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-Beta Functions 5.20 Physical Applications

§5.19 Mathematical Applications

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Keywords:: applications, gamma function, mathematical
Permalink:: http://dlmf.nist.gov/5.19
See also:: Annotations for Ch.5

§5.19(i) Summation of Rational Functions

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Keywords:: applications, mathematical, psi function, rational functions, summation
Permalink:: http://dlmf.nist.gov/5.19.i
See also:: Annotations for §5.19 and Ch.5

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

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See also:: Annotations for §5.19(i), §5.19 and Ch.5

5.19.1	$\displaystyle S$	$\displaystyle=\sum_{k=0}^{\infty}a_{k},$
5.19.1	$\displaystyle a_{k}$	$\displaystyle=\frac{k}{(3k+2)(2k+1)(k+1)}.$
ⓘ Symbols: $k$ : nonnegative integer and $a$ : real or complex variable Permalink: http://dlmf.nist.gov/5.19.E1 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §5.19(i), §5.19(i), §5.19 and Ch.5

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 See also: Annotations for §5.19(i), §5.19(i), §5.19 and Ch.5

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

5.19.3		$S=\psi\left(\tfrac{1}{2}\right)-2\psi\left(\tfrac{2}{3}\right)-\gamma=3\ln 3-2% \ln 2-\tfrac{1}{3}\pi\sqrt{3}.$
ⓘ Symbols: $\gamma$ : Euler’s constant, $\pi$ : the ratio of the circumference of a circle to its diameter, $\psi\left(\NVar{z}\right)$ : psi (or digamma) function and $\ln\NVar{z}$ : principal branch of logarithm function Permalink: http://dlmf.nist.gov/5.19.E3 Encodings: TeX, pMML, png See also: Annotations for §5.19(i), §5.19(i), §5.19 and Ch.5

§5.19(ii) Mellin–Barnes Integrals

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Keywords:: Mellin–Barnes integrals
Referenced by:: §35.8(v)
Permalink:: http://dlmf.nist.gov/5.19.ii
See also:: Annotations for §5.19 and Ch.5

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

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Keywords:: gamma function, $n$ -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 $V$ .
Permalink:: http://dlmf.nist.gov/5.19.iii
See also:: Annotations for §5.19 and Ch.5

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

5.19.4	$\displaystyle V$	$\displaystyle=\frac{\pi^{\frac{1}{2}n}r^{n}}{\Gamma\left(\frac{1}{2}n+1\right)},$
5.19.4	$\displaystyle S$	$\displaystyle=\frac{2\pi^{\frac{1}{2}n}r^{n-1}}{\Gamma\left(\frac{1}{2}n\right% )}=\frac{n}{r}V.$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $n$ : nonnegative integer, $V$ : volume, $S$ : surface and $r$ : radius Permalink: http://dlmf.nist.gov/5.19.E4 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §5.19(iii), §5.19 and Ch.5