{
  "aliases": [
    "big-picard-theorem",
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  "body": "\n## Statement\n\nLet $c \\in \\mathbb{C}$, let $r \u003e 0$, and let $f$ be [[holomorphic|0DG0Z72CQD]] on the punctured disc $0 \u003c |z - c| \u003c r$.\nAssume that $c$ is an [[essential singularity|0YKDJBV26S]] of $f$.\nThen for each $\\delta$ with $0 \u003c \\delta \u003c r$, there is at most one $w_0 \\in \\mathbb{C}$ with the following property. The function $f$ takes the value $w_0$ only finitely many times on $0 \u003c |z - c| \u003c \\delta$.\nThus, on each punctured disc around $c$, $f$ takes each complex value infinitely often, with at most one exception.\n\n## Remarks\n\nThe exception can occur: $e^{1/z}$ never takes the value $0$, and it takes each $w \\neq 0$ infinitely often near $0$, see [[exp(1/z)|0NB31CC4SN]].\nThe theorem is much stronger than the [[Casorati–Weierstrass theorem|02HHNPC8ED]], which gives only that the values come close to each $w$.\nPicard's little theorem is a related result: a nonconstant entire function takes all complex values with at most one exception.\n\n## Explanations\n\n### L1 human\n\nNear an essential singularity, the function does not only come close to each value: it hits almost every value infinitely many times.\nAt most one value can be missed, as $0$ is missed by $e^{1/z}$.\n",
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    "name": "Picard's great theorem",
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  "msc": [
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  "name": "Picard's great theorem",
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