{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $\\Omega$ be a closed discrete subset of $\\C$.\nA power series $\\hat\\varphi(\\zeta) = \\sum_{n \\ge 0} c_n \\zeta^n$ with positive radius of convergence is *$\\Omega$-continuable* if it satisfies both conditions:\n\n1. Its sum is a [[holomorphic function|0DG0Z72CQD]] on a disc $D$ centred at $0$ with $D \\cap \\Omega \\subseteq \\set{0}$.\n2. It has an [[analytic continuation|0M3YQJ81NB]] along every path in $\\C \\setminus \\Omega$ that starts in $D \\setminus \\set{0}$.\n\nA formal series $\\tilde\\varphi(z) = \\sum_{n \\ge 0} a_n z^{-n-1}$ in the variable $z$ near $\\infty$ is an *$\\Omega$-resurgent series* if its formal Borel transform\n$$\\hat\\varphi(\\zeta) = \\sum_{n \\ge 0} \\frac{a_n}{n!} \\zeta^n$$\nis $\\Omega$-continuable.\nThe functions $\\hat\\varphi$ obtained in this way are the *resurgent functions* in the Borel plane.\nThe series $\\tilde\\varphi$ is a *resurgent series* if it is $\\Omega$-resurgent for some closed discrete $\\Omega$.\n\n## Remarks\n\nExample: $\\tilde\\varphi(z) = \\sum_{n \\ge 0} (-1)^n n! \\, z^{-n-1}$ has Borel transform $\\sum_{n \\ge 0} (-1)^n \\zeta^n = \\frac{1}{1 + \\zeta}$.\nThis converges on the unit disc and extends to a holomorphic function on $\\C \\setminus \\set{-1}$, so $\\tilde\\varphi$ is $\\set{-1}$-resurgent.\nNon-example: $\\sum_{n \\ge 0} (n!)^2 z^{-n-1}$ is not resurgent, because its Borel transform $\\sum_{n \\ge 0} n! \\, \\zeta^n$ has radius of convergence $0$, so clause 1 fails.\nEvery convergent series $\\sum a_n z^{-n-1}$ is resurgent, since then $\\hat\\varphi$ is entire.\nÉcalle's general notion allows the singular set to depend on the path of continuation; it is called endless continuability, and the $\\Omega$-continuable germs are a special case.\nThe Borel transform of the formal product of two resurgent series is the convolution of their Borel transforms, and Écalle showed that resurgence is preserved under this product.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "37R4QJCARA",
    "kind": "definition",
    "msc": [
      "34M30",
      "30E15",
      "40G10"
    ],
    "name": "Resurgent function",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Resurgent_function"
      }
    ],
    "tag": "N0QW1RGPR8",
    "uses": [
      "0DG0Z72CQD",
      "0M3YQJ81NB",
      "8DJPSYSHGJ"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/N0QW1RGPR8/",
  "links": [
    {
      "field": "defines",
      "target": "37R4QJCARA"
    },
    {
      "field": "uses",
      "target": "0DG0Z72CQD"
    },
    {
      "field": "uses",
      "target": "0M3YQJ81NB"
    },
    {
      "field": "uses",
      "target": "8DJPSYSHGJ"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/N0QW1RGPR8.md",
  "msc": [
    "34M30",
    "30E15",
    "40G10"
  ],
  "name": "Resurgent function",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/N0Q/N0QW1RGPR8.md",
  "status": null,
  "tag": "N0QW1RGPR8",
  "uri": "m2://N0QW1RGPR8",
  "url": "https://math2.tamnd.com/wiki/resurgent-function-definition/",
  "writing": {
    "tag": "N0QW1RGPR8",
    "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
      }
    ]
  }
}