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  "body": "\n## Definition\n\nLet $\\psi$ be the [[digamma function|HWRRXB0YXS]].\nThe *trigamma function* is the meromorphic function\n$$\\psi_1(z) = \\psi'(z) = \\frac{d^2}{dz^2} \\log \\Gamma(z), \\qquad z \\in \\C \\setminus \\set{0, -1, -2, \\ldots}.$$\n\n## Remarks\n\nEquivalently, $\\psi_1(z) = \\sum_{k=0}^\\infty \\frac{1}{(z+k)^2}$, and $\\psi_1$ is the [[polygamma function|3J3QGKPMZQ]] $\\psi^{(1)}$.\n\nExample: $\\psi_1(1) = \\sum_{k \\ge 1} k^{-2} = \\frac{\\pi^2}{6}$.\nAlso $\\psi_1(\\frac12) = \\sum_{k \\ge 0} \\frac{4}{(2k+1)^2} = 4 \\cdot \\frac{\\pi^2}{8} = \\frac{\\pi^2}{2}$.\n\nNon-example of a valid argument: at $z = 0$ the term $k = 0$ is $1/0^2$, so $\\psi_1$ has a double pole there, as at every non-positive integer.\n\nThe recurrence $\\psi_1(z+1) = \\psi_1(z) - \\frac{1}{z^2}$ and the reflection formula $\\psi_1(1-z) + \\psi_1(z) = \\frac{\\pi^2}{\\sin^2 \\pi z}$ hold for all $z$ that are not integers.\n",
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