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  "body": "\n## Definition\n\nLet $\\Gamma$ be the [[gamma function|0V3FZMVB4K]], extended to a meromorphic function on $\\C$ with poles at $0, -1, -2, \\ldots$ and no zeros.\nThe *reciprocal gamma function* is the function $\\C \\to \\C$ given by\n$$\\frac{1}{\\Gamma}(z) = \\begin{cases} 1/\\Gamma(z) \u0026 \\text{if } z \\notin \\set{0, -1, -2, \\ldots}, \\\\ 0 \u0026 \\text{if } z \\in \\set{0, -1, -2, \\ldots}. \\end{cases}$$\n\n## Remarks\n\nIt is an [[entire function|0419794R0T]]: near a pole $-n$ of $\\Gamma$, the quotient $1/\\Gamma$ has a removable singularity with value $0$, because the pole is simple.\nIts zeros are exactly $0, -1, -2, \\ldots$, and they are simple, with derivative $(-1)^n n!$ at $-n$, the reciprocal of the residue of $\\Gamma$.\nExample: at $z = 3$ the value is $1/\\Gamma(3) = 1/2! = \\tfrac12$.\nNon-example: $\\Gamma$ itself is not entire, since it is not defined at $0$ and $\\abs{\\Gamma(z)} \\to \\infty$ as $z \\to 0$.\nIt is an entire function of [[order|SHF8Z7GMFX]] $1$ and of maximal type.\n\n## Explanations\n\n### L1 human\n\nThe reciprocal gamma function is $1/\\Gamma(z)$, with the value $0$ at the points where $\\Gamma$ blows up.\nThese values make it smooth everywhere in the complex plane.\n",
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