Gudermannian function

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

The Gudermannian function, named after Christoph Gudermann (1798 - 1852), 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

\operatorname{arcgd}(x) ={\rm gd}^{-1}(x)=\int_0^x \frac{dt}{\cos t}\,
=\operatorname{arccosh}(\sec x)=\operatorname{arctanh}(\sin x)\,
=\ln\left(\sec(x)(1+\sin(x))\right)\,
=\ln(\tan x+\sec x)=\ln\left(\tan\left(\frac{\pi}{4}+\frac{x}{2}\right)\right)\,
=\frac{1}{2}\ln\left(\frac{1+\sin x}{1-\sin x} \right).\,

The derivatives of the Gudermannian and its inverse are

{d \over dx}\,\mbox{gd}(x)=\mbox{sech}(x),
{d \over dx}\,\operatorname{arcgd}(x)=\sec(x).

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