{
  "aliases": [
    "mean-value-property-of-harmonic-functions",
    "gauss-mean-value-theorem"
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  "body": "\n## Statement\n\nLet $U \\subseteq \\mathbb{C}$ be an open set, and identify $\\mathbb{C}$ with $\\mathbb{R}^2$ by $x + iy \\leftrightarrow (x, y)$.\nLet $u : U \\to \\mathbb{R}$ be harmonic: $u$ is twice continuously differentiable and $u_{xx} + u_{yy} = 0$ on $U$.\nLet $c \\in \\mathbb{C}$ and $r \u003e 0$ be such that the closed disc $\\{z : |z - c| \\le r\\}$ is contained in $U$.\nThen\n$$u(c) = \\frac{1}{2\\pi} \\int_0^{2\\pi} u(c + r e^{i\\theta}) \\, d\\theta.$$\n\n## Remarks\n\nThe value of $u$ at the center of the circle is the average of $u$ on the circle.\nFor holomorphic functions, the same formula holds, and it follows from [[Cauchy's integral formula|0DG10TVYPN]] with $a = c$. On a disc, $u$ is the [[real part of a holomorphic function|031BG625Y3]], which gives the harmonic case.\nThe property also holds for averages over the solid disc. A continuous function with the mean value property on all small circles is harmonic.\n\n## Explanations\n\n### L1 human\n\nIf $u$ is harmonic, its value at a point is the average of its values on each circle around that point. This is true as long as the closed disc stays in the domain.\nFor example, $u = x^2 - y^2$ at $0$ is $0$, and on the unit circle it is $\\cos 2\\theta$, whose average is $0$.\n",
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        "url": "https://en.wikipedia.org/wiki/Harmonic_function"
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  "msc": [
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  "name": "Mean value property",
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