{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $\\Gamma$ be a finite union of disjoint smooth closed curves in $\\C$.\nLet $a$, $b$ and $f$ be [[Hölder continuous|012PBA0FMD]] functions on $\\Gamma$, and let $k \\colon \\Gamma \\times \\Gamma \\to \\C$ be Hölder continuous.\nA *singular integral equation* with Cauchy kernel on $\\Gamma$ is an equation\n$$a(t) \\, \\varphi(t) + \\frac{b(t)}{\\pi i} \\, \\mathrm{p.v.} \\int_\\Gamma \\frac{\\varphi(\\tau)}{\\tau - t} \\, d\\tau + \\int_\\Gamma k(t, \\tau) \\, \\varphi(\\tau) \\, d\\tau = f(t), \\quad t \\in \\Gamma,$$\nfor an unknown Hölder continuous function $\\varphi$ on $\\Gamma$.\nHere $\\mathrm{p.v.}$ is the [[Cauchy principal value|0SYR10HD4R]]: the limit as $\\varepsilon \\to 0$ of the integral over the part of $\\Gamma$ outside the disc $\\abs{\\tau - t} \u003c \\varepsilon$.\nThe equation is *of normal type* when $a(t)^2 - b(t)^2 \\ne 0$ for all $t \\in \\Gamma$.\nThe equation with $k = 0$ is its *dominant* or *characteristic* equation.\n\n## Remarks\n\nExample: on the unit circle, with $a = 0$, $b = 1$ and $k = 0$, the equation $\\frac{1}{\\pi i} \\, \\mathrm{p.v.} \\int_\\Gamma \\frac{\\varphi(\\tau)}{\\tau - t} \\, d\\tau = f(t)$ is of normal type, since $a^2 - b^2 = -1$.\nFor $\\varphi(\\tau) = \\tau^n$ with $n \\ge 0$ the left side is $t^n$, and for $n \u003c 0$ it is $-t^n$; so $f(t) = t^2$ is solved by $\\varphi(\\tau) = \\tau^2$.\nNon-example: if the kernel is $k(t, \\tau)$ alone, continuous, with $b = 0$ and $a = 1$, the equation is a Fredholm equation of the second kind and is not singular.\nThe term $\\frac{1}{\\tau - t}$ makes the integral divergent as an ordinary integral at $\\tau = t$ unless $\\varphi(t) = 0$; the Hölder condition makes the principal value exist.\nMore generally, the name covers equations with other kernels that are integrated as principal values, including kernels on open arcs and in several variables.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "E30EA6ETZ4",
    "kind": "definition",
    "msc": [
      "45E05",
      "45E99"
    ],
    "name": "Singular integral equation",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Singular_integral"
      }
    ],
    "tag": "AB7W6DBA4P",
    "uses": [
      "R3SG1SN7JQ",
      "0SYR10HD4R",
      "012PBA0FMD"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/AB7W6DBA4P/",
  "links": [
    {
      "field": "defines",
      "target": "E30EA6ETZ4"
    },
    {
      "field": "uses",
      "target": "R3SG1SN7JQ"
    },
    {
      "field": "uses",
      "target": "0SYR10HD4R"
    },
    {
      "field": "uses",
      "target": "012PBA0FMD"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/AB7W6DBA4P.md",
  "msc": [
    "45E05",
    "45E99"
  ],
  "name": "Singular integral equation",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/AB7/AB7W6DBA4P.md",
  "status": null,
  "tag": "AB7W6DBA4P",
  "uri": "m2://AB7W6DBA4P",
  "url": "https://tamnd.github.io/math2/wiki/singular-integral-equation-definition/",
  "writing": {
    "tag": "AB7W6DBA4P",
    "kind": "definition",
    "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
      }
    ]
  }
}