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  "body": "\n## Definition\n\nLet $\\mu$ be a finite positive [[Borel measure|DYR1QFNGNJ]] on $\\R$.\nThe *Stieltjes transform* of $\\mu$ is the function $S_\\mu \\colon \\C \\setminus \\R \\to \\C$ given by\n$$S_\\mu(z) = \\int_\\R \\frac{d\\mu(t)}{t - z}.$$\nIf $\\mu$ is concentrated on $[0, \\infty)$, the same formula defines $S_\\mu$ on $\\C \\setminus [0, \\infty)$.\n\n## Remarks\n\nThe integral converges absolutely, since $\\abs{t - z} \\ge \\abs{\\operatorname{Im} z} \u003e 0$ for real $t$.\nThe function $S_\\mu$ is [[holomorphic|0NAHXR1FRH]], satisfies $S_\\mu(\\bar z) = \\overline{S_\\mu(z)}$, and has $\\operatorname{Im} S_\\mu(z) \u003e 0$ when $\\operatorname{Im} z \u003e 0$ and $\\mu \\ne 0$, because $\\operatorname{Im} \\frac{1}{t - z} = \\frac{\\operatorname{Im} z}{\\abs{t - z}^2}$.\nMany authors use the opposite sign, $\\int \\frac{d\\mu(t)}{z - t}$, or call $S_\\mu$ the Cauchy transform of $\\mu$.\nFor $d\\mu = f(t)\\, dt$ on $[0, \\infty)$ and $x \u003e 0$, $S_\\mu(-x) = \\int_0^\\infty \\frac{f(t)}{x + t}\\, dt$, the classical Stieltjes transform of the function $f$.\n\nExample: for the unit point mass $\\mu = \\delta_0$, $S_\\mu(z) = \\frac{1}{0 - z} = -\\frac{1}{z}$.\nFor the uniform probability measure on $[0, 1]$, $S_\\mu(z) = \\int_0^1 \\frac{dt}{t - z} = \\log\\frac{1 - z}{-z}$ with the principal branch of the logarithm; the argument $1 - 1/z$ is a nonpositive real number only for $z \\in (0, 1]$.\n\nNon-example: Lebesgue measure on $\\R$ is not finite, and $\\int_\\R \\frac{dt}{\\abs{t - z}}$ diverges for every $z$, so the formula does not define a transform.\n",
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