Editing Gudermannian function (section)

== 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>