Beta Integrals
 Euler Beta
 Integral                     Beta Integrals
 Selberg Integral
 A Selberg
  n
 Integral
                                  S. Ole Warnaar
                     Department of Mathematics and Statistics
    Beta Integrals       Euler Beta Integral
  Euler Beta
  Integral
   Wallis formula
   Gamma function
   Euler beta
   integral                         Wallis formula (1656)
   Orthogonal
   polynomials
  Selberg Integral
  A Selberg
   n
  Integral
                                                                     π          22           42          62
                                                                     2 = 1·3 · 3·5 · 5·7 ···
                                                                               ∞                         2
                                                                         =Y                     (2n)
                                                                              n=1 (2n −1)(2n+1)
  Beta Integrals
 Euler Beta
 Integral         Gamma function (Euler 1720s)
  Wallis formula
  Gamma function
  Euler beta
  integral
  Orthogonal
  polynomials
 Selberg Integral
 A Selberg
  n
 Integral
                                     n!nx−1
                 Γ(x) = lim                              x 6= 0;−1;−2;:::
                        n→∞x(x +1)···(x +n−1)
                      =Z0∞tx−1e−tdt                      Re(x) > 0
   Beta Integrals
  Euler Beta
  Integral
  Wallis formula                                    π
  Gamma function       Since                    =        Wallis’ formula is equivalent to
  Euler beta                                         4
  integral
  Orthogonal                            1
  polynomials
  Selberg Integral
  A Selberg                                           Z 1p
   n                                                                     2
  Integral                                          2           1−x dx =Γ(1=2)Γ(3=2)
                                                        0
                       or, by x2 = t, to
                                             Z 1 1=2−1                  3=2−1
                                               0 t           (1 −t)              dt = Γ(1=2)Γ(3=2):
                       This led Euler to the discovery of a more general integral.