{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "AH3G1HDB0D"
    },
    {
      "field": "uses",
      "target": "FGY1XAHVXP"
    },
    {
      "field": "uses",
      "target": "NT65WHM5VV"
    },
    {
      "field": "uses",
      "target": "X1DB08QJHK"
    },
    {
      "field": "defines",
      "target": "BSRJN5M0R4"
    }
  ],
  "body": "\n## Summary\n\nThe Abel equation is the [[iterative functional equation|BCGBQ3HK26]] $\\alpha(f(x)) = \\alpha(x) + 1$, where $f$ is a given map and $\\alpha$ is unknown.\nA solution conjugates the map $f$ to the translation $t \\mapsto t + 1$.\nAn injective solution gives a natural way to define non-integer iterates of $f$, by $f^t(x) = \\alpha^{-1}(\\alpha(x) + t)$.\nIt is closely related to [[Schröder's equation|VTJV7JVJEA]]: if $\\psi$ solves $\\psi(f(x)) = s \\psi(x)$ with $s \u003e 0$, $s \\ne 1$ and $\\psi \u003e 0$, then $\\alpha = \\log_s \\psi$ solves the Abel equation.\n\n## Explanations\n\n### L1 human\n\nThe Abel equation asks for a counter that goes up by exactly one each time the map $f$ is applied.\nFor the doubling map $f(x) = 2x$ on the positive numbers, $\\alpha(x) = \\log_2 x$ is such a counter.\nIn the new coordinate $\\alpha$, applying $f$ becomes simply adding one.\nThen half an application of $f$ is adding one half, which in this example gives $f^{1/2}(x) = \\sqrt{2}\\, x$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "39B12",
      "26A18",
      "30D05"
    ],
    "name": "Abel equation",
    "same_as": {
      "wikidata": "Q4666644",
      "wikipedia": "Abel equation"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Abel_equation"
      }
    ],
    "specializes": [
      "BCGBQ3HK26"
    ],
    "tag": "68SHB7Z721"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/68SHB7Z721/",
  "links": [
    {
      "field": "specializes",
      "target": "BCGBQ3HK26"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/68SHB7Z721.md",
  "msc": [
    "39B12",
    "26A18",
    "30D05"
  ],
  "name": "Abel equation",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/68S/68SHB7Z721.md",
  "status": null,
  "tag": "68SHB7Z721",
  "uri": "m2://68SHB7Z721",
  "url": "https://tamnd.github.io/math2/wiki/abel-equation/",
  "writing": {
    "tag": "68SHB7Z721",
    "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
      }
    ]
  }
}