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  "body": "\n## Statement\n\nLet $n \\ge 1$ be an integer, and let $\\alpha, \\beta, \\gamma \\in \\C$ with $\\operatorname{Re} \\alpha \u003e 0$, $\\operatorname{Re} \\beta \u003e 0$ and\n$$\\operatorname{Re} \\gamma \u003e -\\min\\left( \\frac{1}{n}, \\frac{\\operatorname{Re} \\alpha}{n - 1}, \\frac{\\operatorname{Re} \\beta}{n - 1} \\right),$$\nwhere the last two terms are omitted when $n = 1$.\nLet $\\Gamma$ be the [[gamma function|0H5SYBSED9]].\nThen\n$$\\int_{[0,1]^n} \\prod_{i=1}^{n} t_i^{\\alpha - 1} (1 - t_i)^{\\beta - 1} \\prod_{1 \\le i \u003c j \\le n} \\abs{t_i - t_j}^{2\\gamma} \\, dt_1 \\cdots dt_n = \\prod_{j=0}^{n-1} \\frac{\\Gamma(\\alpha + j\\gamma)\\, \\Gamma(\\beta + j\\gamma)\\, \\Gamma(1 + (j+1)\\gamma)}{\\Gamma(\\alpha + \\beta + (n + j - 1)\\gamma)\\, \\Gamma(1 + \\gamma)}.$$\n\n## Remarks\n\nFor $n = 1$ the formula is Euler's [[beta integral|QHNWVSC6AQ]] $\\int_0^1 t^{\\alpha-1} (1-t)^{\\beta-1} \\, dt = \\frac{\\Gamma(\\alpha)\\Gamma(\\beta)}{\\Gamma(\\alpha + \\beta)}$.\nSelberg proved it in 1944.\nOne proof first treats integer $\\gamma$, where the integrand is a polynomial times the beta weights.\nIt then extends to complex $\\gamma$ by Carlson's theorem on entire functions of exponential type.\nAnother proof, due to Aomoto, derives a recurrence in $\\alpha$ by integrating a derivative and using the symmetry of the integrand.\nExample: $n = 2$, $\\alpha = \\beta = 1$, $\\gamma = 1$ gives $\\int_0^1 \\int_0^1 (t_1 - t_2)^2 \\, dt_1 dt_2 = \\frac{1}{6}$, and the right side is $\\frac{\\Gamma(1)^2 \\Gamma(2)}{\\Gamma(3)\\Gamma(2)} \\cdot \\frac{\\Gamma(2)^2 \\Gamma(3)}{\\Gamma(4)\\Gamma(2)} = \\frac12 \\cdot \\frac{2}{6} = \\frac16$.\nThe bound on $\\operatorname{Re} \\gamma$ is needed for convergence: the factor $\\abs{t_i - t_j}^{2\\gamma}$ is not integrable near the diagonal when $\\gamma$ is too negative.\n\n## Explanations\n\n### L1 human\n\nThe Selberg integral is an $n$-dimensional version of the beta integral.\nIt averages a power of the product of all the pairwise distances between $n$ random points in $[0, 1]$, with beta weights at the endpoints.\nThe surprise is that the answer is an explicit product of gamma values.\n\n### L2 human\n\nThe factor $\\prod_{i\u003cj} \\abs{t_i - t_j}^{2\\gamma}$ is the power of a [[Vandermonde determinant|0P84B554AS]], the same factor that governs the eigenvalue densities of [[random matrices|1HS4RT7HK5]].\nSo the Selberg integral computes the normalizing constants of the Jacobi ensembles, and by limits those of the Laguerre and Gaussian ensembles; the Gaussian case is the [[Mehta integral|4N53T1BZ9S]].\nIt also has $q$-analogues and analogues for other root systems, linked to the [[Macdonald constant term conjecture|5J25W265JA]].\n",
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