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  "body": "\n## Definition\n\nLet $\\Gamma$ be the [[gamma function|0V3FZMVB4K]], holomorphic and without zeros on $\\C \\setminus \\set{0, -1, -2, \\ldots}$, and let $D = \\C \\setminus (-\\infty, 0]$.\nThe *log-gamma function* is the unique [[holomorphic function|0NAHXR1FRH]] $\\operatorname{Log}\\Gamma \\colon D \\to \\C$ such that:\n\n1. $e^{\\operatorname{Log}\\Gamma(z)} = \\Gamma(z)$ for every $z \\in D$;\n2. $\\operatorname{Log}\\Gamma(x)$ is real for every real $x \u003e 0$.\n\nOn $(0, \\infty)$ it equals the real function $x \\mapsto \\ln\\Gamma(x)$, with $\\ln$ the [[natural logarithm|04DXAFN2ES]].\n\n## Remarks\n\nExistence: $D$ is simply connected and $\\Gamma$ has no zeros on $D$, so $\\Gamma$ has a holomorphic logarithm on $D$.\nThis logarithm is unique up to adding $2\\pi i k$, and condition 2 fixes $k$.\nExample: $\\operatorname{Log}\\Gamma(3) = \\ln 2$, which is real, and $e^{\\ln 2} = 2 = \\Gamma(3)$.\nNon-example: the composition $z \\mapsto \\operatorname{Log}(\\Gamma(z))$ with the principal logarithm satisfies condition 1 but is not continuous on $D$.\nThe reason is that the argument of $\\Gamma(x + iy)$ grows without bound as $y \\to \\infty$, so $\\Gamma(z)$ crosses the negative real axis.\nThe functional equation becomes $\\operatorname{Log}\\Gamma(z + 1) = \\operatorname{Log}\\Gamma(z) + \\operatorname{Log} z$ on $D$.\nIts derivative is the digamma function $\\psi = \\Gamma'/\\Gamma$.\n\n## Explanations\n\n### L1 human\n\nThe log-gamma function is the logarithm of the gamma function, chosen so that it is smooth and real on the positive axis.\n",
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