{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\n*Cauchy's power equation* is the [[functional equation|SRAN4EKWS0]]\n$$f(xy) = f(x)\\, f(y) \\quad \\text{for all } x, y \u003e 0,$$\nfor an unknown [[function|0EPAT51DXQ]] $f \\colon (0, \\infty) \\to \\R$.\nA solution is a function $f$ that satisfies this identity for all positive $x$ and $y$.\n\n## Remarks\n\nExample: $f(x) = \\sqrt{x}$ is a solution, since $\\sqrt{xy} = \\sqrt{x} \\sqrt{y}$.\nNon-example: $f(x) = 2x$ is not a solution, since $f(1 \\cdot 1) = 2$ while $f(1) f(1) = 4$.\nThe substitution $x = e^s$, $y = e^t$ turns the equation into [[Cauchy's exponential equation|CEGF8W64VQ]] for $g(s) = f(e^s)$.\nSo the continuous solutions are $f = 0$ and $f(x) = x^c$ with $c \\in \\R$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "MDWN6E58AW",
    "kind": "definition",
    "msc": [
      "39B22"
    ],
    "name": "Cauchy's power equation",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Cauchy%27s_functional_equation"
      }
    ],
    "tag": "STJAXZG1K1",
    "uses": [
      "SRAN4EKWS0",
      "0EPAT51DXQ"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/STJAXZG1K1/",
  "links": [
    {
      "field": "defines",
      "target": "MDWN6E58AW"
    },
    {
      "field": "uses",
      "target": "SRAN4EKWS0"
    },
    {
      "field": "uses",
      "target": "0EPAT51DXQ"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/STJAXZG1K1.md",
  "msc": [
    "39B22"
  ],
  "name": "Cauchy's power equation",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/STJ/STJAXZG1K1.md",
  "status": null,
  "tag": "STJAXZG1K1",
  "uri": "m2://STJAXZG1K1",
  "url": "https://tamnd.github.io/math2/wiki/cauchys-power-equation-definition/",
  "writing": {
    "tag": "STJAXZG1K1",
    "kind": "definition",
    "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
      }
    ]
  }
}