§5.14 Multidimensional Integrals

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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

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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

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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