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  "body": "\n## Definition\n\nFor a real number $s \u003e 0$, the gamma function is\n$$\\Gamma(s) = \\int_0^\\infty t^{s-1} e^{-t} \\, dt.$$\n\n## Remarks\n\nThe [[improper integral|0VCTBJMEJF]] converges at $0$ because $s \u003e 0$, and at $\\infty$ because $e^{-t}$ decays fast. Both claims follow from the [[comparison test|0FR1BDKK10]].\nIntegration by parts gives $\\Gamma(s + 1) = s \\, \\Gamma(s)$. Also $\\Gamma(1) = 1$, so [[$\\Gamma$ extends the factorial|0FDCW77H44]].\nThe function extends to all real numbers except $0, -1, -2, \\dots$, by the recursion. Mathlib defines this extension as `Real.Gamma`.\nThe notation $\\Gamma$ is due to Adrien-Marie Legendre.\n\n## Explanations\n\n### L1 human\n\nThe gamma function is an integral that depends on a parameter $s$.\nAt positive integers it gives factorials, so it interpolates the factorial.\n",
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