math lessons - Gudermannian function

The Gudermannian function relates the circular and hyperbolic trigonometric functions without resorting to complex numbers. It is defined by

${\rm gd}(x)=\int_0^x \frac{dt}{\cosh t}$

${}=2\arctan \left(\tanh\frac{x}{2}\right)$

${}=2\arctan e^x-{\pi\over2}.$

Note that

$\tanh\frac{x}{2} = \tan \frac{\mbox{gd}(x)}{2}.$

The following identities also hold:

$\sinh(x)=\tan(\mbox{gd}(x))\$

$\cosh(x)=\sec(\mbox{gd}(x))\$

$\tanh(x)=\sin(\mbox{gd}(x))\$

$\mbox{sech}(x)=\cos(\mbox{gd}(x))\$

$\mbox{csch}(x)=\cot(\mbox{gd}(x))\$

$\coth(x)=\csc(\mbox{gd}(x))\$

The inverse Gudermannian function is given by

${\rm gd}^{-1}(x)=\int_0^x \frac{dt}{\cos t}=\ln(\tan x+\sec x).\,$

The derivatives of the Gudermannian and its inverse are

${d \over dx}\,\mbox{gd}(x)=\mbox{sech}(x)$

${d \over dx}\,\mbox{gd}^{-1}(x)=\sec(x)$

References