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  "body": "\n## Statement\n\nFor each $s \u003e 0$, the [[gamma function|0V3FZMVB4K]] satisfies $\\Gamma(s + 1) = s \\, \\Gamma(s)$.\nFor each natural number $n$, $\\Gamma(n + 1) = n!$.\n\n## Remarks\n\n[[Integration by parts|08Y1BBQTK9]] on $[\\varepsilon, b]$, with $u = t^s$ and $v = -e^{-t}$, gives the recursion in the limit.\nThe [[integral of $e^{-t}$|039JAF7ZAN]] gives $\\Gamma(1) = 1$. Induction on $n$ then gives $\\Gamma(n + 1) = n!$.\nAlso $\\Gamma(1/2) = \\sqrt{\\pi}$, by the substitution $t = x^2$ and the Gaussian integral.\nMathlib states it as `Real.Gamma_nat_eq_factorial`.\n\n## History\n\nLeonhard Euler announced the integral representation in a letter to Christian Goldbach, dated 8 January 1730.\n\n## Explanations\n\n### L1 human\n\nThe recursion $\\Gamma(s + 1) = s \\, \\Gamma(s)$ is the same as the rule $(n + 1)! = (n + 1) \\cdot n!$.\nSo the gamma function agrees with the factorial at the integers, shifted by one.\n",
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