{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Example\n\nLet $\\tau_1 \u003e 0$ be the optical thickness of a plane-parallel slab.\nLet $\\Psi$ be an even polynomial, the characteristic function, with $\\Psi \\ge 0$ on $[0, 1]$ and $\\int_0^1 \\Psi\\, d\\mu \\le \\frac{1}{2}$.\n*Chandrasekhar's X- and Y-functions* are the functions $X$ and $Y$ on $[0, 1]$ that solve the system\n$$X(\\mu) = 1 + \\mu \\int_0^1 \\frac{\\Psi(\\mu')}{\\mu + \\mu'} \\bigl[X(\\mu) X(\\mu') - Y(\\mu) Y(\\mu')\\bigr]\\, d\\mu',$$\n$$Y(\\mu) = e^{-\\tau_1/\\mu} + \\mu \\int_0^1 \\frac{\\Psi(\\mu')}{\\mu - \\mu'} \\bigl[Y(\\mu) X(\\mu') - X(\\mu) Y(\\mu')\\bigr]\\, d\\mu'.$$\nIn the second equation the bracket vanishes at $\\mu' = \\mu$, so the integrand has no singularity there.\nThis is a system of two coupled [[nonlinear integral equations|BSCYBP6AS6]] for the two unknowns.\n\nCheck of two limits.\nFor $\\tau_1 = 0$, the constant functions $X = 1$ and $Y = 1$ solve the system: both brackets vanish and $e^{0} = 1$.\nThis is a slab of thickness $0$, which lets all light through.\nAs $\\tau_1 \\to \\infty$, the term $e^{-\\tau_1/\\mu}$ tends to $0$; with $Y = 0$ the first equation becomes the equation of [[Chandrasekhar's H-function|99879Z5Y1K]], $X = H$.\n\n## Remarks\n\nThe functions $X$ and $Y$ give the light reflected and transmitted by a slab of finite thickness in the [[radiative transfer equation|EX9YYRMAJA]], while $H$ handles a half-space.\nIn the conservative case $\\int_0^1 \\Psi\\, d\\mu = \\frac{1}{2}$ the system does not determine $X$ and $Y$ uniquely, and an extra condition selects the physical solution.\n\n## Explanations\n\n### L1 human\n\nWhen light falls on a slab of fog of finite thickness, part of it is reflected and part of it goes through.\nChandrasekhar described both parts with two functions, $X$ for the reflected light and $Y$ for the transmitted light.\nThey solve two equations that are tied together.\nFor a slab of zero thickness everything passes, and for an infinitely thick slab nothing passes and $X$ becomes the H-function.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "instance_of": [
      "BSCYBP6AS6"
    ],
    "kind": "example",
    "msc": [
      "45G15",
      "85A25"
    ],
    "name": "Chandrasekhar's X- and Y-functions",
    "same_as": {
      "wikidata": "Q48996583",
      "wikipedia": "Chandrasekhar's X- and Y-function"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Chandrasekhar%27s_X-_and_Y-function"
      }
    ],
    "tag": "YJ1M0FYXJD",
    "uses": [
      "99879Z5Y1K",
      "EX9YYRMAJA",
      "BSCYBP6AS6"
    ]
  },
  "kind": "example",
  "link": "https://tamnd.github.io/math2/node/YJ1M0FYXJD/",
  "links": [
    {
      "field": "instance_of",
      "target": "BSCYBP6AS6"
    },
    {
      "field": "uses",
      "target": "99879Z5Y1K"
    },
    {
      "field": "uses",
      "target": "EX9YYRMAJA"
    },
    {
      "field": "uses",
      "target": "BSCYBP6AS6"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/YJ1M0FYXJD.md",
  "msc": [
    "45G15",
    "85A25"
  ],
  "name": "Chandrasekhar's X- and Y-functions",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/YJ1/YJ1M0FYXJD.md",
  "status": null,
  "tag": "YJ1M0FYXJD",
  "uri": "m2://YJ1M0FYXJD",
  "url": "https://tamnd.github.io/math2/wiki/chandrasekhars-x-and-y-functions/",
  "writing": {
    "tag": "YJ1M0FYXJD",
    "kind": "example",
    "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
      }
    ]
  }
}