- §5.19(i) Summation of Rational Functions
- §5.19(ii) Mellin–Barnes Integrals
- §5.19(iii) $n$-Dimensional Sphere

As shown in Temme (1996b, §3.4), the results given in §5.7(ii) can be used to sum infinite series of rational functions.

5.19.1 | $S$ | $={\displaystyle \sum _{k=0}^{\mathrm{\infty}}}{a}_{k},$ | ||

${a}_{k}$ | $={\displaystyle \frac{k}{(3k+2)(2k+1)(k+1)}}.$ | |||

By decomposition into partial fractions (§1.2(iii))

5.19.2 | $${a}_{k}=\frac{2}{k+\frac{2}{3}}-\frac{1}{k+\frac{1}{2}}-\frac{1}{k+1}=\left(\frac{1}{k+1}-\frac{1}{k+\frac{1}{2}}\right)-2\left(\frac{1}{k+1}-\frac{1}{k+\frac{2}{3}}\right).$$ | ||

Hence from (5.7.6), (5.4.13), and (5.4.19)

5.19.3 | $$S=\psi \left(\frac{1}{2}\right)-2\psi \left(\frac{2}{3}\right)-\gamma =3\mathrm{ln}3-2\mathrm{ln}2-\frac{1}{3}\pi \sqrt{3}.$$ | ||

Many special functions $f(z)$ can be represented as a *Mellin–Barnes
integral*, that is, an integral of a product of gamma functions, reciprocals
of gamma functions, and a power of $z$, the integration contour being
doubly-infinite and eventually parallel to the imaginary axis at both ends. The left-hand
side of (5.13.1) is a typical example. By translating the contour
parallel to itself and summing the residues of the integrand, asymptotic
expansions of $f(z)$ for large $|z|$, or small $|z|$, can be obtained complete
with an integral representation of the error term. For further information
and examples see §2.5 and
Paris and Kaminski (2001, Chapters 5, 6, and 8).

The volume $V$ and surface area $S$ of the $n$-dimensional sphere of radius $r$ are given by

5.19.4 | $V$ | $={\displaystyle \frac{{\pi}^{\frac{1}{2}n}{r}^{n}}{\mathrm{\Gamma}(\frac{1}{2}n+1)}},$ | ||

$S$ | $={\displaystyle \frac{2{\pi}^{\frac{1}{2}n}{r}^{n-1}}{\mathrm{\Gamma}(\frac{1}{2}n)}}={\displaystyle \frac{n}{r}}V.$ | |||