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  "body": "\n## Definition\n\nLet $n \\ge 3$ and let $\\Omega \\subseteq \\R^n$ be an [[open set|02BH3MCBQB]] with [[boundary|05D5EGMFYV]] $\\partial\\Omega$.\nLet $\\sigma_{n-1}$ be the area of the unit sphere, and let $\\Phi(z) = \\frac{1}{(n-2)\\sigma_{n-1}} \\abs{z}^{2-n}$ be the [[fundamental solution|BSJ8X2XRN9]] of $-\\Delta$.\nFix $x \\in \\Omega$.\nAn *image system* for $\\Omega$ and $x$ is a finite or countable family of points $x_j \\in \\R^n \\setminus \\overline{\\Omega}$ and real numbers $c_j$ such that:\n\n1. the series $h(y) = \\sum_j c_j \\Phi(y - x_j)$ converges to a harmonic function of $y$ on $\\Omega$, continuous on $\\overline{\\Omega}$;\n2. $\\Phi(y - x) = h(y)$ for every $y \\in \\partial\\Omega$.\n\nThe *method of images* is the construction of the [[Green's function|M36SD56MWQ]] of $\\Omega$ as $G(x, y) = \\Phi(y - x) - h(y)$ from an image system for each $x \\in \\Omega$.\n\n## Remarks\n\nExample: for the half-space $\\Omega = \\set{y : y_n \u003e 0}$ and $x = (x', x_n)$, the single point $x_1 = (x', -x_n)$ with $c_1 = 1$ is an image system, since $\\abs{y - x} = \\abs{y - x_1}$ when $y_n = 0$.\nThis gives $G(x, y) = \\Phi(y - x) - \\Phi(y - x_1)$.\nFor the unit ball and $x \\ne 0$, the inverse point $x^* = x/\\abs{x}^2$ with $c = \\abs{x}^{2-n}$ is an image system.\nNon-example: for the half-space, the point $(x', -x_n)$ with $c = -1$ fails clause 2, since then $h = -\\Phi(y - x) \\ne \\Phi(y - x)$ on the boundary.\nIn the plane the same idea works with $\\Phi(z) = \\frac{1}{2\\pi} \\log \\frac{1}{\\abs{z}}$, where $h$ may also contain an additive constant; for the unit disc this gives the Green's function $\\frac{1}{2\\pi} \\log \\abs{\\frac{1 - \\bar x y}{y - x}}$ in complex notation.\n",
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