{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "PDG1HMR8GK"
    }
  ],
  "body": "\n## Summary\n\nThe Laguerre–Pólya class is the class of entire functions that are limits, uniformly on compact sets, of real polynomials with only real zeros.\nIts members are described by an explicit Hadamard-type product with real zeros and a Gaussian factor.\nThe class is closed under differentiation, so it is the natural home of real-rootedness for entire functions.\nIt appears in the study of the zeros of the Riemann zeta function and of special functions.\n\n## Explanations\n\n### L1 human\n\nA real polynomial whose roots are all real has a graph that meets the real axis at each of its roots.\nThe functions in the Laguerre–Pólya class are the entire functions that are limits of such polynomials.\nExamples are $\\sin z$, $\\cos z$ and $e^{-z^2}$, the last being a limit of $(1 - z^2/n)^n$.\nThe function $e^{z^2}$ is not in the class, even though it has no zeros at all.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "30D15",
      "30C15",
      "26C10"
    ],
    "name": "Laguerre–Pólya class",
    "same_as": {
      "wikidata": "Q6472739",
      "wikipedia": "Laguerre–Pólya class"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Laguerre%E2%80%93P%C3%B3lya_class"
      }
    ],
    "tag": "X9SRCEENV3"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/X9SRCEENV3/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/X9SRCEENV3.md",
  "msc": [
    "30D15",
    "30C15",
    "26C10"
  ],
  "name": "Laguerre–Pólya class",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/X9S/X9SRCEENV3.md",
  "status": null,
  "tag": "X9SRCEENV3",
  "uri": "m2://X9SRCEENV3",
  "url": "https://math2.tamnd.com/wiki/laguerre-polya-class/",
  "writing": {
    "tag": "X9SRCEENV3",
    "kind": "concept",
    "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
      }
    ]
  }
}