Digital Library of Mathematical Functions
The Gamma Function -- Mathematical Applications

### Mathematical Applications

###### Contents

- Summation of Rational Functions
- Mellin-Barnes Integrals
- n -Dimensional Sphere

#### Summation of Rational Functions

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

##### Example

S = ∑ k = 0 ∞ a k , a k = k ( 3 k + 2 ) ( 2 k + 1 ) ( k + 1 )

By decomposition into partial fractions

a k = 2 k + 2 3 - 1 k + 1 2 - 1 k + 1 = ( 1 k + 1 - 1 k + 1 2 ) - 2 ( 1 k + 1 - 1 k + 2 3 )

Hence from ( Series Expansions 6 ), ( Special Values and Extrema Equation 13 and Equation 19 )

S = ψ ( 1 2 ) - 2 ψ ( 2 3 ) - γ = 3 ln 3 - 2 ln 2 - 1 3 π 3

#### 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. The left-hand side of ( Integral Equation 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.

#### n -Dimensional Sphere

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

V = π 1 2 n r n Γ ( 1 2 n + 1 ) , S = 2 π 1 2 n r n - 1 Γ ( 1 2 n ) = n r V