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  "body": "\n## Statement\n\nLet $U = \\C \\setminus \\set{0, -1, -2, \\ldots}$.\nThere is exactly one [[holomorphic function|0NAHXR1FRH]] $\\Gamma \\colon U \\to \\C$ that agrees with the [[gamma function|0V3FZMVB4K]] on $(0, \\infty)$.\n\n## Remarks\n\nOn the half-plane $\\operatorname{Re} z \u003e 0$, Euler's integral $\\int_0^\\infty t^{z-1} e^{-t}\\,dt$ is holomorphic, see [[Euler integral for the gamma function|Z8D1CZFTWP]].\nOn $\\operatorname{Re} z \u003e -n$, with $z \\notin \\set{0, \\ldots, -n+1}$, the formula $\\Gamma(z + n)/\\big(z (z + 1) \\cdots (z + n - 1)\\big)$ is holomorphic and agrees with the previous one on the overlap; these pieces glue to a holomorphic function on $U$.\nUniqueness follows from the identity theorem, because $U$ is open and connected and $(0, \\infty) \\subset U$ has accumulation points.\nThe function cannot be extended holomorphically to any point of $\\set{0, -1, -2, \\ldots}$, since it has [[poles|QFPFF15PXC]] there.\nIn Mathlib the statement is `Complex.differentiableAt_Gamma`.\n\n## Explanations\n\n### L1 human\n\nThe gamma function, first defined for positive real numbers, extends in a smooth complex way to all complex numbers except $0, -1, -2, \\ldots$.\nThis [[analytic continuation|0M3YQJ81NB]] is unique: there is only one way to do it.\n\n### L2 human\n\nHolomorphy is what lets one prove identities for $\\Gamma$ on the real line and then transfer them to complex arguments, by the identity theorem.\nThe reflection and duplication formulas are usually proved in this way.\nTogether with the description of the poles, it says that $\\Gamma$ is a meromorphic function on $\\C$.\n",
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