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  "body": "\n## Definition\n\nLet $n \\ge 2$ and, for $x \\in \\R^n \\setminus \\set{0}$, let $x^* = x / \\abs{x}^2$ be its inversion in the unit sphere.\nLet $\\Omega \\subseteq \\R^n \\setminus \\set{0}$ be an [[open set|02BH3MCBQB]] and let $\\Omega^* = \\set{x^* : x \\in \\Omega}$.\nThe *Kelvin transform* of a function $u \\colon \\Omega \\to \\R$ is the function $K[u] \\colon \\Omega^* \\to \\R$ given by\n$$K[u](x) = \\abs{x}^{2-n} \\, u(x^*).$$\n\n## Remarks\n\nThe map $x \\mapsto x^*$ is an involution, so $(\\Omega^*)^* = \\Omega$ and $K[K[u]] = u$.\nFor $n = 2$, $K[u] = u \\circ {}^*$ is the composition with the [[inversion in the unit circle|YXST7NGMTC]], $z \\mapsto 1/\\bar z$.\nFor $u$ twice continuously differentiable, $\\Delta(K[u]) = K[\\abs{x}^4 \\Delta u]$, where $\\Delta$ is the [[Laplacian|0E0B6RA8K0]]; so $u$ is [[harmonic|0DEB0JMN0W]] on $\\Omega$ if and only if $K[u]$ is harmonic on $\\Omega^*$.\nExample: in $\\R^3$, $K[x_1](x) = \\abs{x}^{-1} \\cdot x_1 \\abs{x}^{-2} = x_1 / \\abs{x}^3 = -\\partial_1(1/\\abs{x})$, which is harmonic on $\\R^3 \\setminus \\set{0}$, as a derivative of a harmonic function.\nNon-example: without the weight, $u(x^*) = x_1 / \\abs{x}^2$ is not harmonic on $\\R^3 \\setminus \\set{0}$; a direct computation gives $\\Delta(x_1 \\abs{x}^{-2}) = -2 x_1 \\abs{x}^{-4}$.\nThe transform turns the behaviour of $u$ near infinity into the behaviour of $K[u]$ near $0$.\nFor instance, let $n \\ge 3$ and let $u$ be harmonic outside a ball; then $u(x) \\to 0$ at infinity if and only if $K[u]$ has a removable singularity at $0$.\nInversion in a sphere of centre $a$ and radius $\\rho$ gives the analogous transform $\\left(\\rho / \\abs{x - a}\\right)^{n-2} u\\left(a + \\rho^2 (x - a)/\\abs{x - a}^2\\right)$.\n",
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