Selberg integral

Riemann integration, Lebesgue integration

line integral/contour integration

integration of differential forms

integration over supermanifolds, Berezin integral, fermionic path integral

Kontsevich integral, Selberg integral, elliptic Selberg integral

integration in ordinary differential cohomology

integration in differential K-theory

The **Selberg integral** is a higher-dimensional generalization of the integral representation of Euler’s beta function due Alte Selberg. Many further “Selberg-type” generalizations appear in the study of multidimensional generalizations of hypergeometric functions, arrangements of hyperplanes, Knizhnik-Zamolodchikov equation, representation theory of quantum and affine Lie algebras and of vertex operator algebras, random matrix theory etc. There is also an ellitic generalization, see elliptic Selberg integral.

$\int_0^1 \cdots \int_0^1 (u_1\cdots u_p)^{x-1} [(1-u_1)\cdots (1-u_p)]^{y-1} \left( \prod_{i\lt j} (u_j-u_i)\right)^{2 z} d u_1 \ldots d u_p =
\prod_{s = 1}^p \frac{\Gamma(1+s z) \Gamma(x+(s-1)z)\Gamma(y+(s-1)z)}{ \Gamma(1+z)\Gamma(x+y+(p+s-2)z)}$

where $p\gt 0$ is a positive integer, $Re x\gt 0$, $Re y\gt 0$ , $Re z \gt max\{-p^{-1},- Re x/(p-1), - Re y/(p-1)\}$. Notice that the discriminant $\prod_{i\lt j} (u_j-u_i)$ is the value of the standard Vandermonde determinant.

- wikipedia Selberg integral
- Alte Selberg,
*Remarks on a multiple integral*, Norsk Matematisk Tidsskrift**26**: 71–78, MR0018287 - Atle Selberg – utdypning, biography in Norweigian
- Nils A. Baas,
*Atle Selberg (1917-2007)*pdf - Peter J. Forrester, S. Ole Warnaar,
*The importance of the Selberg integral*, Bull. Amer. Math. Soc. (N.S.)**45**(2008), 489-534, pdf, arxiv/0710.3981 - G. E. Andrews, R. Askey, R. Roy,
*Special functions*, Enc. of Math. and its Appl. (1999) - I. M. Gelfand, M. M. Kapranov, A. Zelevinsky,
*Discriminants, resultants and multidimensional determinants*, Birkhäuser 1994, 523 pp. - J.-G. Luque, J.-Y. Thibon,
*Hankel hyperdeterminants and Selberg integrals*, J. Phys.**A36**(2003), 5267–5292, MR1985318 (2004d:15011) - K. Aomoto,
*Jacobi polynomials associated with Selberg integrals*, SIAM J. Math. Anal. 18 (1987) 545–549;*On the complex Selberg integral*, Q. J. Math. Oxford**38**(1987) 385–399. - R. S. Askey,
*Some basic hypergeometric extensions of integrals of Selberg and Andrews*, SIAM J. Math.**11**(1980) 938–951.

Revised on November 7, 2012 22:54:56
by Urs Schreiber
(82.169.65.155)