Editing Gudermannian function (section)

== 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>