§5.19 Mathematical Applications

§5.19(i) Summation of Rational Functions

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

Example

 5.19.1 $\displaystyle S$ $\displaystyle=\sum_{k=0}^{\infty}a_{k},$ $\displaystyle a_{k}$ $\displaystyle=\frac{k}{(3k+2)(2k+1)(k+1)}.$ ⓘ Symbols: $k$: nonnegative integer and $a$: real or complex variable Permalink: http://dlmf.nist.gov/5.19.E1 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 5.19(i), 5.19(i), 5.19 and 5

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

 5.19.2 $a_{k}=\frac{2}{k+\frac{2}{3}}-\frac{1}{k+\frac{1}{2}}-\frac{1}{k+1}=\left(% \frac{1}{k+1}-\frac{1}{k+\frac{1}{2}}\right)-2\left(\frac{1}{k+1}-\frac{1}{k+% \frac{2}{3}}\right).$ ⓘ Symbols: $k$: nonnegative integer and $a$: real or complex variable Permalink: http://dlmf.nist.gov/5.19.E2 Encodings: TeX, pMML, png See also: Annotations for 5.19(i), 5.19(i), 5.19 and 5

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

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

§5.19(ii) Mellin–Barnes Integrals

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).

§5.19(iii) $n$-Dimensional Sphere

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

 5.19.4 $\displaystyle V$ $\displaystyle=\frac{\pi^{\frac{1}{2}n}r^{n}}{\Gamma(\frac{1}{2}n+1)},$ $\displaystyle S$ $\displaystyle=\frac{2\pi^{\frac{1}{2}n}r^{n-1}}{\Gamma(\frac{1}{2}n)}=\frac{n}% {r}V.$