{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "4Z1YXT957Z"
    }
  ],
  "body": "\n## Statement\n\nLet $f$ be an [[entire function|0419794R0T]] that is not constant.\nThen the set $\\C \\setminus f(\\C)$ has at most one point.\nThat is, $f$ takes every complex value with at most one exception.\n\n## Remarks\n\nOne classical proof uses the [[modular lambda function|4Z1YXT957Z]], which gives a holomorphic covering map from the unit disc onto $\\C \\setminus \\set{0, 1}$.\nIf $f$ omits two values, one may assume that they are $0$ and $1$.\nThen $f$ lifts through this covering to an entire function with values in the unit disc, which is constant by [[Liouville's theorem|0EHKVRF4J6]], so $f$ is constant.\nAnother proof uses [[Bloch's theorem|KW31KDX7TB]].\nThe bound is sharp: $e^z$ is not constant and omits exactly the value $0$.\nFor polynomials no value is omitted, by the fundamental theorem of algebra.\n\n## Explanations\n\n### L1 human\n\nLiouville's theorem says that a bounded entire function is constant, so a nonconstant entire function cannot avoid a whole disc of values.\nPicard's little theorem is much stronger: it cannot avoid even two points.\nThe exponential function shows that avoiding one point is possible.\n\n### L2 human\n\nThe theorem has a version for meromorphic functions: a nonconstant meromorphic function on $\\C$ omits at most two values of the Riemann sphere, as $\\tan z$ omits $i$ and $-i$.\n[[Picard's great theorem|08QA8QTM71]] is a local version near an essential singularity and implies the little theorem: if $f$ is not a polynomial, then $f(1/z)$ has an essential singularity at $0$.\nThe [[second main theorem of Nevanlinna theory|GWYE6745JK]] gives a quantitative form: for three distinct values, the solutions of $f(z) = a$ must be numerous.\nThrough Bloch's principle, the theorem is linked to [[Montel's theorem|ZVFVBVJMCB]] on normal families.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "about": [
      "0419794R0T"
    ],
    "kind": "statement",
    "msc": [
      "30D20",
      "30D35"
    ],
    "name": "Picard's little theorem",
    "role": "theorem",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Picard_theorem"
      }
    ],
    "tag": "R2PHM118NG",
    "uses": [
      "0419794R0T"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/R2PHM118NG/",
  "links": [
    {
      "field": "about",
      "target": "0419794R0T"
    },
    {
      "field": "uses",
      "target": "0419794R0T"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/R2PHM118NG.md",
  "msc": [
    "30D20",
    "30D35"
  ],
  "name": "Picard's little theorem",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/R2P/R2PHM118NG.md",
  "status": null,
  "tag": "R2PHM118NG",
  "uri": "m2://R2PHM118NG",
  "url": "https://math2.tamnd.com/wiki/picards-little-theorem/",
  "writing": {
    "tag": "R2PHM118NG",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}