Editing Gudermannian function (section)

=== 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>