</math> valid throughout the [[complex plane]]:

:<math>\begin{aligned}
\operatorname{gd} \psi &= {2\arctan}\bigl(\tanh\tfrac12 \psi \,\bigr), \\[5mu]
\operatorname{gd}^{-1} \phi &= {2\operatorname{artanh}}\bigl(\tan\tfrac12 \phi \,\bigr).
\end{aligned}</math>

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