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  "body": "\n## Definition\n\nLet $\\Omega \\subsetneq \\C$ be a [[simply connected domain|0QW64ARSHB]] and let $a \\in \\Omega$.\nLet $\\mathbb{D}$ be the unit disc.\nBy the [[Riemann mapping theorem|02WHMEG5ZR]] there is a unique [[conformal map|0YYT36R4YD]] $f \\colon \\mathbb{D} \\to \\Omega$ onto $\\Omega$ with $f(0) = a$ and $f'(0) \u003e 0$.\nThe *conformal radius* of $\\Omega$ at $a$ is the number $\\operatorname{rad}(a, \\Omega) = f'(0)$.\n\n## Remarks\n\nEquivalently, $\\operatorname{rad}(a, \\Omega) = 1/\\abs{\\varphi'(a)}$ for any conformal map $\\varphi$ of $\\Omega$ onto $\\mathbb{D}$ with $\\varphi(a) = 0$.\nExample: let $\\Omega = \\mathbb{D}$ and $a \\in \\mathbb{D}$.\nThe map $f(w) = (w + a)/(1 + \\bar a w)$ sends $\\mathbb{D}$ onto $\\mathbb{D}$ with $f(0) = a$ and $f'(0) = 1 - \\abs{a}^2 \u003e 0$, so $\\operatorname{rad}(a, \\mathbb{D}) = 1 - \\abs{a}^2$.\nNon-example: $\\Omega = \\C$ is simply connected, but no conformal map of $\\mathbb{D}$ onto $\\C$ exists, so the definition does not apply.\nWith $d = \\operatorname{dist}(a, \\partial\\Omega)$ one has $d \\le \\operatorname{rad}(a, \\Omega) \\le 4d$.\nThe lower bound follows from the [[Schwarz lemma|0MB48XDBY3]] applied to $f^{-1}$ on the disc of centre $a$ and radius $d$; the upper bound is the [[Koebe quarter theorem|WX0HMW4GT7]].\nThe [[Green's function|M36SD56MWQ]] of $\\Omega$ satisfies $g(z, a) = -\\log\\abs{z - a} + \\log \\operatorname{rad}(a, \\Omega) + o(1)$ as $z \\to a$.\nUnder a conformal map $F$ of $\\Omega$ onto $\\Omega'$, $\\operatorname{rad}(F(a), \\Omega') = \\abs{F'(a)} \\operatorname{rad}(a, \\Omega)$.\n",
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