In mathematics, the Selberg integral is a generalization of Euler beta function to n dimensions introduced by Atle Selberg. It has applications in statistical mechanics, multivariable orthogonal polynomials, random matrix theory, Calogero–Moser–Sutherland model, and Knizhnik–Zamolodchikov equations.

When ${\displaystyle Re(\alpha )>0,Re(\beta )>0,Re(\gamma )>-\min \left({\frac {1}{n}},{\frac {Re(\alpha )}{n-1}},{\frac {Re(\beta )}{n-1}}\right)}$, we have

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

Aomoto proved a slightly more general integral formula. With the same conditions as Selberg's formula,

A proof is found in Chapter 8 of Andrews, Askey & Roy (1999).

When ${\displaystyle Re(\gamma )>-1/n}$,

It is a corollary of Selberg, by setting ${\displaystyle \alpha =\beta }$, and change of variables with ${\displaystyle t_{i}={\frac {1+t'_{i}/{\sqrt {2\alpha }}}{2}}}$, then taking ${\displaystyle \alpha \to \infty }$.

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