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

Selberg's integral formula

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's integral formula

Aomoto proved a slightly more general integral formula. With the same conditions as Selberg's formula, A proof is found in Chapter 8 of.

Mehta's integral

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.

Macdonald's integral

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