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  "body": "\n## Definition\n\nLet $E$ be $\\R$ or $\\C$, let $c \\in \\C$ and let $r \\ge 0$.\nLet $f$ be a function with values in $E$, defined at least on the [[circle|060W5CBZP6]] $\\set{z \\in \\C : \\abs{z - c} = r}$.\nAssume that $\\theta \\mapsto f(c + r e^{i\\theta})$ is [[Lebesgue integrable|DWJF7ABPR1]] on $[0, 2\\pi]$.\nThe *circle average* of $f$ on the circle of centre $c$ and radius $r$ is\n$$M(f; c, r) = \\frac{1}{2\\pi} \\int_0^{2\\pi} f(c + r e^{i\\theta}) \\, d\\theta.$$\n\n## Remarks\n\nExample: let $f(z) = z$, with $c \\in \\C$ and $r \u003e 0$.\nThen $M(f; c, r) = c + \\frac{r}{2\\pi} \\int_0^{2\\pi} e^{i\\theta} \\, d\\theta = c$, since the integral of $e^{i\\theta}$ over a full period is $0$.\nNon-example: let $f(z) = 1/\\abs{z - 1}$ on $\\C \\setminus \\set{1}$, with $c = 0$ and $r = 1$.\nThen $f(e^{i\\theta}) = 1/(2\\abs{\\sin(\\theta/2)})$ for $0 \u003c \\theta \u003c 2\\pi$, which is not integrable near $\\theta = 0$, so the circle average is not defined.\nFor $r = 0$ the circle average is $f(c)$.\nThe [[mean value property|0HYPR64JM2]] says that $M(u; c, r) = u(c)$ when $u$ is harmonic on a neighbourhood of the closed disc of centre $c$ and radius $r$.\nFor $r \u003e 0$, $M(f; c, r)$ is the integral of $f$ against arc length on the circle, divided by the length $2\\pi r$.\n",
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