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  "body": "\n## Definition\n\nLet $U \\subseteq \\R^n$ be an [[open set|02BH3MCBQB]], let $f \\colon U \\to \\R$ and let $a \\in U$.\nThe function $f$ is *harmonic at $a$* if there is an open set $V$ with $a \\in V \\subseteq U$ with two properties.\n\n1. The function $f$ is twice continuously differentiable on $V$.\n2. Its [[Laplacian|0E0B6RA8K0]] satisfies $\\Delta f = 0$ at every point of $V$.\n\n## Remarks\n\nExample: $f(x, y) = \\log(x^2 + y^2)$ on $U = \\R^2 \\setminus \\set{0}$ is harmonic at $a = (1, 0)$.\nIndeed, on $V = U$ one computes $\\partial_x^2 f + \\partial_y^2 f = \\frac{2(y^2 - x^2)}{(x^2 + y^2)^2} + \\frac{2(x^2 - y^2)}{(x^2 + y^2)^2} = 0$.\nNon-example: $f(x, y) = x^3$ on $\\R^2$ is not harmonic at $(0, 0)$.\nIts Laplacian $6x$ vanishes at $(0, 0)$, but it is nonzero at points $(t, 0)$ with $t \\ne 0$ arbitrarily close, so no neighbourhood $V$ works.\nA function is [[harmonic|0DEB0JMN0W]] on $U$ if and only if it is harmonic at every point of $U$.\nIn Mathlib the condition reads: $f$ is $C^2$ at $a$ and $\\Delta f = 0$ on a neighbourhood of $a$.\nThis is the same condition, since a function that is $C^2$ at a point is $C^2$ on a neighbourhood of it.\n",
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