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5 Gamma FunctionApplications

§5.19 Mathematical Applications

  1. §5.19(i) Summation of Rational Functions
  2. §5.19(ii) Mellin–Barnes Integrals
  3. §5.19(iii) n-Dimensional Sphere

§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.


5.19.1 S =k=0ak,
ak =k(3k+2)(2k+1)(k+1).

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

5.19.2 ak=2k+231k+121k+1=(1k+11k+12)2(1k+11k+23).

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

5.19.3 S=ψ(12)2ψ(23)γ=3ln32ln213π3.

§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 V =π12nrnΓ(12n+1),
S =2π12nrn1Γ(12n)=nrV.