{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "KQWJ8SGJEZ"
    },
    {
      "field": "defines",
      "target": "QNS8WFYFXJ"
    }
  ],
  "body": "\n## Summary\n\nThe Cauchy–Riemann equations in several variables are a system of $2n$ real first-order partial differential equations.\nThey are $n$ pairs of [[Cauchy–Riemann equations|0T71SSYZ39]], one pair for each complex coordinate $z_j = x_j + i y_j$.\nIn complex form they read $\\partial f / \\partial \\bar z_j = 0$ for $j = 1, \\dots, n$.\nA continuously differentiable function on an open subset of $\\C^n$ is [[holomorphic|507C34W0ZT]] exactly when it satisfies them.\nFor $n \\ge 2$ the system is overdetermined: $n$ complex equations for one unknown function, which is a source of the rigidity of the theory.\n\n## Explanations\n\n### L1 human\n\nA function of one complex variable is holomorphic when its real and imaginary parts satisfy two linked equations.\nIn several variables, the same two equations must hold in each variable separately.\nSo there are two equations for each coordinate, $2n$ in all.\nFor example, $f(z_1, z_2) = z_1 z_2$ satisfies all of them, while $f(z_1, z_2) = z_1 \\bar z_2$ fails the pair for $z_2$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "32W05",
      "32A10"
    ],
    "name": "Cauchy–Riemann equations in several variables",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Cauchy%E2%80%93Riemann_equations"
      }
    ],
    "tag": "9PT772R94A"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/9PT772R94A/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/9PT772R94A.md",
  "msc": [
    "32W05",
    "32A10"
  ],
  "name": "Cauchy–Riemann equations in several variables",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/9PT/9PT772R94A.md",
  "status": null,
  "tag": "9PT772R94A",
  "uri": "m2://9PT772R94A",
  "url": "https://tamnd.github.io/math2/wiki/cauchy-riemann-equations-in-several-variables/",
  "writing": {
    "tag": "9PT772R94A",
    "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
      }
    ]
  }
}