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.
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.
In particular, when , the term on the right is .
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.
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