DLMF: GA.14 Multidimensional Integrals

§GA.14. Multidimensional Integrals

Sources:

Andrews et.al.(1999)(§§1.8, 8.1–8.3, and 8.7), Mehta(2004)(pp. 224–227).

Notes:

Andrews et.al.(1999)(pp. 32–34, 401–410, and 426)

GA.14.7

Mehta(2004)(pp. 224–227)

Keywords:

gamma function, multidimensional integral

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

GA.14.1 $\int _{{V_{n}}}t_{1}^{{z_{1}-1}}t_{2}^{{z_{2}-1}}\cdots t_{n}^{{z_{n}-1}}dt_{1}dt_{2}\cdots dt_{n}=\frac{\Gamma\!\left(z_{1}\right)\Gamma\!\left(z_{2}\right)\cdots\Gamma\!\left(z_{n}\right)}{\Gamma\!\left(1+z_{1}+z_{2}+\dots+z_{n}\right)},$

Notations:: $\Gamma\!\left(z\right)$ : Gamma function, $n$ : nonnegative integer, $z$ : complex variable and $V_{n}$ : simplex
Encodings:: pMathML, png, TeX

GA.14.2 $\int _{{V_{n}}}\left(1-\sum _{{k=1}}^{n}t_{k}\right)^{{z_{{n+1}}-1}}\prod _{{k=1}}^{n}t_{k}^{{z_{k}-1}}dt_{k}=\frac{\Gamma\!\left(z_{1}\right)\Gamma\!\left(z_{2}\right)\cdots\Gamma\!\left(z_{{n+1}}\right)}{\Gamma\!\left(z_{1}+z_{2}+\dots+z_{{n+1}}\right)}.$

Notations:: $\Gamma\!\left(z\right)$ : Gamma function, $n$ : nonnegative integer, $k$ : nonnegative integer, $z$ : complex variable and $V_{n}$ : simplex
Encodings:: pMathML, png, TeX

Selberg-type Integrals

Keywords:: multidimensional integral, Selberg-type integral

Let

GA.14.3 $\Delta(t_{1},t_{2},\dots,t_{n})=\prod _{{1\le j<k\le n}}(t_{j}-t_{k}).$

Notations:: $j$ : nonnegative integer, $n$ : nonnegative integer, $k$ : nonnegative integer and $\Delta$ : product
Encodings:: pMathML, pMathML, png, png, TeX, TeX

Then

GA.14.4 $\int _{{[0,1]^{n}}}t_{1}t_{2}\cdots t_{m}|\Delta(t_{1},\dots,t_{n})|^{{2c}}\prod _{{k=1}}^{n}t_{k}^{{a-1}}(1-t_{k})^{{b-1}}dt_{k}=\frac{1}{(\Gamma\!\left(1+c\right))^{n}}\prod _{{k=1}}^{m}\frac{a+(n-k)c}{a+b+(2n-k-1)c}\*\prod _{{k=1}}^{n}\frac{\Gamma\!\left(a+(n-k)c\right)\Gamma\!\left(b+(n-k)c\right)\Gamma\!\left(1+kc\right)}{\Gamma\!\left(a+b+(2n-k-1)c\right)},$

Notations:: $\Gamma\!\left(z\right)$ : Gamma function, $m$ : nonnegative integer, $n$ : nonnegative integer, $k$ : nonnegative integer, $a$ : real or complex variable, $b$ : real or complex variable and $\Delta$ : product
Encodings:: pMathML, png, TeX

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

Secondly,

GA.14.5 $\int _{{[0,\infty)^{n}}}t_{1}t_{2}\cdots t_{m}|\Delta(t_{1},\dots,t_{n})|^{{2c}}\prod _{{k=1}}^{n}t_{k}^{{a-1}}e^{{-t_{k}}}dt_{k}=\prod _{{k=1}}^{m}(a+(n-k)c)\frac{\prod _{{k=1}}^{n}\Gamma\!\left(a+(n-k)c\right)\Gamma\!\left(1+kc\right)}{(\Gamma\!\left(1+c\right))^{n}},$

Notations:: $\Gamma\!\left(z\right)$ : Gamma function, $m$ : nonnegative integer, $n$ : nonnegative integer, $k$ : nonnegative integer, $a$ : real or complex variable and $\Delta$ : product
Encodings:: pMathML, png, TeX

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

Thirdly,

GA.14.6 $\frac{1}{(2\pi)^{{n/2}}}\int _{{(-\infty,\infty)^{n}}}|\Delta(t_{1},\dots,t_{n})|^{{2c}}\prod _{{k=1}}^{n}\mathrm{exp}\!\left(-\tfrac{1}{2}t_{k}^{2}\right)dt_{k}=\frac{\prod _{{k=1}}^{n}\Gamma\!\left(1+kc\right)}{(\Gamma\!\left(1+c\right))^{n}},$ $\realpart{c}>-1/n$ .

Notations:: $\Gamma\!\left(z\right)$ : Gamma function, $n$ : nonnegative integer, $k$ : nonnegative integer and $\Delta$ : product
Referenced by:: §GA.20
Encodings:: pMathML, png, TeX

Dyson's Integral

Keywords:: Dyson's integral, multidimensional integral

GA.14.7 $\frac{1}{(2\pi)^{n}}\int _{{[-\pi,\pi]^{n}}}\prod _{{1\le j<k\le n}}|e^{{i\theta _{j}}}-e^{{i\theta _{k}}}|^{{2b}}d\theta _{1}\cdots d\theta _{n}=\frac{\Gamma\!\left(1+bn\right)}{(\Gamma\!\left(1+b\right))^{n}},$ $\realpart{b}>-1/n$ .

Notations:: $\Gamma\!\left(z\right)$ : Gamma function, $j$ : nonnegative integer, $n$ : nonnegative integer, $k$ : nonnegative integer and $b$ : real or complex variable
Referenced by:: §GA.14, §GA.20
Encodings:: pMathML, png, TeX