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  "body": "\n## Definition\n\nLet $\\Omega$ be a bounded [[domain|0EWMDD6C14]] in $\\C$ with [[boundary|05D5EGMFYV]] $\\partial\\Omega$, and let $w \\in \\Omega$.\nA *Green's function of $\\Omega$ with pole at $w$* is a function $g(\\cdot, w) \\colon \\Omega \\setminus \\set{w} \\to \\R$ such that:\n\n1. $g(\\cdot, w)$ is [[harmonic|0DEB0JMN0W]] on $\\Omega \\setminus \\set{w}$.\n2. The function $z \\mapsto g(z, w) + \\log\\abs{z - w}$ extends to a harmonic function on a neighbourhood of $w$.\n3. $g(z, w) \\to 0$ as $z \\to \\zeta$, for every $\\zeta \\in \\partial\\Omega$.\n\n## Remarks\n\nIf such a function exists, it is unique, by the maximum principle applied to the difference of two of them.\nIt exists whenever the [[Dirichlet problem|8D56GQ4AG8]] on $\\Omega$ is solvable for continuous boundary data: take $g(z, w) = h(z) - \\log\\abs{z - w}$, where $h$ is harmonic with boundary values $\\log\\abs{\\zeta - w}$.\nExample: for the unit disc, $g(z, w) = \\log\\abs{\\dfrac{1 - \\bar w z}{z - w}}$.\nHere $\\log\\abs{1 - \\bar w z}$ is harmonic on the disc because $1 - \\bar w z \\ne 0$ there, and on $\\abs{z} = 1$ one has $\\abs{1 - \\bar w z} = \\abs{\\bar z - \\bar w} = \\abs{z - w}$, so $g = 0$ on the boundary.\nNon-example: for $w \\ne 0$, the function $z \\mapsto -\\log\\abs{z - w}$ on the unit disc satisfies clauses 1 and 2, but at the boundary point $\\zeta = -w/\\abs{w}$ its limit is $-\\log(1 + \\abs{w}) \\ne 0$, so clause 3 fails.\nIn the sense of distributions, $\\Delta_z g(z, w) = -2\\pi\\delta_w$, where $\\Delta$ is the [[Laplacian|0E0B6RA8K0]].\nThe Green's function is positive on $\\Omega \\setminus \\set{w}$ and symmetric: $g(z, w) = g(w, z)$.\nIn $\\R^n$ with $n \\ge 3$, the analogous definition uses the singularity $\\abs{x - y}^{2 - n}$ in place of $-\\log\\abs{z - w}$, up to normalizing constants.\n",
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