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  "body": "\n## Statement\n\nLet $\\Gamma$ be the [[gamma function|0V3FZMVB4K]], extended to a [[meromorphic function|0NSD1TPHR0]] on $\\C$ that is holomorphic on $\\C \\setminus \\set{0, -1, -2, \\ldots}$.\nFor every $z \\in \\C \\setminus \\set{0, -1, -2, \\ldots}$,\n$$\\Gamma(z + 1) = z \\, \\Gamma(z).$$\n\n## Remarks\n\nFor $\\operatorname{Re} z \u003e 0$, integration by parts in $\\Gamma(z + 1) = \\int_0^\\infty t^{z} e^{-t} \\, dt$ gives $z \\int_0^\\infty t^{z-1} e^{-t} \\, dt$; the boundary terms vanish.\nBoth sides are holomorphic on the connected set $\\C \\setminus \\set{0, -1, -2, \\ldots}$, so the identity theorem extends the equation from the half-plane to this set.\nConversely the equation is the usual way to build the extension: for $\\operatorname{Re} z \u003e -n$ one sets $\\Gamma(z) = \\Gamma(z + n)/\\big(z (z + 1) \\cdots (z + n - 1)\\big)$.\nWith $\\Gamma(1) = 1$ the equation gives $\\Gamma(n + 1) = n!$, see [[the gamma function extends the factorial|0FDCW77H44]].\nFor example $\\Gamma(\\tfrac32) = \\tfrac12 \\Gamma(\\tfrac12) = \\tfrac{\\sqrt{\\pi}}{2}$.\nIn Mathlib the statement is `Complex.Gamma_add_one`.\n\n## Explanations\n\n### L1 human\n\nThe gamma function obeys the same rule as the factorial: moving the argument up by one multiplies the value by the old argument.\nFor whole numbers this is the rule $(n+1)! = (n+1) \\cdot n!$.\nThe rule holds at every complex number where $\\Gamma$ is defined.\n\n### L2 human\n\nThe functional equation reduces the values of $\\Gamma$ anywhere to its values on a strip of width one, such as $1 \\le \\operatorname{Re} z \u003c 2$.\nIt also shows where the poles are: dividing by $z$ at $z = 0$ creates a pole, and each further step to the left creates another.\nTogether with log-convexity it characterizes $\\Gamma$ on $(0, \\infty)$, by the [[Bohr–Mollerup theorem|QFF87AXWN2]], and together with boundedness on a strip it characterizes $\\Gamma$ on the right half-plane, by [[Wielandt's theorem|WCM87QEEZW]].\n",
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