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, CalogeroâÂÂMoserâÂÂSutherland model, and KnizhnikâÂÂZamolodchikov equations.
When , 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 .
When ,
It is a corollary of Selberg, by setting , and change of variables with , then taking .
This was conjectured by , who were unaware of Selberg's earlier work.
It is the partition function for a gas of point charges moving on a line that are attracted to the origin.
In particular, when , the term on the right is .
conjectured the following extension of Mehta's integral to all finite root systems, Mehta's original case corresponding to the A<sub>n−1</sub> root system.
The product is over the roots r of the roots system and the numbers d<sub>j</sub> are the degrees of the generators of the ring of invariants of the reflection group. gave a uniform proof for all crystallographic reflection groups. Several years later he proved it in full generality, making use of computer-aided calculations by Garvan.