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  "body": "\n## Summary\n\nA Runge domain is an open set in $\\C^n$ on which every [[holomorphic function|507C34W0ZT]] can be approximated by polynomials, uniformly on each compact subset.\nThe name comes from [[Runge's theorem|6DG4CTGXJD]], which describes such open sets in the plane: they are the open sets whose connected components are simply connected.\nIn several variables, convex domains and balls are Runge domains, and the [[Oka–Weil theorem|7AWZJNQGT2]] is the main tool to recognise them.\n\n## Explanations\n\n### L1 human\n\nPolynomials are the simplest holomorphic functions, so one asks on which regions they are enough to approximate all holomorphic functions.\nOn a disc, Taylor polynomials do the job.\nOn an annulus they do not: the function $1/z$ cannot be approximated by polynomials on a circle around the hole.\nA Runge domain is a region where polynomials always suffice, at least on each compact part.\n",
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