{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Proof\n\nWe differentiate the Cauchy–Riemann equations once more and use the symmetry of mixed partial derivatives.\n\n1. Since $f$ is [[holomorphic|0DG0Z72CQD]] on $U$, [[holomorphic functions are infinitely differentiable|0QEBN8DGVE]]. So, as a function of $(x, y)$, $f$ has continuous partial derivatives of all orders on $U$.\n2. The partial derivatives of $f$ are those of $u$ plus $i$ times those of $v$, by the definition of [[real and imaginary parts|02ZE6JHJ7G]]. So all partial derivatives of $u$ and $v$ exist and are continuous on $U$.\n3. In particular, $u$ and $v$ are of [[class $C^2$|0TCBZMEYGN]] on $U$. This is the first claim.\n4. Since $f$ is complex differentiable at each point of $U$, the [[Cauchy–Riemann equations|0T71SSYZ39]] hold at each point of $U$: $u_x = v_y$ and $u_y = -v_x$ on $U$.\n5. These are equalities of functions on the open set $U$. So we may differentiate both sides. Differentiating $u_x = v_y$ in $x$ gives $u_{xx} = (v_y)_x$.\n6. Differentiating $u_y = -v_x$ in $y$ gives $u_{yy} = -(v_x)_y$.\n7. Since $v$ is of class $C^2$, [[Schwarz's theorem|0NNQY8WCW6]] gives $(v_y)_x = (v_x)_y$ on $U$.\n8. Adding steps 5 and 6 and using step 7 gives $u_{xx} + u_{yy} = (v_y)_x - (v_x)_y = 0$ on $U$.\n9. Differentiating $u_y = -v_x$ in $x$ gives $(u_y)_x = -v_{xx}$. Differentiating $u_x = v_y$ in $y$ gives $(u_x)_y = v_{yy}$.\n10. Schwarz's theorem for $u$ gives $(u_y)_x = (u_x)_y$. So $v_{xx} + v_{yy} = -(u_y)_x + (u_x)_y = 0$ on $U$.\n11. By steps 3, 8 and 10, $u$ and $v$ are of class $C^2$ with zero Laplacian. So they are [[harmonic|0W5NJQ8MKY]] on $U$.\n\n## Idea\n\nThe Cauchy–Riemann equations tie the first derivatives of $u$ to those of $v$. Differentiating once more, the two second derivatives $u_{xx}$ and $u_{yy}$ become the same mixed derivative of $v$ with opposite signs. The real work is hidden in step 1: we need to know that $f$ has second derivatives at all.\n",
  "commit": "4a3290d299c8b2cd6493fce2296920ba7f82d942",
  "front": {
    "kind": "proof",
    "msc": [
      "31A05"
    ],
    "name": "Proof that the real part of a holomorphic function is harmonic",
    "proves": "031BG625Y3",
    "sources": [
      {
        "locator": "Connections with complex function theory",
        "url": "https://en.wikipedia.org/wiki/Harmonic_function"
      }
    ],
    "tag": "XY7D3X10QR",
    "uses": [
      "0DG0Z72CQD",
      "0QEBN8DGVE",
      "0T71SSYZ39",
      "0NNQY8WCW6",
      "0TCBZMEYGN",
      "0W5NJQ8MKY",
      "02ZE6JHJ7G"
    ]
  },
  "kind": "proof",
  "link": "https://math2.tamnd.com/node/XY7D3X10QR/",
  "links": [
    {
      "field": "proves",
      "target": "031BG625Y3"
    },
    {
      "field": "uses",
      "target": "0DG0Z72CQD"
    },
    {
      "field": "uses",
      "target": "0QEBN8DGVE"
    },
    {
      "field": "uses",
      "target": "0T71SSYZ39"
    },
    {
      "field": "uses",
      "target": "0NNQY8WCW6"
    },
    {
      "field": "uses",
      "target": "0TCBZMEYGN"
    },
    {
      "field": "uses",
      "target": "0W5NJQ8MKY"
    },
    {
      "field": "uses",
      "target": "02ZE6JHJ7G"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/XY7D3X10QR.md",
  "msc": [
    "31A05"
  ],
  "name": "Proof that the real part of a holomorphic function is harmonic",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/XY7/XY7D3X10QR.md",
  "status": null,
  "tag": "XY7D3X10QR",
  "uri": "m2://XY7D3X10QR",
  "url": "https://math2.tamnd.com/wiki/proof-that-the-real-part-of-a-holomorphic-function-is-harmonic/",
  "writing": {
    "tag": "XY7D3X10QR",
    "kind": "proof",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}