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  "body": "\n## Definition\n\nIdentify $\\R^2$ with $\\C$.\nLet $\\mu$ be a positive [[Borel measure|DYR1QFNGNJ]] on $\\C$ with finite total mass and compact [[support|FQTXV6BQZG]].\nThe *logarithmic potential* of $\\mu$ is the function $p_\\mu \\colon \\C \\to (-\\infty, +\\infty]$,\n$$p_\\mu(z) = \\int \\log \\frac{1}{\\abs{z - w}} \\, d\\mu(w).$$\nIt is the [[potential|JHKSF3XFY2]] of $\\mu$ with respect to the [[logarithmic kernel|VDZ5FKYC2T]].\n\n## Remarks\n\nExample: let $\\mu$ be the uniform probability measure on the unit circle.\nThen $p_\\mu(z) = 0$ for $\\abs{z} \\le 1$ and $p_\\mu(z) = -\\log\\abs{z}$ for $\\abs{z} \\ge 1$, by the mean value property.\nFor $\\abs{z} \u003e 1$ apply it to the harmonic function $w \\mapsto \\log\\abs{z - w}$ on the disc $\\abs{w} \u003c \\abs{z}$.\nFor $\\abs{z} \u003c 1$ write $\\abs{z - e^{it}} = \\abs{1 - z e^{-it}}$ and apply it to the harmonic function $\\zeta \\mapsto \\log\\abs{1 - z\\zeta}$ on the disc $\\abs{\\zeta} \u003c 1/\\abs{z}$; the case $\\abs{z} = 1$ follows from $\\int_0^{2\\pi} \\log\\abs{1 - e^{it}} \\, dt = 0$.\nExample: for $\\mu = \\sum_{j=1}^d \\delta_{z_j}$ one gets $p_\\mu(z) = -\\log\\abs{(z - z_1)\\cdots(z - z_d)}$.\nNon-example: for Lebesgue measure on all of $\\C$ the integral diverges at every point, which is why the definition asks for a finite measure with compact support.\nThe function $p_\\mu$ is [[harmonic|0DEB0JMN0W]] on $\\C \\setminus \\operatorname{supp}\\mu$, and $p_\\mu(z) = -\\mu(\\C) \\log\\abs{z} + o(1)$ as $\\abs{z} \\to \\infty$.\nIn the sense of distributions, $\\Delta p_\\mu = -2\\pi\\mu$.\nSome authors use the opposite sign, $\\int \\log\\abs{z - w} \\, d\\mu(w)$, which is subharmonic.\n",
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