{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "7D7XWA8WWJ"
    },
    {
      "field": "uses",
      "target": "FG5448NZ18"
    }
  ],
  "body": "\n## Summary\n\nThe torus integral of a function $f$ on $\\C^n$ is the iterated contour integral of $f(\\zeta) \\, d\\zeta_1 \\cdots d\\zeta_n$ over a product of $n$ circles.\nThis product is the [[distinguished boundary|A6HN207Y2T]] of a [[polydisc|DJBWP7D3M8]].\nIt is computed by parametrising each circle by an angle and integrating over the cube $[0, 2\\pi]^n$.\nIt appears in the [[Cauchy integral formula on a polydisc|FG5448NZ18]].\nIt also gives the coefficients of a [[power series in several variables|R3QTCPTJK7]].\nFor $n = 1$ it is the [[contour integral|06ZGPYTWK8]] over a circle.\n\n## Explanations\n\n### L1 human\n\nIn one variable, one integrates a function around a circle by walking around the circle once.\nIn $n$ variables, each coordinate $\\zeta_j$ walks around its own circle, independently of the others.\nThe points visited form a torus, like the surface of a doughnut when $n = 2$.\nFor example, the torus integral of $1/(\\zeta_1 \\zeta_2)$ over the product of two unit circles is $(2\\pi i)^2$, the product of two one-variable integrals equal to $2\\pi i$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "32A26",
      "32A10"
    ],
    "name": "Torus integral",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Cauchy%27s_integral_formula"
      }
    ],
    "tag": "R49Q9M519Z"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/R49Q9M519Z/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/R49Q9M519Z.md",
  "msc": [
    "32A26",
    "32A10"
  ],
  "name": "Torus integral",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/R49/R49Q9M519Z.md",
  "status": null,
  "tag": "R49Q9M519Z",
  "uri": "m2://R49Q9M519Z",
  "url": "https://tamnd.github.io/math2/wiki/torus-integral/",
  "writing": {
    "tag": "R49Q9M519Z",
    "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
      }
    ]
  }
}