# §5.14 Multidimensional Integrals

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 $\int_{V_{n}}t_{1}^{z_{1}-1}t_{2}^{z_{2}-1}\cdots t_{n}^{z_{n}-1}\,\mathrm{d}t_% {1}\,\mathrm{d}t_{2}\cdots\,\mathrm{d}t_{n}=\frac{\Gamma\left(z_{1}\right)% \Gamma\left(z_{2}\right)\cdots\Gamma\left(z_{n}\right)}{\Gamma\left(1+z_{1}+z_% {2}+\dots+z_{n}\right)},$
 5.14.2 $\int_{V_{n}}\left(1-\sum_{k=1}^{n}t_{k}\right)^{z_{n+1}-1}\prod_{k=1}^{n}t_{k}% ^{z_{k}-1}\,\mathrm{d}t_{k}=\frac{\Gamma\left(z_{1}\right)\Gamma\left(z_{2}% \right)\cdots\Gamma\left(z_{n+1}\right)}{\Gamma\left(z_{1}+z_{2}+\dots+z_{n+1}% \right)}.$

## Selberg-type Integrals

Let

 5.14.3 $\Delta(t_{1},t_{2},\dots,t_{n})=\prod_{1\leq j ⓘ 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 $\int_{[0,1]^{n}}t_{1}t_{2}\cdots t_{m}|\Delta(t_{1},\dots,t_{n})|^{2c}\prod_{k% =1}^{n}t_{k}^{a-1}(1-t_{k})^{b-1}\,\mathrm{d}t_{k}=\frac{1}{(\Gamma\left(1+c% \right))^{n}}\prod_{k=1}^{m}\frac{a+(n-k)c}{a+b+(2n-k-1)c}\*\prod_{k=1}^{n}% \frac{\Gamma\left(a+(n-k)c\right)\Gamma\left(b+(n-k)c\right)\Gamma\left(1+kc% \right)}{\Gamma\left(a+b+(2n-k-1)c\right)},$

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

Secondly,

 5.14.5 $\int_{[0,\infty)^{n}}t_{1}t_{2}\cdots t_{m}|\Delta(t_{1},\dots,t_{n})|^{2c}% \prod_{k=1}^{n}t_{k}^{a-1}e^{-t_{k}}\,\mathrm{d}t_{k}=\prod_{k=1}^{m}(a+(n-k)c% )\frac{\prod_{k=1}^{n}\Gamma\left(a+(n-k)c\right)\Gamma\left(1+kc\right)}{(% \Gamma\left(1+c\right))^{n}},$

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

Thirdly,

 5.14.6 $\frac{1}{(2\pi)^{n/2}}\int_{(-\infty,\infty)^{n}}|\Delta(t_{1},\dots,t_{n})|^{% 2c}\prod_{k=1}^{n}\exp\left(-\tfrac{1}{2}t_{k}^{2}\right)\,\mathrm{d}t_{k}=% \frac{\prod_{k=1}^{n}\Gamma\left(1+kc\right)}{(\Gamma\left(1+c\right))^{n}},$ $\Re c>-1/n$.

## Dyson’s Integral

 5.14.7 $\frac{1}{(2\pi)^{n}}\int_{[-\pi,\pi]^{n}}\prod_{1\leq j $\Re b>-1/n$.