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  "body": "\n## Statement\n\nLet $K \\subseteq \\C$ be a [[compact set|00R0X1WGPX]].\nLet $f$ be a [[holomorphic function|0DG0Z72CQD]] on an open set containing $K$.\nLet $A \\subseteq \\C \\setminus K$ be a set that contains at least one point of each bounded connected component of $\\C \\setminus K$.\nThen for every $\\varepsilon \u003e 0$ there is a [[rational function|0WCZH4AFMT]] $r$ whose poles all lie in $A$ such that\n$$\\abs{f(z) - r(z)} \u003c \\varepsilon \\quad \\text{for all } z \\in K.$$\nIn particular, if $\\C \\setminus K$ is [[connected|0D939CMME7]], then $f$ is a [[uniform limit|08HJRTG5N7]] on $K$ of polynomials.\n\n## Remarks\n\nProof idea: by the Cauchy integral formula over a cycle around $K$, $f$ is a uniform limit on $K$ of finite sums of terms $c/(z - a)$ with $a$ off $K$.\nEach pole $a$ can then be moved to a point of $A$ in the same component of $\\C \\setminus K$, by expanding in power series.\nFor the unbounded component, the pole is moved to $\\infty$ instead.\nA pole at $\\infty$ means a polynomial.\n\nThe hypothesis on $A$ is necessary.\nLet $K$ be the unit circle and $f(z) = 1/z$; the bounded component of $\\C \\setminus K$ is the open disc.\nFor every polynomial $p$ one has $\\oint_K p(z) \\, dz = 0$, while $\\oint_K z^{-1} \\, dz = 2\\pi i$.\nIf $\\abs{p - 1/z} \u003c \\varepsilon$ on $K$, then $2\\pi = \\abs{\\oint_K (z^{-1} - p(z)) \\, dz} \\le 2\\pi \\varepsilon$, which fails for $\\varepsilon \u003c 1$.\nWith $A = \\set{0}$ the rational function $r(z) = 1/z$ itself has its only pole in $A$ and approximates $f$ exactly.\n\n## Explanations\n\n### L1 human\n\nA holomorphic function near a compact set can be approximated, uniformly on the set, by rational functions.\nThe poles of the rational functions are only needed inside the holes of the set, one chosen point per hole being enough.\nIf the set has no holes, polynomials are enough.\nThe circle example shows that holes really matter.\n\n### L2 human\n\nRunge's theorem is the starting point of approximation theory in the complex plane.\nIt is used to build holomorphic functions with prescribed properties, for example sequences of polynomials that converge pointwise on $\\C$ to a discontinuous limit.\nIt gives the existence of solutions of the $\\bar\\partial$-equation and the Mittag-Leffler theorem on general open sets.\n[[Mergelyan's theorem|5KKCSZJC29]] strengthens the polynomial case to functions that are only continuous on $K$ and holomorphic in its interior.\n",
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