5 Gamma Function Properties 5.13 Integrals 5.15 Polygamma Functions

§5.14 Multidimensional Integrals

ⓘ

Keywords:: beta function, gamma function, integral representations, multidimensional
Notes:: See Andrews et al. (1999, pp. 32–34, 401–410, and 426). For (5.14.7) see Mehta (2004, pp. 224–227).
Permalink:: http://dlmf.nist.gov/5.14
See also:: Annotations for Ch.5

Let $V_{n}$ be the simplex: $t_{1}+t_{2}+\dots+t_{n}\leq 1$ , $t_{k}\geq 0$ . Then for $\Re z_{k}>0$ , $k=1,2,\dots,n+1$ ,

5.14.1		$\int_{V_{n}}t_{1}^{z_{1}-1}t_{2}^{z_{2}-1}\cdots t_{n}^{z_{n}-1}\,\mathrm{d}t_% {1}\,\mathrm{d}t_{2}\cdots\,\mathrm{d}t_{n}=\frac{\Gamma\left(z_{1}\right)% \Gamma\left(z_{2}\right)\cdots\Gamma\left(z_{n}\right)}{\Gamma\left(1+z_{1}+z_% {2}+\dots+z_{n}\right)},$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $n$ : nonnegative integer, $z$ : complex variable and $V_{n}$ : simplex Referenced by: §5.19(iii) Permalink: http://dlmf.nist.gov/5.14.E1 Encodings: TeX, pMML, png See also: Annotations for §5.14 and Ch.5

5.14.2		$\int_{V_{n}}\left(1-\sum_{k=1}^{n}t_{k}\right)^{z_{n+1}-1}\prod_{k=1}^{n}t_{k}% ^{z_{k}-1}\,\mathrm{d}t_{k}=\frac{\Gamma\left(z_{1}\right)\Gamma\left(z_{2}% \right)\cdots\Gamma\left(z_{n+1}\right)}{\Gamma\left(z_{1}+z_{2}+\dots+z_{n+1}% \right)}.$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $n$ : nonnegative integer, $k$ : nonnegative integer, $z$ : complex variable and $V_{n}$ : simplex Referenced by: §5.19(iii) Permalink: http://dlmf.nist.gov/5.14.E2 Encodings: TeX, pMML, png See also: Annotations for §5.14 and Ch.5

Selberg-type Integrals

ⓘ

Keywords:: Selberg-type integrals, gamma function
See also:: Annotations for §5.14 and Ch.5

Let

5.14.3		$\Delta(t_{1},t_{2},\dots,t_{n})=\prod_{1\leq j<k\leq n}(t_{j}-t_{k}).$
ⓘ Defines: $\Delta$ : product (locally) Symbols: $j$ : nonnegative integer, $n$ : nonnegative integer and $k$ : nonnegative integer Permalink: http://dlmf.nist.gov/5.14.E3 Encodings: TeX, pMML, png See also: Annotations for §5.14, §5.14 and Ch.5

Then

5.14.4		$\int_{[0,1]^{n}}t_{1}t_{2}\cdots t_{m}\|\Delta(t_{1},\dots,t_{n})\|^{2c}\prod_{k% =1}^{n}t_{k}^{a-1}(1-t_{k})^{b-1}\,\mathrm{d}t_{k}=\frac{1}{(\Gamma\left(1+c% \right))^{n}}\prod_{k=1}^{m}\frac{a+(n-k)c}{a+b+(2n-k-1)c}\*\prod_{k=1}^{n}% \frac{\Gamma\left(a+(n-k)c\right)\Gamma\left(b+(n-k)c\right)\Gamma\left(1+kc% \right)}{\Gamma\left(a+b+(2n-k-1)c\right)},$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $[\NVar{a},\NVar{b}]$ : closed interval, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $m$ : nonnegative integer, $n$ : nonnegative integer, $k$ : nonnegative integer, $a$ : real or complex variable, $b$ : real or complex variable and $\Delta$ : product Permalink: http://dlmf.nist.gov/5.14.E4 Encodings: TeX, pMML, png See also: Annotations for §5.14, §5.14 and Ch.5

provided that $\Re a$ , $\Re b>0$ , $\Re c>-\min(1/n,\Re a/(n-1),\Re b/(n-1))$ .

Secondly,

5.14.5		$\int_{[0,\infty)^{n}}t_{1}t_{2}\cdots t_{m}\|\Delta(t_{1},\dots,t_{n})\|^{2c}% \prod_{k=1}^{n}t_{k}^{a-1}e^{-t_{k}}\,\mathrm{d}t_{k}=\prod_{k=1}^{m}(a+(n-k)c% )\frac{\prod_{k=1}^{n}\Gamma\left(a+(n-k)c\right)\Gamma\left(1+kc\right)}{(% \Gamma\left(1+c\right))^{n}},$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $[\NVar{a},\NVar{b})$ : half-closed interval, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $m$ : nonnegative integer, $n$ : nonnegative integer, $k$ : nonnegative integer, $a$ : real or complex variable and $\Delta$ : product Permalink: http://dlmf.nist.gov/5.14.E5 Encodings: TeX, pMML, png See also: Annotations for §5.14, §5.14 and Ch.5

when $\Re a>0$ , $\Re c>-\min(1/n,\Re a/(n-1))$ .

Thirdly,

5.14.6		$\frac{1}{(2\pi)^{n/2}}\int_{(-\infty,\infty)^{n}}\|\Delta(t_{1},\dots,t_{n})\|^{% 2c}\prod_{k=1}^{n}\exp\left(-\tfrac{1}{2}t_{k}^{2}\right)\,\mathrm{d}t_{k}=% \frac{\prod_{k=1}^{n}\Gamma\left(1+kc\right)}{(\Gamma\left(1+c\right))^{n}},$
$\Re c>-1/n$ .
ⓘ 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 $x$ , $\exp\NVar{z}$ : exponential function, $\int$ : integral, $(\NVar{a},\NVar{b})$ : open interval, $\Re$ : real part, $n$ : nonnegative integer, $k$ : nonnegative integer and $\Delta$ : product Referenced by: §5.20 Permalink: http://dlmf.nist.gov/5.14.E6 Encodings: TeX, pMML, png See also: Annotations for §5.14, §5.14 and Ch.5

Dyson’s Integral

ⓘ

Keywords:: Dyson’s integral, gamma function, integral representations, multidimensional
See also:: Annotations for §5.14 and Ch.5

5.14.7		$\frac{1}{(2\pi)^{n}}\int_{[-\pi,\pi]^{n}}\prod_{1\leq j<k\leq n}\|e^{i\theta_{j% }}-e^{i\theta_{k}}\|^{2b}\,\mathrm{d}\theta_{1}\cdots\,\mathrm{d}\theta_{n}=% \frac{\Gamma\left(1+bn\right)}{(\Gamma\left(1+b\right))^{n}},$
$\Re b>-1/n$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $[\NVar{a},\NVar{b}]$ : closed interval, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\Re$ : real part, $j$ : nonnegative integer, $n$ : nonnegative integer, $k$ : nonnegative integer and $b$ : real or complex variable Referenced by: §5.14, §5.20 Permalink: http://dlmf.nist.gov/5.14.E7 Encodings: TeX, pMML, png See also: Annotations for §5.14, §5.14 and Ch.5

5.13 Integrals 5.15 Polygamma Functions

Site Privacy Accessibility Privacy Program Copyrights Vulnerability Disclosure No Fear Act Policy FOIA Environmental Policy Scientific Integrity Information Quality Standards Commerce.gov Science.gov USA.gov

Note: This service is not intended for secure transactions such as banking, social media, email, or purchasing. Use at your own risk. We assume no liability whatsoever for broken pages.

§5.14 Multidimensional Integrals

Selberg-type Integrals

Dyson’s Integral

Pfad - The Proxy pFad of © 2024 Garber Painting. All rights reserved.