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  "body": "\n## Statement\n\nLet $n \\ge 1$ be an integer and let $\\gamma \\in \\C$ with $\\operatorname{Re} \\gamma \u003e -\\frac{1}{n}$.\nLet $\\Gamma$ be the [[gamma function|0H5SYBSED9]].\nThen\n$$\\frac{1}{(2\\pi)^{n/2}} \\int_{\\R^n} \\prod_{1 \\le i \u003c j \\le n} \\abs{x_i - x_j}^{2\\gamma}\\, e^{-(x_1^2 + \\cdots + x_n^2)/2} \\, dx_1 \\cdots dx_n = \\prod_{j=1}^{n} \\frac{\\Gamma(1 + j\\gamma)}{\\Gamma(1 + \\gamma)}.$$\n\n## Remarks\n\nThe formula follows from the [[Selberg integral|S8VTVSJ6EX]] by a limit.\nPut $\\alpha = \\beta = \\lambda$ and $t_i = \\frac12 + \\frac{x_i}{2\\sqrt{2\\lambda}}$; then $(t_i(1 - t_i))^{\\lambda - 1}$ tends, after rescaling, to $e^{-x_i^2/2}$ as $\\lambda \\to \\infty$, and Stirling's formula evaluates the limit of the right side.\nFor $\\gamma = 0$ both sides equal $1$, by the [[Gaussian integral|0DFT1GP1JP]].\nExample: $n = 2$ and $\\gamma = 1$.\nThen $x_1 - x_2$ is normal with mean $0$ and variance $2$ under the Gaussian weight, so the left side is $\\mathbb{E}[(x_1 - x_2)^2] = 2$, and the right side is $\\frac{\\Gamma(2)}{\\Gamma(2)} \\cdot \\frac{\\Gamma(3)}{\\Gamma(2)} = 2$.\nThe bound $\\operatorname{Re} \\gamma \u003e -\\frac1n$ is needed: for smaller $\\gamma$ the singularity where all $x_i$ coincide makes the integral diverge.\n\n## Explanations\n\n### L1 human\n\nTake $n$ independent standard normal numbers and look at the product of all their pairwise distances, raised to the power $2\\gamma$.\nThe Mehta integral gives the average of this quantity as a short product of gamma values.\n\n### L2 human\n\nFor $\\gamma = \\frac12, 1, 2$ the integrand is, up to the Gaussian factor, the joint density of the eigenvalues of the Gaussian orthogonal, unitary and symplectic ensembles of [[random matrices|1HS4RT7HK5]].\nSo the formula gives the normalizing constants of these densities.\nIt is the Gaussian member of a family of integrals over root systems, the Macdonald–Mehta integrals, of which this is the case of type $A_{n-1}$.\n",
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  "name": "Mehta integral",
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