5 Gamma Function Properties 5.13 Integrals 5.15 Polygamma Functions

§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 $\int_{V_{n}}t_{1}^{z_{1}-1}t_{2}^{z_{2}-1}\cdots t_{n}^{z_{n}-1}\,\mathrm{d}t_% {1}\,\mathrm{d}t_{2}\cdots\,\mathrm{d}t_{n}=\frac{\Gamma\left(z_{1}\right)% \Gamma\left(z_{2}\right)\cdots\Gamma\left(z_{n}\right)}{\Gamma\left(1+z_{1}+z_% {2}+\dots+z_{n}\right)},$

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, : nonnegative integer, : 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 $\int_{V_{n}}\left(1-\sum_{k=1}^{n}t_{k}\right)^{z_{n+1}-1}\prod_{k=1}^{n}t_{k}% ^{z_{k}-1}\,\mathrm{d}t_{k}=\frac{\Gamma\left(z_{1}\right)\Gamma\left(z_{2}% \right)\cdots\Gamma\left(z_{n+1}\right)}{\Gamma\left(z_{1}+z_{2}+\dots+z_{n+1}% \right)}.$

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, : nonnegative integer, : nonnegative integer, : 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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Keywords:: Selberg-type integrals, gamma function
See also:: Annotations for §5.14 and Ch.5

Let

5.14.3 $\Delta(t_{1},t_{2},\dots,t_{n})=\prod_{1\leq j<k\leq n}(t_{j}-t_{k}).$

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Defines:: $\Delta$ : product (locally)
Symbols:: : nonnegative integer, : nonnegative integer and : 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 $\int_{[0,1]^{n}}t_{1}t_{2}\cdots t_{m}|\Delta(t_{1},\dots,t_{n})|^{2c}\prod_{k% =1}^{n}t_{k}^{a-1}(1-t_{k})^{b-1}\,\mathrm{d}t_{k}=\frac{1}{(\Gamma\left(1+c% \right))^{n}}\prod_{k=1}^{m}\frac{a+(n-k)c}{a+b+(2n-k-1)c}\*\prod_{k=1}^{n}% \frac{\Gamma\left(a+(n-k)c\right)\Gamma\left(b+(n-k)c\right)\Gamma\left(1+kc% \right)}{\Gamma\left(a+b+(2n-k-1)c\right)},$

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $[\NVar{a},\NVar{b}]$ : closed interval, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, : nonnegative integer, : nonnegative integer, : nonnegative integer, : real or complex variable, : 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 $\int_{[0,\infty)^{n}}t_{1}t_{2}\cdots t_{m}|\Delta(t_{1},\dots,t_{n})|^{2c}% \prod_{k=1}^{n}t_{k}^{a-1}e^{-t_{k}}\,\mathrm{d}t_{k}=\prod_{k=1}^{m}(a+(n-k)c% )\frac{\prod_{k=1}^{n}\Gamma\left(a+(n-k)c\right)\Gamma\left(1+kc\right)}{(% \Gamma\left(1+c\right))^{n}},$

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $[\NVar{a},\NVar{b})$ : half-closed interval, $\,\mathrm{d}\NVar{x}$ : differential of , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, : nonnegative integer, : nonnegative integer, : nonnegative integer, : 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 $\frac{1}{(2\pi)^{n/2}}\int_{(-\infty,\infty)^{n}}|\Delta(t_{1},\dots,t_{n})|^{% 2c}\prod_{k=1}^{n}\exp\left(-\tfrac{1}{2}t_{k}^{2}\right)\,\mathrm{d}t_{k}=% \frac{\prod_{k=1}^{n}\Gamma\left(1+kc\right)}{(\Gamma\left(1+c\right))^{n}},$ $\Re c>-1/n$ .

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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 , $\exp\NVar{z}$ : exponential function, $\int$ : integral, $(\NVar{a},\NVar{b})$ : open interval, $\Re$ : real part, : nonnegative integer, : 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

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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 $\frac{1}{(2\pi)^{n}}\int_{[-\pi,\pi]^{n}}\prod_{1\leq j<k\leq n}|e^{i\theta_{j% }}-e^{i\theta_{k}}|^{2b}\,\mathrm{d}\theta_{1}\cdots\,\mathrm{d}\theta_{n}=% \frac{\Gamma\left(1+bn\right)}{(\Gamma\left(1+b\right))^{n}},$ $\Re b>-1/n$ .

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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 , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\Re$ : real part, : nonnegative integer, : nonnegative integer, : nonnegative integer and : 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

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