{
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  "body": "\n## Statement\n\nLet $U \\subseteq \\mathbb{C}$ be an open set and let $f_1, f_2, \\dots$ be functions $U \\to \\mathbb{C}$ that are [[holomorphic|0DG0Z72CQD]] on $U$.\nLet $f : U \\to \\mathbb{C}$ be a function such that $f_n \\to f$ uniformly on each compact subset of $U$.\nThen $f$ is holomorphic on $U$. Moreover, $f_n' \\to f'$ uniformly on each compact subset of $U$.\n\n## Remarks\n\nProof idea: $f$ is continuous. On each disc in $U$, [[Cauchy's integral theorem|01MPZXYVYF]] gives $\\oint_{\\partial T} f_n = 0$ for each triangle $T$, so $\\oint_{\\partial T} f = 0$ by the uniform convergence, and [[Morera's theorem|0VDD7W0EKZ]] applies.\nA main use: if a series $\\sum g_k$ of holomorphic functions converges uniformly on compact subsets of $U$, then its sum is holomorphic on $U$.\nThere is no real analogue. On $[-1, 1]$, the polynomials converge uniformly to $|x|$, which is not differentiable at $0$.\n\n## Explanations\n\n### L1 human\n\nIf holomorphic functions get close to a function $f$ uniformly, then $f$ is holomorphic too, and their derivatives get close to the derivative of $f$.\nThis lets us build new holomorphic functions as limits of series, for example the zeta function $\\sum n^{-z}$.\n",
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  "msc": [
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  "name": "Uniform limits of holomorphic functions are holomorphic",
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  "url": "https://math2.tamnd.com/wiki/uniform-limits-of-holomorphic-functions-are-holomorphic/",
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