§5.19 Mathematical Applications

Keywords:: gamma function
Permalink:: http://dlmf.nist.gov/5.19

§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

Keywords:: psi function, rational functions
Permalink:: http://dlmf.nist.gov/5.19.i

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

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

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

Symbols:: $k$ : nonnegative integer and $a$ : real or complex variable
Permalink:: http://dlmf.nist.gov/5.19.E1
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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

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

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

Symbols:: $\EulerConstant$ : Euler’s constant, $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function and $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function
Permalink:: http://dlmf.nist.gov/5.19.E3
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§5.19(ii) Mellin–Barnes Integrals

Keywords:: Mellin–Barnes integrals
Referenced by:: §35.8(v)
Permalink:: http://dlmf.nist.gov/5.19.ii

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 $V$ .
Keywords:: gamma function, $n$ -dimensional sphere
Permalink:: http://dlmf.nist.gov/5.19.iii