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  "body": "\n## Definition\n\nLet $K \\subset \\C$ be a [[compact|0QPHW703PV]] set.\nLet $\\mathcal{A}(K)$ be the set of [[holomorphic|0DG0Z72CQD]] functions $f \\colon \\C \\setminus K \\to \\C$ with $\\abs{f(z)} \\le 1$ for all $z$ and $f(z) \\to 0$ as $z \\to \\infty$.\nFor $f \\in \\mathcal{A}(K)$ put $f'(\\infty) = \\lim_{z \\to \\infty} z f(z)$; this limit exists.\nThe *analytic capacity* of $K$ is\n$$\\gamma(K) = \\sup \\set{\\abs{f'(\\infty)} : f \\in \\mathcal{A}(K)}.$$\n\n## Remarks\n\nThe limit $f'(\\infty)$ exists because $w \\mapsto f(1/w)$ has a removable singularity at $w = 0$ with value $0$.\nExample: for the closed disc $K = \\set{\\abs{z} \\le r}$, the function $r/z$ lies in $\\mathcal{A}(K)$ with $f'(\\infty) = r$.\nFor any $f \\in \\mathcal{A}(K)$, the function $g(w) = f(r/w)$ extends to the unit disc with $g(0) = 0$ and $g'(0) = f'(\\infty)/r$.\nThe [[Schwarz lemma|0MB48XDBY3]] gives $\\abs{g'(0)} \\le 1$, so $\\gamma(K) = r$.\nNon-example of positive capacity: for a single point $K = \\set{a}$, each $f \\in \\mathcal{A}(K)$ extends to a bounded entire function by [[Riemann's theorem on removable singularities|0BSQDZWF34]].\nBy Liouville's theorem it is constant, hence $0$, so $\\gamma(\\set{a}) = 0$.\nA segment of length $\\ell$ has analytic capacity $\\ell/4$.\nAhlfors proved that $\\gamma(K) = 0$ if and only if every bounded holomorphic function on $U \\setminus K$ extends holomorphically to $U$, for each open $U \\supseteq K$.\n",
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