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In mathematics the '''Selberg integral''' is a generalization of [[Beta function|Euler beta function]] to ''n'' dimensions introduced by  {{harvs|txt|authorlink=Atle Selberg|first=Atle|last= Selberg|year=1944}}.

==Selberg's integral formula==
:<math> \begin{align}
S_{n} (\alpha, \beta, \gamma) & =
\int_0^1 \cdots \int_0^1 \prod_{i=1}^n t_i^{\alpha-1}(1-t_i)^{\beta-1}
\prod_{1 \le i < j \le n} |t_i - t_j |^{2 \gamma}\,dt_1 \cdots dt_n \\

 & = \prod_{j = 0}^{n-1} 
\frac {\Gamma(\alpha + j \gamma) \Gamma(\beta + j \gamma) \Gamma (1 + (j+1)\gamma)} 
{\Gamma(\alpha + \beta + (n+j-1)\gamma) \Gamma(1+\gamma)}
\end{align}
</math>

Selberg's formula  implies [[Dixon's identity]] for well poised hypergeometric series, and some special cases of [[Dyson's conjecture]].

==Aomoto's integral formula==
{{harvtxt|Aomoto|1987}} proved a slightly more general integral formula:
:<math>
\int_0^1 \cdots \int_0^1 \left(\prod_{i=1}^k t_i\right)\prod_{i=1}^n t_i^{\alpha-1}(1-t_i)^{\beta-1}
\prod_{1 \le i < j \le n} |t_i - t_j |^{2 \gamma}\,dt_1 \cdots dt_n 
</math>
:<math>=
S_n(\alpha,\beta,\gamma) \prod_{j=1}^k\frac{\alpha+(n-j)\gamma}{\alpha+\beta+(2n-j-1)\gamma}.
</math>

==Mehta's integral==
Mehta's integral is 
:<math>
\frac{1}{(2\pi)^{n/2}}\int_{-\infty}^{\infty} \cdots \int_{-\infty}^{\infty} \prod_{i=1}^n e^{-t_i^2/2}
\prod_{1 \le i < j \le n} |t_i - t_j |^{2 \gamma}\,dt_1 \cdots dt_n.
</math>
It is the partition function for a gas of point charges moving on a line that are attracted to the origin {{harv|Mehta|2004}}. 
Its value can be deduced from that of the Selberg integral, and is 
:<math>\prod_{j=1}^n\frac{\Gamma(1+j\gamma)}{\Gamma(1+\gamma)}.</math>
This was conjectured by {{harvtxt|Mehta|Dyson|1963}}, who were unaware of Selberg's earlier work.

==Macdonald's integral==
{{harvtxt|Macdonald|1982}} conjectured the following extension of Mehta's integral to all finite root systems, Mehta's original case corresponding to the ''A''<sub>''n''&minus;1</sub> root system. 
:<math>\frac{1}{(2\pi)^{n/2}}\int\cdots\int \left|\prod_r\frac{2(x,r)}{(r,r)}\right|^{\gamma}e^{-(x_1^2+\cdots+x_n^2)/2}dx_1\cdots dx_n 
=\prod_{j=1}^n\frac{\Gamma(1+d_j\gamma)}{\Gamma(1+\gamma)}</math>
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. 
{{harvtxt|Opdam|1989}} gave a uniform proof for all crystallographic reflection groups. Several years later he proved it in full generality ({{harvtxt|Opdam|1993}}), making use of computer-aided calculations by Garvan.

==References==
*{{Citation | author1-link = George Andrews (mathematician) | author2-link = Richard Askey | last1=Andrews | first1=George E. | last2=Askey | first2=Richard | last3=Roy | first3=Ranjan | title=Special functions | publisher=[[Cambridge University Press]] | series=Encyclopedia of Mathematics and its Applications | isbn=978-0-521-62321-6 | mr=1688958 | year=1999 | volume=71}} (Chapter 8)
*{{citation
   |last1=Aomoto
   |first1=K
   |title=On the complex Selberg integral
   |journal=The Quarterly Journal of Mathematics
   |year=1987
   |volume=38 | issue = 4
   |pages=385–399
   |doi=10.1093/qmath/38.4.385}}  
*{{citation|url=http://www.ams.org/bull/2008-45-04/S0273-0979-08-01221-4/home.html 	   	
|title=The importance of the Selberg integral
|first= Peter J.|last= Forrester|first2= S. Ole |last2=Warnaar
|journal= Bull. Amer. Math. Soc. |volume=45 |year=2008|pages= 489–534|doi=10.1090/S0273-0979-08-01221-4|issue=4  }}
*{{Citation | last1=Macdonald | first1=I. G. | title=Some conjectures for root systems | doi=10.1137/0513070 | mr=674768 | year=1982 | journal=SIAM Journal on Mathematical Analysis | issn=0036-1410 | volume=13 | issue=6 | pages=988–1007}}
*{{Citation | last1=Mehta | first1=Madan Lal | title=Random matrices | publisher=Elsevier/Academic Press, Amsterdam | edition=3rd | series=Pure and Applied Mathematics (Amsterdam) | isbn=978-0-12-088409-4 | mr=2129906 | year=2004 | volume=142}}
*{{Citation | last1=Mehta | first1=Madan Lal | last2=Dyson | first2=Freeman J. | title=Statistical theory of the energy levels of complex systems. V | doi=10.1063/1.1704009 | mr=0151232 | year=1963 | journal=[[Journal of Mathematical Physics]] | issn=0022-2488 | volume=4 | pages=713–719 | issue=5}}
*{{citation|last=Opdam|first=E.M.|year=1989|title=Some applications of hypergeometric shift operators|issue=1|journal=Invent. Math.|volume=98|doi=10.1007/BF01388841|mr=1010152|pages=275&ndash;282}}
*{{citation|last=Opdam|first=E.M.|year=1993|title=Dunkl operators, Bessel functions and the discriminant of a finite Coxeter group|journal=Compositio Math.|volume=85|issue=3|pages=333&ndash;373|zbl=0778.33009|mr=1214452|url= http://www.numdam.org/item?id=CM_1993__85_3_333_0}}
*{{Citation | last1=Selberg | first1=Atle | title=Remarks on a multiple integral | mr=0018287 | year=1944 | journal=Norsk Mat. Tidsskr. | volume=26 | pages=71–78}}

[[Category:Special functions]]