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  "body": "\n## Summary\n\nThe digamma function $\\psi$ is the logarithmic derivative of the [[gamma function|0H5SYBSED9]], $\\psi(z) = \\Gamma'(z)/\\Gamma(z)$.\nIt turns the product rule $\\Gamma(z+1) = z\\,\\Gamma(z)$ into the sum rule $\\psi(z+1) = \\psi(z) + 1/z$.\nAt positive integers it gives the [[harmonic numbers|0NANDX7V02]] shifted by the [[Euler–Mascheroni constant|00C5P9JZBK]].\nIt is the first member of the family of [[polygamma functions|3J3QGKPMZQ]].\n\n## Explanations\n\n### L1 human\n\nThe gamma function grows very fast, so it is often easier to look at its logarithm.\nThe digamma function is the slope of $\\log \\Gamma$.\nBecause $\\Gamma(n+1) = n!$, the slope of $\\log \\Gamma$ changes by $1/n$ when one steps from $n$ to $n+1$.\nSo the values at whole numbers add up the fractions $1, \\frac12, \\frac13, \\ldots$.\nFor example, $\\psi(1) = -\\gamma \\approx -0.5772$ and $\\psi(4) = 1 + \\frac12 + \\frac13 - \\gamma$.\n",
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