§ 5.19. Mathematical Applications

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

§ 5.19(i). Summation of Rational Functions

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

¶ Example

5.19.1

$S=\sum _{{k=0}}^{\infty}a_{k},$

$a_{k}=\frac{k}{(3k+2)(2k+1)(k+1)}.$

Symbols:: $k$ : nonnegative integer and $a$ : real or complex variable
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By decomposition into partial fractions (§Ch.1)

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

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

Symbols:: $\EulerConstant$ : Euler's constant and $\psi\!\left(z\right)$ : Psi or digamma function
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§ 5.19(ii). Mellin-Barnes Integrals

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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

Notes:: See Stein and Shakarchi (2003, pp. 208–209) and Robnik (1980). The formula for $V$ can also be verified by setting $t_{k}=(x_{k}/r)^{2}$ and $z_{k}=\tfrac{1}{2},k=1,2,\dots,n$ , in (5.14.1). The formula for $S$ can be verified in a similar way from (5.14.2), or derived by differentiating the formula for $V$ .
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