Gudermannian function

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Gudermannian function with its asymptotes y = ±π/2 marked in blue.

The Gudermannian function, named after Christoph Gudermann (1798–1852), relates the circular functions and hyperbolic functions without using complex numbers.

It is defined by

$\begin{align}{\rm{gd}}(x)&=\int_0^x\frac{dp}{\cosh p} \\ &=\arcsin\left(\tanh x \right) =\mbox{arctan}\left(\sinh x \right) \\ &=2\arctan\left(\tanh\left(\frac{x}{2}\right)\right) =2\arctan(e^x)-\frac{\pi}{2}. \end{align}\,\!$

Some related formulas don't quite work as definitions. For example, for real x, arccos(sech x) = |gd(x)| = arcsec(cosh x). (See inverse trigonometric functions.)

The following identities hold:

$\begin{align}{\color{white}\dot{{\color{black} \sin(\mbox{gd}(x))}}}&=\tanh x ;\quad \csc(\mbox{gd}(x))=\coth x ;\\ \cos(\mbox{gd}(x))&=\mbox{sech}\, x ;\quad\, \sec(\mbox{gd}(x))=\cosh x ;\\ \tan(\mbox{gd}(x))&=\sinh x ;\quad\, \cot(\mbox{gd}(x))=\mbox{csch}\, x ;\\ {}_{\color{white}.}\tan\left(\frac{\mbox{gd}(x)}{2}\right)&=\tanh\left(\frac{x}{2}\right). \end{align}\,\!$

The inverse Gudermannian function.

The inverse Gudermannian function, which is defined on the interval -π/2 < x < π/2, is given by

$\begin{align} \operatorname{gd}^{-1}(x) & = \int_0^x\frac{dp}{\cos p} \\ & = \ln\left| \frac{1 + \sin x}{\cos x} \right| = \frac{1}{2}\ln \left| \frac{1 + \sin x}{1 - \sin x} \right| \\ & = \ln\left| \tan x +\sec x \right| = \ln \left| \tan\left(\frac{\pi}{4} + \frac{x}{2}\right) \right| \\ & = \mbox{artanh}\,(\sin x) = \mbox{arsinh}\,(\tan x). \end{align}$