{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Summary\n\nSpecial functions are the named functions of analysis that occur again and again in mathematics and physics, beyond the elementary functions.\nExamples are the [[gamma function|0H5SYBSED9]], the [[hypergeometric function|463X5V31D9]], the [[Bessel functions|CZCMRDWDYW]], the [[Legendre polynomials|QFYGJVA996]] and the [[elliptic integrals|NXMCYGCB59]].\nThe term has no formal definition: a function counts as special when it has an established name, notation and theory.\nTheir study collects identities, integral representations, series, asymptotic formulas and differential equations that make these functions usable in computation.\n\n## Explanations\n\n### L1 human\n\nPolynomials, exponentials, logarithms and trigonometric functions are the everyday functions of school mathematics.\nMany problems lead to functions that are not of this kind, but appear so often that they get a name, like a new word in a language.\nFor example, the area under the bell curve $e^{-x^2}$ from $0$ to $x$ cannot be written with elementary functions, so it gets its own name, the error function.\nTables, formulas and computer libraries then treat these functions as well as the familiar ones.\nMany special functions solve a simple differential equation or extend a sequence such as the factorials to non-integer arguments.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "33-00",
      "33E99"
    ],
    "name": "Special functions",
    "same_as": {
      "wikidata": "Q867707",
      "wikipedia": "Special functions"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Special_functions"
      }
    ],
    "tag": "HWY72ZK273"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/HWY72ZK273/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/HWY72ZK273.md",
  "msc": [
    "33-00",
    "33E99"
  ],
  "name": "Special functions",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/HWY/HWY72ZK273.md",
  "status": null,
  "tag": "HWY72ZK273",
  "uri": "m2://HWY72ZK273",
  "url": "https://tamnd.github.io/math2/wiki/special-functions/",
  "writing": {
    "tag": "HWY72ZK273",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}