§ 5.14. Multidimensional Integrals

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

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

5.14.1 $\int _{{V_{n}}}t_{1}^{{z_{1}-1}}t_{2}^{{z_{2}-1}}\cdots t_{n}^{{z_{n}-1}}dt_{1}dt_{2}\cdots dt_{n}=\frac{\Gamma\!\left(z_{1}\right)\Gamma\!\left(z_{2}\right)\cdots\Gamma\!\left(z_{n}\right)}{\Gamma\!\left(1+z_{1}+z_{2}+\dots+z_{n}\right)},$

Defines:: $V_{n}$ : simplex
Symbols:: $\Gamma\!\left(z\right)$ : Gamma function, $n$ : nonnegative integer and $z$ : complex variable
Referenced by:: §5.19(iii)
Permalink:: http://dlmf.nist.gov/5.14.E1
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5.14.2 $\int _{{V_{n}}}\left(1-\sum _{{k=1}}^{n}t_{k}\right)^{{z_{{n+1}}-1}}\prod _{{k=1}}^{n}t_{k}^{{z_{k}-1}}dt_{k}=\frac{\Gamma\!\left(z_{1}\right)\Gamma\!\left(z_{2}\right)\cdots\Gamma\!\left(z_{{n+1}}\right)}{\Gamma\!\left(z_{1}+z_{2}+\dots+z_{{n+1}}\right)}.$

Defines:: $V_{n}$ : simplex
Symbols:: $\Gamma\!\left(z\right)$ : Gamma function, $n$ : nonnegative integer, $k$ : nonnegative integer and $z$ : complex variable
Referenced by:: §5.19(iii)
Permalink:: http://dlmf.nist.gov/5.14.E2
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¶ Selberg-type Integrals

Let

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

Defines:: $\Delta$ : product
Symbols:: $j$ : nonnegative integer, $n$ : nonnegative integer and $k$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/5.14.E3
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Then

5.14.4 $\int _{{[0,1]^{n}}}t_{1}t_{2}\cdots t_{m}|\Delta(t_{1},\dots,t_{n})|^{{2c}}\prod _{{k=1}}^{n}t_{k}^{{a-1}}(1-t_{k})^{{b-1}}dt_{k}=\frac{1}{(\Gamma\!\left(1+c\right))^{n}}\prod _{{k=1}}^{m}\frac{a+(n-k)c}{a+b+(2n-k-1)c}\*\prod _{{k=1}}^{n}\frac{\Gamma\!\left(a+(n-k)c\right)\Gamma\!\left(b+(n-k)c\right)\Gamma\!\left(1+kc\right)}{\Gamma\!\left(a+b+(2n-k-1)c\right)},$

Symbols:: $\Gamma\!\left(z\right)$ : Gamma function, $m$ : 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
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provided that $\realpart{a}$ , $\realpart{b}>0$ , $\realpart{c}>-\min(1/n,\realpart{a/(n-1)},\realpart{b/(n-1)})$ .

Secondly,

5.14.5 $\int _{{[0,\infty)^{n}}}t_{1}t_{2}\cdots t_{m}|\Delta(t_{1},\dots,t_{n})|^{{2c}}\prod _{{k=1}}^{n}t_{k}^{{a-1}}e^{{-t_{k}}}dt_{k}=\prod _{{k=1}}^{m}(a+(n-k)c)\frac{\prod _{{k=1}}^{n}\Gamma\!\left(a+(n-k)c\right)\Gamma\!\left(1+kc\right)}{(\Gamma\!\left(1+c\right))^{n}},$

Symbols:: $\Gamma\!\left(z\right)$ : Gamma function, $m$ : nonnegative integer, $n$ : nonnegative integer, $k$ : nonnegative integer, $a$ : real or complex variable and $\Delta$ : product
Permalink:: http://dlmf.nist.gov/5.14.E5
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when $\realpart{a}>0$ , $\realpart{c}>-\min(1/n,\realpart{a/(n-1)})$ .

Thirdly,

5.14.6 $\frac{1}{(2\pi)^{{n/2}}}\int _{{(-\infty,\infty)^{n}}}|\Delta(t_{1},\dots,t_{n})|^{{2c}}\prod _{{k=1}}^{n}\exp\!\left(-\tfrac{1}{2}t_{k}^{2}\right)dt_{k}=\frac{\prod _{{k=1}}^{n}\Gamma\!\left(1+kc\right)}{(\Gamma\!\left(1+c\right))^{n}},$ $\realpart{c}>-1/n$ .

Symbols:: $\Gamma\!\left(z\right)$ : Gamma function, $n$ : nonnegative integer, $k$ : nonnegative integer and $\Delta$ : product
Referenced by:: §5.20
Permalink:: http://dlmf.nist.gov/5.14.E6
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¶ Dyson's Integral

5.14.7 $\frac{1}{(2\pi)^{n}}\int _{{[-\pi,\pi]^{n}}}\prod _{{1\le j<k\le n}}|e^{{i\theta _{j}}}-e^{{i\theta _{k}}}|^{{2b}}d\theta _{1}\cdots d\theta _{n}=\frac{\Gamma\!\left(1+bn\right)}{(\Gamma\!\left(1+b\right))^{n}},$ $\realpart{b}>-1/n$ .

Symbols:: $\Gamma\!\left(z\right)$ : Gamma function, $j$ : 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
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