# Selberg integral

In mathematics the Selberg integral is a generalization of Euler beta function to n dimensions introduced by Atle Selberg (1944).

## Selberg's integral formula

{\displaystyle {\begin{aligned}S_{n}(\alpha ,\beta ,\gamma )&=\int _{0}^{1}\cdots \int _{0}^{1}\prod _{i=1}^{n}t_{i}^{\alpha -1}(1-t_{i})^{\beta -1}\prod _{1\leq i

Selberg's formula implies Dixon's identity for well poised hypergeometric series, and some special cases of Dyson's conjecture.

## Aomoto's integral formula

Aomoto (1987) proved a slightly more general integral formula:

${\displaystyle \int _{0}^{1}\cdots \int _{0}^{1}\left(\prod _{i=1}^{k}t_{i}\right)\prod _{i=1}^{n}t_{i}^{\alpha -1}(1-t_{i})^{\beta -1}\prod _{1\leq i
${\displaystyle =S_{n}(\alpha ,\beta ,\gamma )\prod _{j=1}^{k}{\frac {\alpha +(n-j)\gamma }{\alpha +\beta +(2n-j-1)\gamma }}.}$

## Mehta's integral

Mehta's integral is

${\displaystyle {\frac {1}{(2\pi )^{n/2}}}\int _{-\infty }^{\infty }\cdots \int _{-\infty }^{\infty }\prod _{i=1}^{n}e^{-t_{i}^{2}/2}\prod _{1\leq i

It is the partition function for a gas of point charges moving on a line that are attracted to the origin (Mehta 2004). Its value can be deduced from that of the Selberg integral, and is

${\displaystyle \prod _{j=1}^{n}{\frac {\Gamma (1+j\gamma )}{\Gamma (1+\gamma )}}.}$

This was conjectured by Mehta & Dyson (1963), who were unaware of Selberg's earlier work.

## Macdonald's integral

Macdonald (1982) conjectured the following extension of Mehta's integral to all finite root systems, Mehta's original case corresponding to the An−1 root system.

${\displaystyle {\frac {1}{(2\pi )^{n/2}}}\int \cdots \int \left|\prod _{r}{\frac {2(x,r)}{(r,r)}}\right|^{\gamma }e^{-(x_{1}^{2}+\cdots +x_{n}^{2})/2}dx_{1}\cdots dx_{n}=\prod _{j=1}^{n}{\frac {\Gamma (1+d_{j}\gamma )}{\Gamma (1+\gamma )}}}$

The product is over the roots r of the roots system and the numbers dj are the degrees of the generators of the ring of invariants of the reflection group. Opdam (1989) gave a uniform proof for all crystallographic reflection groups. Several years later he proved it in full generality (Opdam (1993)), making use of computer-aided calculations by Garvan.