5 Gamma Function Properties 5.12 Beta Function 5.14 Multidimensional Integrals

§5.13 Integrals

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Keywords:: beta function, gamma function, integral representations, integrals
Notes:: For (5.13.1) see Paris and Kaminski (2001, p. 96); (5.13.2) follows by taking $c=0$ , $b=a$ , $s=\mathrm{i}t$ and $z=e^{\mathrm{i}(\pi-2b)}$ . For (5.13.3)–(5.13.4) see Titchmarsh (1986a, pp. 188 and 194), and for (5.13.5) see Andrews et al. (1999, p. 152).
Permalink:: http://dlmf.nist.gov/5.13
See also:: Annotations for Ch.5

In (5.13.1) the integration path is a straight line parallel to the imaginary axis.

5.13.1		${\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\Gamma\left(s+a\right)\Gamma\left% (b-s\right)z^{-s}\,\mathrm{d}s=\frac{\Gamma\left(a+b\right)z^{a}}{(1+z)^{a+b}}},$
$\Re\left(a+b\right)>0$ , $-\Re a<c<\Re b$ , $\|\operatorname{ph}z\|<\pi$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\operatorname{ph}$ : phase, $\Re$ : real part, $s$ : real or complex variable, $z$ : complex variable, $a$ : real or complex variable and $b$ : real or complex variable Referenced by: §5.13, §5.13, §5.19(ii) Permalink: http://dlmf.nist.gov/5.13.E1 Encodings: TeX, pMML, png See also: Annotations for §5.13 and Ch.5

5.13.2		${\frac{1}{2\pi}\int_{-\infty}^{\infty}\|\Gamma\left(a+it\right)\|^{2}e^{(2b-\pi)% t}\,\mathrm{d}t=\frac{\Gamma\left(2a\right)}{(2\sin b)^{2a}}},$
$a>0$ , $0<b<\pi$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\sin\NVar{z}$ : sine function, $a$ : real or complex variable and $b$ : real or complex variable Referenced by: §5.13 Permalink: http://dlmf.nist.gov/5.13.E2 Encodings: TeX, pMML, png See also: Annotations for §5.13 and Ch.5

Barnes’ Beta Integral

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Keywords:: Barnes’ beta integral
See also:: Annotations for §5.13 and Ch.5

5.13.3		$\frac{1}{2\pi}\int_{-\infty}^{\infty}\Gamma\left(a+it\right)\Gamma\left(b+it% \right)\Gamma\left(c-it\right)\Gamma\left(d-it\right)\,\mathrm{d}t=\frac{% \Gamma\left(a+c\right)\Gamma\left(a+d\right)\Gamma\left(b+c\right)\Gamma\left(% b+d\right)}{\Gamma\left(a+b+c+d\right)},$
$\Re a,\Re b,\Re c,\Re d>0$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\Re$ : real part, $a$ : real or complex variable and $b$ : real or complex variable Referenced by: §5.13 Permalink: http://dlmf.nist.gov/5.13.E3 Encodings: TeX, pMML, png See also: Annotations for §5.13, §5.13 and Ch.5

Ramanujan’s Beta Integral

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Keywords:: Ramanujan’s beta integral
See also:: Annotations for §5.13 and Ch.5

5.13.4		$\int_{-\infty}^{\infty}\frac{\,\mathrm{d}t}{\Gamma\left(a+t\right)\Gamma\left(% b+t\right)\Gamma\left(c-t\right)\Gamma\left(d-t\right)}=\frac{\Gamma\left(a+b+% c+d-3\right)}{\Gamma\left(a+c-1\right)\Gamma\left(a+d-1\right)\Gamma\left(b+c-% 1\right)\Gamma\left(b+d-1\right)},$
$\Re\left(a+b+c+d\right)>3$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\Re$ : real part, $a$ : real or complex variable and $b$ : real or complex variable Referenced by: §5.13 Permalink: http://dlmf.nist.gov/5.13.E4 Encodings: TeX, pMML, png See also: Annotations for §5.13, §5.13 and Ch.5

de Branges–Wilson Beta Integral

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Keywords:: beta integrals, de Branges–Wilson beta integral
See also:: Annotations for §5.13 and Ch.5

5.13.5		$\frac{1}{4\pi}\int_{-\infty}^{\infty}\frac{\prod_{k=1}^{4}\Gamma\left(a_{k}+it% \right)\Gamma\left(a_{k}-it\right)}{\Gamma\left(2it\right)\Gamma\left(-2it% \right)}\,\mathrm{d}t=\frac{\prod_{1\leq j<k\leq 4}\Gamma\left(a_{j}+a_{k}% \right)}{\Gamma\left(a_{1}+a_{2}+a_{3}+a_{4}\right)},$
$\Re\left(a_{k}\right)>0$ , $k=1,2,3,4$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\Re$ : real part, $j$ : nonnegative integer, $k$ : nonnegative integer and $a$ : real or complex variable Referenced by: §5.13 Permalink: http://dlmf.nist.gov/5.13.E5 Encodings: TeX, pMML, png See also: Annotations for §5.13, §5.13 and Ch.5

For compendia of integrals of gamma functions see Apelblat (1983, pp. 124–127 and 129–130), Erdélyi et al. (1954a, b), Gradshteyn and Ryzhik (2015, §6.4), Oberhettinger (1974, pp. 191–204), Oberhettinger and Badii (1973, pp. 307–316), Prudnikov et al. (1986b, pp. 57–64), Prudnikov et al. (1992a, pp. 127–130), and Prudnikov et al. (1992b, pp. 113–123).

5.12 Beta Function 5.14 Multidimensional Integrals

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§5.13 Integrals

Barnes’ Beta Integral

Ramanujan’s Beta Integral

de Branges–Wilson Beta Integral

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