{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "CD7JZ43JBA"
    }
  ],
  "body": "\n## Summary\n\nClifford analysis extends complex analysis to $\\R^n$ by letting functions take values in a [[Clifford algebra|A38B1QFK20]].\nIts central objects are the [[monogenic functions|646PNPT1R6]], the functions annihilated by a [[Dirac operator|3DXGJTEQZH]].\nThe square of the Dirac operator is the Laplacian up to sign.\nSo the theory refines harmonic analysis, as holomorphic functions refine harmonic functions in the plane.\nIt has a Cauchy integral formula, Taylor and Laurent series, and links to singular integrals, spin geometry and representation theory.\nIt grew out of [[quaternionic analysis|SAT7ST5WP7]], which is closely related to its low-dimensional cases.\n\n## Explanations\n\n### L1 human\n\nIn the plane, holomorphic functions are the solutions of the Cauchy–Riemann equations, a first-order system whose \"square\" is the Laplace equation.\nIn higher dimensions there is no such system with complex numbers alone.\nA Clifford algebra supplies anticommuting units $e_1, \\ldots, e_n$ with $e_j^2 = -1$, and with them the operator $\\sum_j e_j \\partial_{x_j}$ squares to minus the Laplacian.\nFunctions killed by this operator behave much like holomorphic functions: they have an integral formula, power series, and a maximum principle.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "30G35",
      "15A66"
    ],
    "name": "Clifford analysis",
    "same_as": {
      "wikidata": "Q5133083",
      "wikipedia": "Clifford analysis"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Clifford_analysis"
      }
    ],
    "tag": "DKVSAZQWF7"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/DKVSAZQWF7/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/DKVSAZQWF7.md",
  "msc": [
    "30G35",
    "15A66"
  ],
  "name": "Clifford analysis",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/DKV/DKVSAZQWF7.md",
  "status": null,
  "tag": "DKVSAZQWF7",
  "uri": "m2://DKVSAZQWF7",
  "url": "https://math2.tamnd.com/wiki/clifford-analysis/",
  "writing": {
    "tag": "DKVSAZQWF7",
    "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
      }
    ]
  }
}