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  "body": "\n## Definition\n\nLet $n \\ge 2$ and let $\\omega_n$ be the volume of the unit ball of $\\R^n$.\nThe *fundamental solution of the Laplacian* on $\\R^n$ is the function $\\Phi \\colon \\R^n \\setminus \\set{0} \\to \\R$ given by\n$$\\Phi(x) = -\\frac{1}{2\\pi} \\log \\abs{x} \\quad (n = 2), \\qquad \\Phi(x) = \\frac{1}{n(n-2)\\omega_n} \\, \\frac{1}{\\abs{x}^{n-2}} \\quad (n \\ge 3).$$\n\n## Remarks\n\nThe function $\\Phi$ is locally integrable on $\\R^n$, and in the sense of distributions it satisfies $-\\Delta \\Phi = \\delta$, where $\\Delta$ is the [[Laplacian|0E0B6RA8K0]] and $\\delta$ is the [[Dirac delta|0KPHENZ5S7]] at $0$.\nSo $-\\Phi$ is a [[fundamental solution|K2NK9A8M5F]] of $\\Delta$ in the general sense; many texts use this sign convention so that $\\Phi \u003e 0$ near the origin.\nExample: for $n = 3$, $\\omega_3 = \\frac{4\\pi}{3}$ and $\\Phi(x) = \\frac{1}{4\\pi\\abs{x}}$.\nFor a radial function $f(r)$ with $r = \\abs{x}$ one has $\\Delta f = f'' + \\frac{n-1}{r} f'$, and $f = r^{2-n}$ gives $(2-n)(1-n) r^{-n} + (n-1)(2-n) r^{-n} = 0$, so $\\Phi$ is [[harmonic|0DEB0JMN0W]] on $\\R^n \\setminus \\set{0}$.\nNon-example: $\\abs{x}^{-1}$ on $\\R^4$ is not a multiple of $\\Phi$, and it is not harmonic there, since $\\Delta \\abs{x}^{-1} = -\\abs{x}^{-3}$.\nThe constant is chosen so that the flux of $-\\nabla \\Phi$ out of every sphere about $0$ equals $1$; here $n\\omega_n$ is the area of the unit sphere.\nOther normalisations, such as $\\abs{x}^{2-n}$ or $\\log(1/\\abs{x})$ without constants, are common in potential theory.\n",
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