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 and hyperbolic trigonometric functions without using complex numbers.

It is defined by

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

The following identities also hold:

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

The inverse Gudermannian function.

The inverse Gudermannian function is given by

$\begin{align} \mbox{arcgd}(x)&={\rm {gd}}^{-1}(x)=\int_0^x\frac{dp}{\cos(p)},\\ &={}\mbox{arccosh}(\sec(x))=\mbox{arctanh}(\sin(x)),\\ &={}\ln\left(\sec(x)(1+\sin(x))\right),\\ &={}\ln(\tan(x)+\sec(x))=\ln\tan\left(\frac{\pi}{4}+\frac{x}{2}\right),\\ &={}\frac{1}{2}\ln \frac{1+\sin(x)}{1-\sin(x)} .\end{align}\,\!$