5 Gamma Function Properties 5.13 Integrals 5.15 Polygamma Functions

§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).
Keywords:: beta function, gamma function
Permalink:: http://dlmf.nist.gov/5.14

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

5.14.1 $\int_{{V_{n}}}t_{1}^{{z_{1}-1}}t_{2}^{{z_{2}-1}}\cdots t_{n}^{{z_{n}-1}}dt_{1}% dt_{2}\cdots dt_{n}=\frac{\mathop{\Gamma\/}\nolimits\!\left(z_{1}\right)% \mathop{\Gamma\/}\nolimits\!\left(z_{2}\right)\cdots\mathop{\Gamma\/}\nolimits% \!\left(z_{n}\right)}{\mathop{\Gamma\/}\nolimits\!\left(1+z_{1}+z_{2}+\dots+z_% {n}\right)},$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, : 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
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5.14.2 $\int_{{V_{n}}}\left(1-\sum_{{k=1}}^{n}t_{k}\right)^{{z_{{n+1}}-1}}\prod_{{k=1}% }^{n}t_{k}^{{z_{k}-1}}dt_{k}=\frac{\mathop{\Gamma\/}\nolimits\!\left(z_{1}% \right)\mathop{\Gamma\/}\nolimits\!\left(z_{2}\right)\cdots\mathop{\Gamma\/}% \nolimits\!\left(z_{{n+1}}\right)}{\mathop{\Gamma\/}\nolimits\!\left(z_{1}+z_{% 2}+\dots+z_{{n+1}}\right)}.$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, : 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
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¶ Selberg-type Integrals

Keywords:: Selberg-type integrals, gamma function

Let

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

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

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, : closed interval, : 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
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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 $\int_{{[0,\infty)^{n}}}t_{1}t_{2}\cdots t_{m}|\Delta(t_{1},\dots,t_{n})|^{{2c}% }\prod_{{k=1}}^{n}t_{k}^{{a-1}}e^{{-t_{k}}}dt_{k}=\prod_{{k=1}}^{m}(a+(n-k)c)% \frac{\prod_{{k=1}}^{n}\mathop{\Gamma\/}\nolimits\!\left(a+(n-k)c\right)% \mathop{\Gamma\/}\nolimits\!\left(1+kc\right)}{(\mathop{\Gamma\/}\nolimits\!% \left(1+c\right))^{n}},$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, : half-closed interval, : differential of , : base of exponential function, $\int$ : integral, : nonnegative integer, : nonnegative integer, : nonnegative integer, : 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 $\frac{1}{(2\pi)^{{n/2}}}\int_{{(-\infty,\infty)^{n}}}|\Delta(t_{1},\dots,t_{n}% )|^{{2c}}\prod_{{k=1}}^{n}\mathop{\exp\/}\nolimits\!\left(-\tfrac{1}{2}t_{k}^{% 2}\right)dt_{k}=\frac{\prod_{{k=1}}^{n}\mathop{\Gamma\/}\nolimits\!\left(1+kc% \right)}{(\mathop{\Gamma\/}\nolimits\!\left(1+c\right))^{n}},$ $\realpart{c}>-1/n$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, : differential of , $\mathop{\exp\/}\nolimits z$ : exponential function, $\int$ : integral, : open interval, $\realpart{}$ : real part, : nonnegative integer, : nonnegative integer and $\Delta$ : product
Referenced by:: ¶ ‣ §5.20
Permalink:: http://dlmf.nist.gov/5.14.E6
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¶ Dyson’s Integral

Keywords:: Dyson’s integral, gamma function

5.14.7 $\frac{1}{(2\pi)^{n}}\int_{{[-\pi,\pi]^{n}}}\prod_{{1\leq j<k\leq n}}|e^{{i% \theta_{j}}}-e^{{i\theta_{k}}}|^{{2b}}d\theta_{1}\cdots d\theta_{n}=\frac{% \mathop{\Gamma\/}\nolimits\!\left(1+bn\right)}{(\mathop{\Gamma\/}\nolimits\!% \left(1+b\right))^{n}},$ $\realpart{b}>-1/n$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, : closed interval, : differential of , : base of exponential function, $\int$ : integral, $\realpart{}$ : 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

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