{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "9DYE6V628H"
    },
    {
      "field": "uses",
      "target": "F7A335JYKF"
    }
  ],
  "body": "\n## Summary\n\nThe Whittaker functions $M_{\\kappa, \\mu}(z)$ and $W_{\\kappa, \\mu}(z)$ are solutions of Whittaker's equation $w'' + \\left( -\\frac14 + \\frac{\\kappa}{z} + \\frac{\\frac14 - \\mu^2}{z^2} \\right) w = 0$.\nThey are the [[confluent hypergeometric functions|6AZ4ZBN1M8]] rewritten so that the equation has no first derivative term.\nThis normal form is convenient in physics and in asymptotic analysis, and it makes the symmetry $\\mu \\mapsto -\\mu$ visible.\n\n## Explanations\n\n### L1 human\n\nKummer's equation has a first derivative term; multiplying the unknown by $e^{-z/2} z^{b/2}$ removes it.\nThe result is Whittaker's equation, which looks like a Schrödinger equation with a constant, a Coulomb-type term $\\kappa / z$ and a centrifugal-type term $(\\frac14 - \\mu^2)/z^2$.\nThe function $M_{\\kappa, \\mu}$ is the solution that behaves like $z^{\\mu + 1/2}$ near $0$, and $W_{\\kappa, \\mu}$ is the solution that decays like $e^{-z/2} z^{\\kappa}$ at $+\\infty$.\nA simple case is $W_{0, 1/2}(z) = e^{-z/2}$, which indeed satisfies $w'' - \\frac14 w = 0$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "33C15"
    ],
    "name": "Whittaker function",
    "same_as": {
      "wikidata": "Q3754618",
      "wikipedia": "Whittaker function"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Whittaker_function"
      }
    ],
    "specializes": [
      "6AZ4ZBN1M8"
    ],
    "tag": "7EFDMJSVDP"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/7EFDMJSVDP/",
  "links": [
    {
      "field": "specializes",
      "target": "6AZ4ZBN1M8"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/7EFDMJSVDP.md",
  "msc": [
    "33C15"
  ],
  "name": "Whittaker function",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/7EF/7EFDMJSVDP.md",
  "status": null,
  "tag": "7EFDMJSVDP",
  "uri": "m2://7EFDMJSVDP",
  "url": "https://tamnd.github.io/math2/wiki/whittaker-function/",
  "writing": {
    "tag": "7EFDMJSVDP",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}