Gudermannian Function

Denoted either $\gamma(x)$ or $\mathop{\rm gd}\nolimits (x)$.

$$\mathop{\rm gd}\nolimits (x) \equiv \tan^{-1}(\sinh x) = 2\tan^{-1}(e^x)-{\textstyle{1\over 2}}\pi$$ (1)

$$\mathop{\rm gd}\nolimits ^{-1}(x) = \ln[\tan({\textstyle{1\over 4}}\pi+{\textstyle{1\over 2}}x)] = \ln(\sec x+\tan x).$$ (2)

The derivatives are given by

$${d\over dx} \mathop{\rm gd}\nolimits (x) = \mathop{\rm sech}\nolimits x$$ (3)

$${d\over dx} \mathop{\rm gd}\nolimits ^{-1}(x) = \sec x.$$ (4)