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  "body": "\n## Statement\n\nLet $K \\subseteq \\C$ be a [[compact set|00R0X1WGPX]] such that $\\C \\setminus K$ is [[connected|0D939CMME7]].\nLet $f \\colon K \\to \\C$ be [[continuous|0P27W9RW0G]] on $K$ and [[holomorphic|0DG0Z72CQD]] on the [[interior|0C38S0KMJK]] of $K$.\nThen for every $\\varepsilon \u003e 0$ there is a polynomial $p$ such that\n$$\\abs{f(z) - p(z)} \u003c \\varepsilon \\quad \\text{for all } z \\in K.$$\n\n## Remarks\n\nThe conditions on $f$ are necessary, since a [[uniform limit|08HJRTG5N7]] of polynomials on $K$ is continuous on $K$ and holomorphic on its interior.\nThe condition on $K$ is also necessary.\nOn the unit circle, $f(z) = 1/z$ is continuous and the interior is empty, yet $f$ is not a uniform limit of polynomials.\nIndeed $\\oint p(z) \\, dz = 0$ for polynomials while $\\oint z^{-1} \\, dz = 2\\pi i$.\n\nTwo special cases were known before.\nIf $f$ is holomorphic on a neighbourhood of $K$, the result is [[Runge's theorem|6DG4CTGXJD]].\nIf $K$ is the closed unit disc, it suffices to approximate $f$ by $f(rz)$ with $r \u003c 1$ and then truncate its Taylor series.\nThe general proof is much harder; it uses estimates on Cauchy integrals that are uniform over the shape of $K$.\n\nExample: let $K = [0, 1]$, a segment with empty interior and connected complement.\nThe theorem says every continuous function on $[0, 1]$ is a uniform limit of complex polynomials, which recovers the Weierstrass approximation theorem.\n\n## Explanations\n\n### L1 human\n\nIf a compact set in the plane has no holes, then every function that is continuous on it and holomorphic inside it can be approximated by polynomials.\nThe approximation is uniform: the error is small at every point of the set at once.\nWithout holes there is no \"hidden\" Cauchy integral that polynomials cannot imitate.\nThis is the best possible result of its kind.\n\n### L2 human\n\nMergelyan's theorem completes the theory of polynomial approximation on compact sets with connected complement.\nIt contains the Weierstrass approximation theorem and its complex analogue for sets with empty interior.\nFor sets with holes, the question of approximation by rational functions is harder and is described by analytic capacity (Vitushkin's theory).\nOn unbounded closed sets the corresponding statement is [[Arakelyan's theorem|1S20A3R659]].\n",
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