# 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

\begin{align} 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 \le i < j \le n} |t_i - t_j |^{2 \gamma}\,dt_1 \cdots dt_n \\ & = \prod_{j = 0}^{n-1} \frac {\Gamma(\alpha + j \gamma) \Gamma(\beta + j \gamma) \Gamma (1 + (j+1)\gamma)} {\Gamma(\alpha + \beta + (n+j-1)\gamma) \Gamma(1+\gamma)} \end{align}

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:

$\int_0^1 \cdots \int_0^1 \left(\prod_{i=1}^k t_i\right)\prod_{i=1}^n t_i^{\alpha-1}(1-t_i)^{\beta-1} \prod_{1 \le i < j \le n} |t_i - t_j |^{2 \gamma}\,dt_1 \cdots dt_n$
$= S_n(\alpha,\beta,\gamma) \prod_{j=1}^k\frac{\alpha+(n-j)\gamma}{\alpha+\beta+(2n-j-1)\gamma}.$

## Mehta's integral

Mehta's integral is

$\frac{1}{(2\pi)^{n/2}}\int_{-\infty}^{\infty} \cdots \int_{-\infty}^{\infty} \prod_{i=1}^n e^{-t_i^2/2} \prod_{1 \le i < j \le n} |t_i - t_j |^{2 \gamma}\,dt_1 \cdots dt_n.$

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

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

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.

$\frac{1}{(2\pi)^{n/2}}\int\cdots\int \left|\prod_r\frac{2(x,r)}{(r,r)}\right|^{\gamma}e^{-(x_1^2+\cdots+x_n^2)/2}dx_1\cdots dx_n =\prod_{j=1}^n\frac{\Gamma(1+d_j\gamma)}{\Gamma(1+\gamma)}$

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.