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  "body": "\n## Statement\n\nLet $\\mathcal{H}$ be a Hankel contour.\nIt is a path that comes from $-\\infty$ below the negative real axis and goes once around $0$ counterclockwise.\nThen it returns to $-\\infty$ above the negative real axis.\nFor $t \\notin (-\\infty, 0]$ let $t^{-z} = e^{-z \\operatorname{Log} t}$, with $\\operatorname{Log}$ the principal [[complex logarithm|02QGGD5G87]].\nOn the two sides of the cut, $t^{-z}$ takes its limits from below and from above.\nThen for every $z \\in \\C$,\n$$\\frac{1}{\\Gamma(z)} = \\frac{1}{2\\pi i} \\int_{\\mathcal{H}} e^{t} \\, t^{-z} \\, dt,$$\nwhere $1/\\Gamma$ is the [[reciprocal gamma function|PY6K9AWR33]].\n\n## Remarks\n\nThe integral converges for every $z$ because $e^t$ decays fast as $\\operatorname{Re} t \\to -\\infty$, and it is an entire function of $z$.\nFor $\\operatorname{Re} z \u003c 1$, the path can be squeezed onto the cut, with $t = s e^{-i\\pi}$ on the lower side and $t = s e^{i\\pi}$ on the upper side, $s \u003e 0$.\nThe small circle around $0$ contributes nothing in the limit.\nThe two sides give $\\frac{1}{2\\pi i}(e^{i\\pi z} - e^{-i\\pi z}) \\int_0^\\infty e^{-s} s^{-z}\\, ds = \\frac{\\sin(\\pi z)}{\\pi}\\,\\Gamma(1 - z)$, which equals $1/\\Gamma(z)$ by the [[reflection formula|WF7KQAQTS2]].\nThe identity theorem then extends the formula to all $z \\in \\C$.\nAt a positive integer $z = n$ the integrand $e^t t^{-n}$ is single-valued, and the formula reduces to the residue at $0$, which is $1/(n-1)!$.\n\n## Explanations\n\n### L1 human\n\nEuler's integral defines the gamma function only for arguments with positive real part.\nHankel's idea is to integrate along a loop around the origin instead of along a half-line.\nThe resulting formula for $1/\\Gamma$ works for every complex number at once.\n\n### L2 human\n\nThe Hankel integral gives a direct proof that $1/\\Gamma$ is entire, without first building the meromorphic continuation of $\\Gamma$.\nSimilar loop integrals represent other functions with branch points, such as the Riemann zeta function, and the same technique gives their continuations.\nIt is also a starting point for asymptotic estimates of $1/\\Gamma$ by the saddle point method.\n",
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