# §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

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

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

## §5.19(ii) Mellin–Barnes Integrals

Many special functions 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 , 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 for large , or small , 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) -Dimensional Sphere

The volume and surface area of the -dimensional sphere of radius are given by