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{{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>

__TOC__
{{clear}}

== Circular–hyperbolic identities ==

We can evaluate the integral of the hyperbolic secant using the stereographic projection ([[tangent half-angle substitution#Hyperbolic functions|hyperbolic half-tangent]]) as a [[Integration by substitution|change of variables]]:<ref>{{harvp|Masson|2021}}</ref>

:<math>\begin{align}
\operatorname{gd} \psi
&\equiv \int_0^\psi \frac{1}{\operatorname{cosh} t}\mathrm{d}t
= \int_0^{\tanh\frac12\psi} \frac{1-u^2}{1 + u^2}\frac{2\,\mathrm{d}u}{1 - u^2} \qquad \bigl(u = \tanh\tfrac12 t \bigr) \\[8mu]
&= 2\int_0^{\tanh\frac12\psi} \frac{1}{1 + u^2} \mathrm{d}u
= {2\arctan}\bigl(\tanh\tfrac12\psi\,\bigr), \\[5mu]
\tan\tfrac12{\operatorname{gd} \psi} &= \tanh\tfrac12\psi.
\end{align}</math>

Letting <math display=inline>\phi = \operatorname{gd} \psi</math> and <math display=inline>s = \tan \tfrac12 \phi = \tanh \tfrac12 \psi</math> we can derive a number of identities between hyperbolic functions of <math display=inline>\psi</math> and circular functions of <math display=inline>\phi.</math><ref>{{harvp|Gottschalk|2003}} pp. 23–27</ref>

[[File:Gudermannian identities.png|frameless|left|upright=2|Identities related to the Gudermannian function represented graphically.]]
{{clear|left}}
:<math>\begin{align}
s &= \tan \tfrac12 \phi = \tanh \tfrac12 \psi,                   \\[6mu]
\frac{2s}{1 + s^2} &= \sin \phi      = \tanh \psi,               \quad &
\frac{1 + s^2}{2s} &= \csc \phi      = \coth \psi,               \\[10mu]
\frac{1 - s^2}{1 + s^2} &= \cos \phi = \operatorname{sech} \psi, \quad &
\frac{1 + s^2}{1 - s^2} &= \sec \phi = \cosh \psi,               \\[10mu]
\frac{2s}{1 - s^2}      &= \tan \phi = \sinh \psi,               \quad &
\frac{1 - s^2}{2s}      &= \cot \phi = \operatorname{csch} \psi. \\[8mu]
\end{align}</math>

These are commonly used as expressions for <math>\operatorname{gd}</math> and <math>\operatorname{gd}^{-1}</math> for real values of <math>\psi</math> and <math>\phi</math> with <math>|\phi| < \tfrac12\pi.</math> For example, the numerically well-behaved formulas

:<math>\begin{align}
\operatorname{gd} \psi &= \operatorname{arctan} (\sinh \psi), \\[6mu]
\operatorname{gd}^{-1} \phi &= \operatorname{arsinh} (\tan \phi).
\end{align}</math>

(Note, for <math>|\phi| > \tfrac12\pi</math> and for complex arguments, care must be taken choosing [[Branch point|branches]] of the inverse functions.)<ref>{{harvp|Masson|2021}} draws complex-valued plots of several of these, demonstrating that naïve implementations that choose the principal branch of inverse trigonometric functions yield incorrect results.</ref>

We can also express <math display=inline>\psi</math> and <math display=inline>\phi</math> in terms of <math display=inline>s\colon</math>

:<math>\begin{align}
2\arctan s &= \phi = \operatorname{gd} \psi,                     \\[6mu]
2\operatorname{artanh} s &= \operatorname{gd}^{-1} \phi = \psi.  \\[6mu]
\end{align}</math>

If we expand <math display=inline>\tan\tfrac12</math> and <math display=inline>\tanh\tfrac12</math> in terms of the [[Exponential function#Complex plane|exponential]], then we can see that <math display=inline>s,</math> <math>\exp \phi i,</math> and <math>\exp \psi</math> are all [[Möbius transformation]]s of each-other (specifically, rotations of the [[Riemann sphere]]):

:<math>\begin{align}
s &= i\frac{1-e^{\phi i}}{1+e^{\phi i}} = \frac{e^\psi - 1}{e^\psi + 1}, \\[10mu]
i \frac{s - i}{s + i} &= \exp \phi i \quad = \frac{e^\psi - i}{e^\psi + i}, \\[10mu]
\frac{1 + s}{1 - s} &= i\frac{i+e^{\phi i}}{i-e^{\phi i}} \,= \exp \psi.
\end{align}</math>

For real values of <math display=inline>\psi</math> and <math display=inline>\phi</math> with <math>|\phi| < \tfrac12\pi</math>, these Möbius transformations can be written in terms of trigonometric functions in several ways,

:<math>\begin{align}
\exp \psi
&= \sec \phi + \tan \phi
= \tan\tfrac12 \bigl(\tfrac12\pi + \phi \bigr) \\[6mu]
&= \frac{1 + \tan\tfrac12 \phi}{1 - \tan\tfrac12 \phi}
= \sqrt{\frac{1+\sin \phi}{1-\sin \phi}}, \\[12mu]

\exp \phi i
&= \operatorname{sech} \psi + i \tanh \psi
= \tanh\tfrac12 \bigl({-\tfrac12}\pi i + \psi \bigr) \\[6mu]
&= \frac{1 + i \tanh\tfrac12 \psi}{1 - i \tanh\tfrac12 \psi}
= \sqrt{\frac{1 + i \sinh \psi}{1 - i \sinh \psi}}.
\end{align}</math>

These give further expressions for <math>\operatorname{gd}</math> and <math>\operatorname{gd}^{-1}</math> for real arguments with <math>|\phi| < \tfrac12\pi.</math> For example,<ref name=weinstein>{{mathworld|urlname=Gudermannian|title=Gudermannian}}</ref>

:<math>\begin{align}
\operatorname{gd} \psi &= 2 \arctan e^\psi - \tfrac12\pi, \\[6mu]
\operatorname{gd}^{-1} \phi &= \log (\sec \phi + \tan \phi).
\end{align}</math>

== Complex values ==
[[File:Gudermannian conformal map.png|thumb|right|upright=1.5|The Gudermannian function {{math|''z'' ↦ gd ''z''}} is a conformal map from an infinite strip to an infinite strip. It can be broken into two parts: a map {{math|''z'' ↦ tanh {{sfrac|1|2}}''z''}} from one infinite strip to the complex unit disk and a map {{math|''ζ'' ↦ 2 arctan ''ζ''}} from the disk to the other infinite strip.]]

As a [[Complex analysis|function of a complex variable]], <math display=inline>z \mapsto w = \operatorname{gd} z</math> [[conformal map|conformally maps]] the infinite strip <math display=inline>\left|\operatorname{Im}z\right| \leq \tfrac12\pi</math> to the infinite strip <math display=inline>\left|\operatorname{Re}w\right| \leq \tfrac12\pi,</math> while <math display=inline>w \mapsto z = \operatorname{gd}^{-1} w</math> conformally maps the infinite strip <math display=inline>\left|\operatorname{Re}w\right| \leq \tfrac12\pi</math> to the infinite strip <math display=inline> \left|\operatorname{Im}z\right| \leq \tfrac12\pi.</math>

[[Analytic continuation|Analytically continued]] by [[Schwarz reflection principle|reflections]] to the whole complex plane, <math display=inline>z \mapsto w = \operatorname{gd} z</math> is a periodic function of period <math display=inline>2\pi i</math> which sends any infinite strip of "height" <math display=inline>2\pi i</math> onto the strip <math display=inline>-\pi< \operatorname{Re}w \leq \pi.</math> Likewise, extended to the whole complex plane, <math display=inline>w \mapsto z = \operatorname{gd}^{-1} w</math> is a periodic function of period <math display=inline>2\pi</math> which sends any infinite strip of "width" <math display=inline>2\pi</math> onto the strip <math display=inline>-\pi < \operatorname{Im}z \leq \pi.</math><ref>{{harvp|Kennelly|1929}}</ref> For all points in the complex plane, these functions can be correctly written as:

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

For the <math display=inline>\operatorname{gd}</math> and <math display=inline>\operatorname{gd}^{-1}</math> functions to remain invertible with these extended domains, we might consider each to be a [[multivalued function]] (perhaps <math display=inline>\operatorname{Gd}</math> and <math display=inline>\operatorname{Gd}^{-1}</math>, with <math display=inline>\operatorname{gd}</math> and <math display=inline>\operatorname{gd}^{-1}</math> the [[principal branch]]) or consider their domains and codomains as [[Riemann surface]]s.

If <math display=inline>u + iv = \operatorname{gd}(x + iy),</math> then the real and imaginary components <math display=inline>u</math> and <math display=inline>v</math> can be found by:<ref>{{harvp|Kennelly|1929}} [https://archive.org/details/dli.ministry.19102/page/181 p. 181];
{{harvp|Beyer|1987}} [https://archive.org/details/crchandbookofmat00beye/page/269/mode/1up p. 269]</ref>

:<math>
\tan u = \frac{\sinh x}{\cos y}, \quad \tanh v = \frac{\sin y}{\cosh x}.
</math>

(In practical implementation, make sure to use the [[atan2|2-argument arctangent]], {{nobr|<math display=inline>u = \operatorname{atan2}(\sinh x, \cos y)</math>.)}}

Likewise, if <math display=inline>x + iy = \operatorname{gd}^{-1}(u + iv),</math> then components <math display=inline>x</math> and <math display=inline>y</math> can be found by:<ref>{{harvp|Beyer|1987}} [https://archive.org/details/crchandbookofmat00beye/page/269/mode/1up p. 269] – note the typo.</ref>

:<math>
\tanh x = \frac{\sin u}{\cosh v}, \quad \tan y =  \frac{\sinh v}{\cos u}.
</math>

Multiplying these together reveals the additional identity<ref name=weinstein/>
:<math>
\tanh x\, \tan y = \tan u\, \tanh v.
</math>

=== Symmetries ===

The two functions can be thought of as rotations or reflections of each-other, with a similar relationship as <math display=inline>\sinh iz = i \sin z</math> [[Hyperbolic functions#Hyperbolic functions for complex numbers|between sine and hyperbolic sine]]:<ref>{{harvp|Legendre|1817}} [https://archive.org/details/exercicescalculi02legerich/page/n165/ §4.2.8(163) pp. 144–145]</ref>
:<math>\begin{aligned}
\operatorname{gd} iz      &= i \operatorname{gd}^{-1} z, \\[5mu]
\operatorname{gd}^{-1} iz &= i \operatorname{gd} z.
\end{aligned}</math>

The functions are both [[even and odd functions|odd]] and they commute with [[complex conjugate|complex conjugation]]. That is, a reflection across the real or imaginary axis in the domain results in the same reflection in the codomain:
:<math>\begin{aligned}
\operatorname{gd} (-z)       &= -\operatorname{gd} z, &\quad
\operatorname{gd} \bar z   &= \overline{\operatorname{gd} z}, &\quad
\operatorname{gd} (-\bar z)  &= -\overline{\operatorname{gd} z}, \\[5mu]
\operatorname{gd}^{-1} (-z) &= -\operatorname{gd}^{-1} z, &\quad
\operatorname{gd}^{-1} \bar z   &= \overline{\operatorname{gd}^{-1} z}, &\quad
\operatorname{gd}^{-1} (-\bar z)  &= -\overline{\operatorname{gd}^{-1} z}.
\end{aligned}</math>

The functions are [[periodic function|periodic]], with periods <math display=inline>2\pi i</math> and <math display=inline>2\pi</math>:
:<math>\begin{aligned}
\operatorname{gd} (z + 2\pi i) &= \operatorname{gd} z, \\[5mu]
\operatorname{gd}^{-1} (z + 2\pi) &= \operatorname{gd}^{-1} z.
\end{aligned}</math>

A translation in the domain of <math display=inline>\operatorname{gd}</math> by <math display=inline>\pm\pi i</math> results in a half-turn rotation and translation in the codomain by one of <math display=inline>\pm\pi,</math> and vice versa for <math display=inline>\operatorname{gd}^{-1}\colon</math><ref>{{harvp|Kennelly|1929}} [https://archive.org/details/dli.ministry.19102/page/182 p. 182]</ref>

:<math>\begin{aligned}
\operatorname{gd} ({\pm \pi i} + z) &= \begin{cases}
\pi - \operatorname{gd} z
  \quad &\mbox{if }\ \  \operatorname{Re} z \geq 0, \\[5mu]
 -\pi - \operatorname{gd} z
  \quad &\mbox{if }\ \  \operatorname{Re} z < 0,
\end{cases} \\[15mu]

\operatorname{gd}^{-1} ({\pm \pi} + z) &= \begin{cases}
\pi i - \operatorname{gd}^{-1} z
  \quad &\mbox{if }\ \  \operatorname{Im} z \geq 0, \\[3mu]
-\pi i - \operatorname{gd}^{-1} z
  \quad &\mbox{if }\ \  \operatorname{Im} z < 0.
\end{cases}
\end{aligned}</math>

A reflection in the domain of <math display=inline>\operatorname{gd}</math> across either of the lines <math display=inline>x \pm \tfrac12\pi i</math> results in a reflection in the codomain across one of the lines <math display=inline>\pm \tfrac12\pi + yi,</math> and vice versa for <math display=inline>\operatorname{gd}^{-1}\colon</math>

:<math>\begin{aligned}
\operatorname{gd} ({\pm \pi i} + \bar z) &= \begin{cases}

\pi - \overline{\operatorname{gd} z}
  \quad &\mbox{if }\ \  \operatorname{Re} z \geq 0, \\[5mu]
 -\pi - \overline{\operatorname{gd} z}
  \quad &\mbox{if }\ \  \operatorname{Re} z < 0,
\end{cases} \\[15mu]

\operatorname{gd}^{-1} ({\pm \pi} - \bar z) &= \begin{cases}
\pi i + \overline{\operatorname{gd}^{-1} z}
  \quad &\mbox{if }\ \  \operatorname{Im} z \geq 0, \\[3mu]
-\pi i + \overline{\operatorname{gd}^{-1} z}
  \quad &\mbox{if }\ \  \operatorname{Im} z < 0.
\end{cases}
\end{aligned}</math>

This is related to the identity
:<math>
\tanh\tfrac12 ({\pi i} \pm z) = \tan\tfrac12 ({\pi} \mp \operatorname{gd} z).
</math>

=== Specific values ===

A few specific values (where <math display=inline>\infty</math> indicates the limit at one end of the infinite strip):<ref>{{harvp|Kahlig|Reich|2013}}</ref>
:<math>\begin{align}
\operatorname{gd}(0) &= 0,  &\quad
  {\operatorname{gd}}\bigl({\pm {\log}\bigl(2 + \sqrt3\bigr)}\bigr) &= \pm \tfrac13\pi, \\[5mu]
\operatorname{gd}(\pi i) &= \pi, &\quad
  {\operatorname{gd}}\bigl({\pm \tfrac13}\pi i\bigr) &= \pm {\log}\bigl(2 + \sqrt3\bigr)i, \\[5mu]
\operatorname{gd}({\pm \infty}) &= \pm\tfrac12\pi, &\quad
  {\operatorname{gd}}\bigl({\pm {\log}\bigl(1 + \sqrt2\bigr)}\bigr) &= \pm \tfrac14\pi, \\[5mu]
{\operatorname{gd}}\bigl({\pm \tfrac12}\pi i\bigr) &= \pm \infty i, &\quad
  {\operatorname{gd}}\bigl({\pm \tfrac14}\pi i\bigr) &= \pm {\log}\bigl(1 + \sqrt2\bigr)i, \\[5mu]

&& {\operatorname{gd}}\bigl({\log}\bigl(1 + \sqrt2\bigr) \pm \tfrac12\pi i\bigr) &= \tfrac12\pi \pm {\log}\bigl(1 + \sqrt2\bigr)i, \\[5mu]
&& {\operatorname{gd}}\bigl({-\log}\bigl(1 + \sqrt2\bigr) \pm \tfrac12\pi i\bigr) &= -\tfrac12\pi \pm {\log}\bigl(1 + \sqrt2\bigr)i.

\end{align}</math>

==Derivatives==
As the Gudermannian and inverse Gudermannian functions can be defined as the antiderivatives of the hyperbolic secant and circular secant functions, respectively, their derivatives are those secant functions:

:<math>\begin{align}
\frac{\mathrm d}{\mathrm d z} \operatorname{gd} z      &= \operatorname{sech} z , \\[10mu]
\frac{\mathrm d}{\mathrm d z} \operatorname{gd}^{-1} z &= \sec z .
\end{align}</math>

== Argument-addition identities ==

By combining [[hyperbolic functions#Sums of arguments|hyperbolic]] and [[trigonometric functions#Sum and difference formulas|circular]] argument-addition identities,

:<math>\begin{align}
\tanh(z + w) &= \frac{\tanh z + \tanh w}{1 + \tanh z \, \tanh w }, \\[10mu]
\tan(z + w)  &= \frac{\tan z  + \tan w }{1 - \tan z  \, \tan w  },
\end{align}</math>

with the [[#Circular–hyperbolic identities|circular–hyperbolic identity]],
:<math>
\tan \tfrac12 (\operatorname{gd} z) = \tanh \tfrac12 z,
</math>

we have the Gudermannian argument-addition identities:

:<math>\begin{align}
\operatorname{gd}(z + w) &= 2 \arctan \frac
  {\tan \tfrac12(\operatorname{gd} z) + \tan\tfrac12(\operatorname{gd} w)}
  {1 + \tan\tfrac12(\operatorname{gd} z) \, \tan\tfrac12(\operatorname{gd} w)}, \\[12mu]
\operatorname{gd}^{-1}(z + w) &= 2 \operatorname{artanh} \frac
  {\tanh\tfrac12(\operatorname{gd}^{-1} z) + \tanh\tfrac12(\operatorname{gd}^{-1} w)}
  {1 - \tanh\tfrac12(\operatorname{gd}^{-1} z) \, \tanh\tfrac12(\operatorname{gd}^{-1} w)}.
\end{align}</math>

Further argument-addition identities can be written in terms of other circular functions,<ref>{{harvp|Cayley|1862}} [https://archive.org/details/londonedinburg4241862lond/page/21 p. 21]</ref> but they require greater care in choosing branches in inverse functions. Notably,

:<math>\begin{align}
\operatorname{gd}(z + w) &= u + v, \quad \text{where}\ \tan u = \frac{\sinh z}{\cosh w},\ \tan v = \frac{\sinh w}{\cosh z}, \\[10mu]
\operatorname{gd}^{-1}(z + w) &= u + v, \quad \text{where}\ \tanh u = \frac{\sin z}{\cos w},\ \tanh v = \frac{\sin w}{\cos z},
\end{align}</math>

which can be used to derive the [[Gudermannian function#Complex values|per-component computation]] for the complex Gudermannian and inverse Gudermannian.<ref>{{harvp|Kennelly|1929}} [https://archive.org/details/dli.ministry.19102/page/180 pp. 180–183]</ref>

In the specific case <math display=inline>z = w,</math> double-argument identities are

:<math>\begin{align}
\operatorname{gd}(2z)
&= 2 \arctan (\sin(\operatorname{gd} z)), \\[5mu]
\operatorname{gd}^{-1}(2z) &= 2 \operatorname{artanh}(\sinh(\operatorname{gd}^{-1}z)).
\end{align}</math>

== Taylor series ==
The [[Taylor series]] near zero, valid for complex values <math display=inline>z</math> with <math display=inline>|z| < \tfrac12\pi,</math> are<ref>{{harvp|Legendre|1817}} [https://archive.org/details/exercicescalculi02legerich/page/n165/ §4.2.7(162) pp. 143–144]</ref>

:<math>\begin{align}
\operatorname{gd} z
&= \sum_{k=0}^\infty \frac{E_k}{(k+1)!}z^{k+1}
= z - \frac16z^3 + \frac1{24}z^5 - \frac{61}{5040}z^7 + \frac{277}{72576}z^9 - \dots, \\[10mu]
\operatorname{gd}^{-1} z
&= \sum_{k=0}^\infty \frac{|E_k|}{(k+1)!}z^{k+1}
= z + \frac16z^3 + \frac1{24}z^5 + \frac{61}{5040}z^7 + \frac{277}{72576}z^9 + \dots,
\end{align}</math>

where the numbers <math display=inline>E_{k}</math> are the [[Euler numbers|Euler secant numbers]], 1, 0, -1, 0, 5, 0, -61, 0, 1385 ... (sequences {{OEIS link|A122045}}, {{OEIS link|A000364}}, and {{OEIS link|A028296}} in the [[On-Line Encyclopedia of Integer Sequences|OEIS]]). These series were first computed by [[James Gregory (mathematician)|James Gregory]] in 1671.<ref>{{cite book |editor-last=Turnbull |editor-first=Herbert Westren |year=1939 |title=James Gregory; Tercentenary Memorial Volume |publisher=G. Bell & Sons |page=170 }}</ref>

Because the Gudermannian and inverse Gudermannian functions are the integrals of the hyperbolic secant and secant functions, the numerators <math display=inline>E_{k}</math> and <math display=inline>|E_{k}|</math> are same as the numerators of the [[Hyperbolic functions#Taylor series expressions|Taylor series for {{math|sech}}]] and [[Trigonometric functions#Power series expansion|{{math|sec}}]], respectively, but shifted by one place.

The reduced unsigned numerators are 1, 1, 1, 61, 277, ... and the reduced denominators are 1, 6, 24, 5040, 72576, ... (sequences {{OEIS link|A091912}} and {{OEIS link|A136606}} in the [[On-Line Encyclopedia of Integer Sequences|OEIS]]).

==History==
{{Broader| Mercator projection#History| Integral of the secant function}}
The function and its inverse are related to the [[Mercator projection]]. The vertical coordinate in the Mercator projection is called [[Latitude#Isometric latitude|isometric latitude]], and is often denoted <math display=inline>\psi.</math> In terms of [[latitude]] <math display=inline>\phi</math> on the sphere (expressed in [[radian]]s) the isometric latitude can be written

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

The inverse from the isometric latitude to spherical latitude is <math display=inline>\phi = \operatorname{gd} \psi.</math> (Note: on an [[ellipsoid of revolution]], the relation between geodetic latitude and isometric latitude is slightly more complicated.)

[[Gerardus Mercator]] plotted his celebrated map in 1569, but the precise method of construction was not revealed. In 1599, [[Edward Wright (mathematician)|Edward Wright]] described a method for constructing a Mercator projection numerically from trigonometric tables, but did not produce a closed formula. The closed formula was published in 1668 by [[James Gregory (mathematician)|James Gregory]].

The Gudermannian function per se 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 [[Christoph Gudermann]]'s work in the 1830s on the theory of special functions.<ref>{{harvp|Becker|Van Orstrand|1909}}</ref>
Gudermann had published articles in ''[[Crelle's Journal]]'' that were later collected in a book<ref>{{harvp|Gudermann|1833}}</ref>
which expounded <math display=inline>\sinh</math> and <math display=inline>\cosh</math> to a wide audience (although represented by the symbols <math display=inline>\mathfrak{Sin}</math> and <math display=inline>\mathfrak{Cos}</math>).

The notation <math display=inline>\operatorname{gd}</math> was introduced by Cayley who starts by calling <math display=inline>\phi = \operatorname{gd} u</math> the [[Jacobi elliptic functions#am|Jacobi elliptic amplitude]] <math display=inline>\operatorname{am} u</math> in the degenerate case where the elliptic modulus is <math display=inline>m = 1,</math> so that <math display=inline>\sqrt{1 + m\sin\!^2\,\phi}</math> reduces to <math display=inline>\cos \phi.</math><ref>{{harvp|Cayley|1862}}</ref> This is the inverse of the [[integral of the secant function]]. Using Cayley's notation,

:<math>
u = \int_0 \frac{d\phi}{\cos \phi}
= {\log\, \tan}\bigl(\tfrac14\pi + \tfrac12 \phi\bigr).
</math>

He then derives "the definition of the transcendent",

:<math>
\operatorname{gd} u = {\frac1i \log\, \tan} \bigl(\tfrac14\pi + \tfrac12 ui\bigr),
</math>

observing that "although exhibited in an imaginary form, [it] is a real function of {{nobr|<math display=inline> u</math>".}}

The Gudermannian and its inverse were used to make [[trigonometric tables]] of circular functions also function as tables of hyperbolic functions. Given a hyperbolic angle <math display=inline>\psi</math>, hyperbolic functions could be found by first looking up <math display=inline>\phi = \operatorname{gd} \psi</math> in a Gudermannian table and then looking up the appropriate circular function of <math display=inline>\phi</math>, or by directly locating <math display=inline>\psi</math> in an auxiliary <math>\operatorname{gd}^{-1}</math> column of the trigonometric table.<ref>For example Hoüel labels the hyperbolic functions across the top in Table XIV of: {{cite book |last=Hoüel |first=Guillaume Jules |year=1885 |title=Recueil de formules et de tables numériques |publisher=Gauthier-Villars |page=36 |url=https://archive.org/details/recueildeformul00hogoog/page/n115/}} </ref>

== Generalization ==

The Gudermannian function can be thought of mapping points on one branch of a hyperbola to points on a semicircle. Points on one sheet of an ''n''-dimensional [[hyperboloid|hyperboloid of two sheets]] can be likewise mapped onto a ''n''-dimensional hemisphere via stereographic projection. The [[Hyperbolic geometry#The hemisphere model|hemisphere model of hyperbolic space]] uses such a map to represent hyperbolic space.

==Applications==

[[File:Distance in the half-plane model 3.png|thumb|right|upright=1.5|Distance in the [[Poincaré half-plane model]] of the [[hyperbolic plane]] from the apex of a semicircle to another point on it is the inverse Gudermannian function of the central angle.]]

*The [[angle of parallelism]] function in [[hyperbolic geometry]] is the [[Angle#Combining angle pairs|complement]] of the gudermannian, <math>\mathit{\Pi}(\psi) = \tfrac12\pi - \operatorname{gd} \psi.</math>
* On a [[Mercator projection]] a line of constant latitude is parallel to the equator (on the projection) at a distance proportional to the anti-gudermannian of the latitude.
* The Gudermannian function (with a complex argument) may be used to define the [[transverse Mercator projection]].<ref>{{harvp|Osborne|2013}} p. 74</ref>
* The Gudermannian function appears in a non-periodic solution of the [[inverted pendulum]].<ref>{{harvp|Robertson|1997}}</ref>
* The Gudermannian function appears in a moving mirror solution of the dynamical [[Casimir effect]].<ref>{{harvp|Good|Anderson|Evans|2013}}</ref>
* If an infinite number of infinitely long, equidistant, parallel, coplanar, straight wires are kept at equal [[electric potential|potentials]] with alternating signs, the potential-flux distribution in a cross-sectional plane perpendicular to the wires is the complex Gudermannian function.<ref>{{harvp|Kennelly|1928}}</ref>
* The Gudermannian function is a [[sigmoid function]], and as such is sometimes used as an [[activation function]] in machine learning.
* The (scaled and shifted) Gudermannian function is the [[cumulative distribution function]] of the [[hyperbolic secant distribution]].
* A function based on the Gudermannian provides a good model for the shape of [[spiral galaxy]] arms.<ref>{{harvp|Ringermacher|Mead|2009}}</ref>

== See also ==
*[[Tractrix]]
*{{slink|Catenary#Catenary of equal strength}}

== Notes ==
{{reflist|25em}}

== References ==
{{sfn whitelist |CITEREFRoyOlver2010}}
{{refbegin|30em}}
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{{refend}}

== External links ==

* Penn, Michael (2020) [https://www.youtube.com/watch?v=rypoQgdF5cM "the Gudermannian function!"] on YouTube.


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