{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "0EHKVRF4J6"
    },
    {
      "field": "uses",
      "target": "0G2YJN0H5S"
    }
  ],
  "body": "\n## Definition\n\nAn entire function is a function $f : \\mathbb{C} \\to \\mathbb{C}$ that is [[holomorphic|0DG0Z72CQD]] on all of $\\mathbb{C}$.\n\n## Remarks\n\nPolynomials, the [[complex exponential|054VXHMCA3]] and the complex [[sine and cosine|0A7YDKRNR6]] are entire.\nSums, products and compositions of entire functions are entire.\nAn entire function is equal to one power series $\\sum_{n \\ge 0} a_n z^n$ that converges for every $z \\in \\mathbb{C}$.\nBy [[Liouville's theorem|0EHKVRF4J6]], each bounded entire function is constant.\n\n## Explanations\n\n### L1 human\n\nAn entire function is a function that is holomorphic at every complex number, with no exceptions.\nFor example, $z^3 - 2z + 5$ and $e^z$ are entire, but $1/z$ is not, because it is not defined at $0$.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "defines": "0G4F0XB9X2",
    "kind": "definition",
    "msc": [
      "30D20"
    ],
    "name": "Entire function",
    "sources": [
      {
        "locator": "lead section",
        "url": "https://en.wikipedia.org/wiki/Entire_function"
      }
    ],
    "tag": "0419794R0T",
    "uses": [
      "0DG0Z72CQD"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/0419794R0T/",
  "links": [
    {
      "field": "defines",
      "target": "0G4F0XB9X2"
    },
    {
      "field": "uses",
      "target": "0DG0Z72CQD"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0419794R0T.md",
  "msc": [
    "30D20"
  ],
  "name": "Entire function",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/041/0419794R0T.md",
  "status": null,
  "tag": "0419794R0T",
  "uri": "m2://0419794R0T",
  "url": "https://math2.tamnd.com/wiki/entire-function-definition/",
  "writing": {
    "tag": "0419794R0T",
    "kind": "definition",
    "level": "W1",
    "next": [
      {
        "what": "edge instance_of in",
        "note": "one example",
        "have": 0,
        "want": 1
      },
      {
        "what": "edge not_instance_of in",
        "note": "one non-example that fails one clause",
        "have": 0,
        "want": 1
      }
    ]
  }
}