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## Article 24Series for Gudermanian and its Inverse.

Substitute for $sechu,secv$ in (49), (48) their expansions, Art. 16, and integrate, then

No constants of integration appear, since $gdu$ vanishes with $u$, and ${gd}^{-1}v$ with $v$. These series are seldom used in computation, as $gdu$ is best found and tabulated by means of tables of natural tangents and hyperbolic sines, from the equation

and a table of the direct function can be used to furnish the numerical values of the inverse function; or the latter can be obtained from the equation,

To obtain a logarithmic expression for ${gd}^{-1}v$, let

Prob. 63.
Evaluate ${\frac{gdu-u}{{u}^{3}}]}_{u\doteq 0}$, ${\frac{{gd}^{-1}v-v}{{v}^{3}}]}_{v\doteq 0}$.
Prob. 64.
Prove that $gdu-sinu$ is an inﬁnitesimal of the ﬁfth order, when $u\doteq 0$.
Prob. 65.
Prove the relations $\frac{1}{4}\pi +\frac{1}{2}v{tan}^{-1}{e}^{u}$, $\frac{1}{4}\pi -\frac{1}{2}v={tan}^{-1}{e}^{-u}$.

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