{
  "aliases": [
    "cauchy-riemann-conditions"
  ],
  "backlinks": [
    {
      "field": "uses",
      "target": "031BG625Y3"
    }
  ],
  "body": "\n## Statement\n\nLet $U \\subseteq \\mathbb{C}$ be an open set and let $f : U \\to \\mathbb{C}$ be a function.\nIdentify $\\mathbb{C}$ with $\\mathbb{R}^2$ by $x + i y \\leftrightarrow (x, y)$.\nLet $u = \\operatorname{Re} f$ and $v = \\operatorname{Im} f$ be the [[real and imaginary parts|02ZE6JHJ7G]] of $f$, as real functions of $(x, y)$ on $U$.\nLet $z_0 = x_0 + i y_0 \\in U$.\nThen $f$ is [[complex differentiable|0D1C9Y1EXA]] at $z_0$ if and only if the following two conditions hold. First, $u$ and $v$ are differentiable at $(x_0, y_0)$ as functions of two real variables. Second, they satisfy\n$$\\frac{\\partial u}{\\partial x} = \\frac{\\partial v}{\\partial y}, \\qquad \\frac{\\partial u}{\\partial y} = - \\frac{\\partial v}{\\partial x}$$\nat $(x_0, y_0)$.\nIn this case $f'(z_0) = \\frac{\\partial u}{\\partial x}(x_0, y_0) + i \\frac{\\partial v}{\\partial x}(x_0, y_0)$.\n\n## Remarks\n\nFor $f(z) = z^2$ we have $u = x^2 - y^2$ and $v = 2xy$. Then $u_x = 2x = v_y$ and $u_y = -2y = -v_x$.\nThe real differentiability of $u$ and $v$ is necessary. The function $u(x, y) = \\sqrt{|xy|}$, $v = 0$ satisfies the two equations at $(0, 0)$, but $u + iv$ is not complex differentiable at $0$.\nThe two equations together are the one complex equation $\\frac{\\partial f}{\\partial y} = i \\frac{\\partial f}{\\partial x}$.\n\n## Explanations\n\n### L1 human\n\nA complex derivative must be the same in each direction. If you compare the derivative along the real axis with the derivative along the imaginary axis, you get the two equations.\nSo the equations are a test with partial derivatives for a complex derivative.\nFor example, $\\bar z = x - i y$ has $u_x = 1$ and $v_y = -1$, so it fails the test.\n",
  "commit": "f591989ed15704e9d8755a7a5a6426d73de7a3ae",
  "front": {
    "aliases": [
      "cauchy-riemann-conditions"
    ],
    "kind": "statement",
    "msc": [
      "30A99"
    ],
    "name": "Cauchy–Riemann equations",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q622741",
      "wikipedia": "Cauchy–Riemann equations"
    },
    "sources": [
      {
        "locator": "Complex differentiability",
        "url": "https://en.wikipedia.org/wiki/Cauchy%E2%80%93Riemann_equations"
      },
      {
        "locator": "lead section",
        "url": "https://en.wikipedia.org/wiki/Cauchy%E2%80%93Riemann_equations"
      }
    ],
    "tag": "0T71SSYZ39",
    "uses": [
      "0D1C9Y1EXA",
      "02ZE6JHJ7G"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/0T71SSYZ39/",
  "links": [
    {
      "field": "uses",
      "target": "0D1C9Y1EXA"
    },
    {
      "field": "uses",
      "target": "02ZE6JHJ7G"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0T71SSYZ39.md",
  "msc": [
    "30A99"
  ],
  "name": "Cauchy–Riemann equations",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0T7/0T71SSYZ39.md",
  "status": null,
  "tag": "0T71SSYZ39",
  "uri": "m2://0T71SSYZ39",
  "url": "https://math2.tamnd.com/wiki/cauchy-riemann-equations/",
  "writing": {
    "tag": "0T71SSYZ39",
    "kind": "statement",
    "level": "W1",
    "next": [
      {
        "what": "edge about out",
        "note": "the concepts that the statement is about",
        "have": 0,
        "want": 1
      },
      {
        "what": "edge proves in",
        "note": "at least one proof node",
        "have": 0,
        "want": 1
      },
      {
        "what": "edge instance_of or illustrates in",
        "note": "one example that shows the statement at work",
        "have": 0,
        "want": 1
      },
      {
        "what": "explanation L2",
        "have": 0,
        "want": 1
      }
    ]
  }
}