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  "body": "\n## Definition\n\nLet $n \\ge 2$ and let $\\mu$ and $\\nu$ be finite positive [[Borel measures|DYR1QFNGNJ]] on $\\R^n$ with compact [[supports|FQTXV6BQZG]].\nLet $k$ be the [[Newtonian kernel|6TJTWQ5HCR]] $k(x) = \\abs{x}^{2-n}$ when $n \\ge 3$, and the [[logarithmic kernel|VDZ5FKYC2T]] $k(x) = \\log(1/\\abs{x})$ when $n = 2$, with $k(0) = +\\infty$.\nThe *mutual energy* of $\\mu$ and $\\nu$ is\n$$I(\\mu, \\nu) = \\iint k(x - y)\\, d\\mu(x)\\, d\\nu(y) \\in (-\\infty, +\\infty].$$\n\n## Remarks\n\nThe mutual energy is symmetric, $I(\\mu, \\nu) = I(\\nu, \\mu)$, and $I(\\mu, \\mu)$ is the [[energy|SFSRQCM3EG]] $I(\\mu)$.\nIt equals $\\int U^\\mu \\, d\\nu$, where $U^\\mu$ is the [[potential|JHKSF3XFY2]] of $\\mu$.\nFor a signed measure $\\lambda = \\mu - \\nu$ with $I(\\mu)$, $I(\\nu)$ and $I(\\mu, \\nu)$ finite, one puts $I(\\lambda) = I(\\mu) - 2 I(\\mu, \\nu) + I(\\nu)$.\n\nExample: for Dirac measures $\\delta_a$ and $\\delta_b$ with $a \\ne b$ in $\\R^3$, $I(\\delta_a, \\delta_b) = 1/\\abs{a - b}$.\nFor the uniform probability measure $\\sigma$ on the unit sphere in $\\R^3$ and $\\delta_0$, $I(\\sigma, \\delta_0) = U^\\sigma(0) = 1$.\n\nNon-example: for $\\mu = \\nu = \\delta_0$, the integrand is $k(0) = +\\infty$ on the only point of the support, so $I(\\mu, \\nu) = +\\infty$ is not a finite mutual energy.\n",
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