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Gudermannian Function

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

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


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

The derivatives are given by
\begin{displaymath}
{d\over dx} \mathop{\rm gd}\nolimits (x) = \mathop{\rm sech}\nolimits x
\end{displaymath} (3)


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




© 1996-9 Eric W. Weisstein
1999-05-25