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  "body": "\n## Definition\n\nLet $\\Gamma$ be the [[gamma function|0H5SYBSED9]], viewed as a [[meromorphic function|0ZZ7ZYDR8W]] on $\\C$ with poles at $0, -1, -2, \\ldots$ and no zeros.\nThe *digamma function* is the meromorphic function\n$$\\psi(z) = \\frac{\\Gamma'(z)}{\\Gamma(z)} = \\frac{d}{dz} \\log \\Gamma(z), \\qquad z \\in \\C \\setminus \\set{0, -1, -2, \\ldots}.$$\n\n## Remarks\n\nExample: from $\\Gamma(z+1) = z\\,\\Gamma(z)$, taking logarithmic derivatives gives $\\psi(z+1) = \\psi(z) + \\frac1z$.\nWith $\\psi(1) = \\Gamma'(1) = -\\gamma$, where $\\gamma$ is the [[Euler–Mascheroni constant|00C5P9JZBK]], this gives $\\psi(n+1) = H_n - \\gamma$ for every integer $n \\ge 1$, where $H_n = 1 + \\frac12 + \\cdots + \\frac1n$ is the [[harmonic number|0NANDX7V02]].\nAnother value is $\\psi(\\frac12) = -\\gamma - 2\\ln 2$.\n\nNon-example of a valid argument: $z = 0$ is a pole of $\\Gamma$, so $\\psi$ is not defined there.\nIn fact $\\psi$ has a simple pole with residue $-1$ at each of $0, -1, -2, \\ldots$.\n\nThe function $\\psi$ has no other poles, because $\\Gamma$ has no zeros.\n",
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