Selberg Integral

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

Selberg's integral formula

When $Re(\alpha) > 0, Re(\beta) > 0, Re(\gamma) > -\min \left(\frac 1n , \frac, \frac\right)$, we have
:$\begin S_ (\alpha, \beta, \gamma) & = \int_0^1 \cdots \int_0^1 \prod_^n t_i^(1-t_i)^ \prod_ , t_i - t_j , ^\,dt_1 \cdots dt_n \\ & = \prod_^ \frac \end$

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

proved a slightly more general integral formula. With the same conditions as Selberg's formula,
:$\int_0^1 \cdots \int_0^1 \left(\prod_^k t_i\right)\prod_^n t_i^(1-t_i)^ \prod_ , t_i - t_j , ^\,dt_1 \cdots dt_n$
:$= S_n(\alpha,\beta,\gamma) \prod_^k\frac.$
A proof is found in Chapter 8 of .

Mehta's integral

When $Re(\gamma) > -1/n$,
:$\frac\int_^ \cdots \int_^ \prod_^n e^ \prod_ , t_i - t_j , ^\,dt_1 \cdots dt_n = \prod_^n\frac.$
It is a corollary of Selberg, by setting $\alpha = \beta$, and change of variables with $t_i = \frac$, then taking $\alpha \to \infty$.

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 ''A''_''n''−1 root system.
:$\frac\int\cdots\int \left, \prod_r\frac\^e^dx_1\cdots dx_n =\prod_^n\frac$
The product is over the roots ''r'' of the roots system and the numbers ''d''_''j'' 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.

References

* (Chapter 8)
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*{{Citation , last1=Selberg , first1=Atle , title=Remarks on a multiple integral , mr=0018287 , year=1944 , journal=Norsk Mat. Tidsskr. , volume=26 , pages=71–78

Special functions