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  "body": "\n## Statement\n\nLet $\\Gamma$ be the [[gamma function|0V3FZMVB4K]], extended to a [[meromorphic function|0NSD1TPHR0]] on $\\C$.\nThe [[poles|066E7PJCYE]] of $\\Gamma$ are exactly the points $z = -n$ with $n \\in \\set{0, 1, 2, \\ldots}$.\nEach of them is a simple pole, and the [[residue|0PHN07KVQT]] of $\\Gamma$ at $-n$ is\n$$\\operatorname{Res}_{z = -n} \\Gamma(z) = \\frac{(-1)^n}{n!}.$$\n\n## Remarks\n\nIterating the functional equation gives $\\Gamma(z) = \\dfrac{\\Gamma(z + n + 1)}{z (z + 1) \\cdots (z + n)}$ near $z = -n$.\nThe numerator is holomorphic at $-n$ with value $\\Gamma(1) = 1$, and only the factor $z + n$ of the denominator vanishes there, to first order.\nSo the pole is simple, and the residue is $1/\\big((-n)(-n + 1) \\cdots (-1)\\big) = (-1)^n/n!$.\nAway from these points $\\Gamma$ is holomorphic, see [[holomorphy of the gamma function|ZKJF80DTNB]].\nFor example, near $z = 0$ one has $\\Gamma(z) = \\frac{1}{z} - \\gamma + O(z)$, where $\\gamma$ is the [[Euler–Mascheroni constant|00C5P9JZBK]].\n\n## Explanations\n\n### L1 human\n\nThe gamma function blows up at $0$ and at every negative integer.\nNear each such point it behaves like a constant divided by the distance to the point, and the constants $1, -1, \\tfrac12, -\\tfrac16, \\ldots$ are $(-1)^n/n!$.\n\n### L2 human\n\nThe residues explain contour integral formulas such as the Mellin–Barnes integrals, where closing the contour to the left picks up the poles of $\\Gamma$ and produces power series.\nFor instance, summing the residues of $\\Gamma(s) x^{-s}$ at $s = -n$ gives $\\sum_n (-1)^n x^n / n! = e^{-x}$.\nThe fact that the poles are simple and $\\Gamma$ has no zeros makes $1/\\Gamma$ an entire function with simple zeros.\n",
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