{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "M28SB04806"
    }
  ],
  "body": "\n## Summary\n\nThe envelope of holomorphy of a domain $D \\subseteq \\C^n$ is the largest domain to which every [[holomorphic function|507C34W0ZT]] on $D$ continues.\nThis domain may be spread over $\\C^n$ in several sheets.\nIt exists for every domain and is unique up to isomorphism.\nIn general it is a [[Riemann domain over $\\C^n$|A15BF6QHDR]] and not a subset of $\\C^n$.\nA domain is a [[domain of holomorphy|4487CSC0MB]] exactly when it is its own envelope of holomorphy.\n\n## Explanations\n\n### L1 human\n\nGiven a region, ask how far all of its holomorphic functions can be pushed at once.\nThe answer is the envelope of holomorphy: the region together with all its hidden extensions.\nFor the [[Hartogs figure|GHJC53RFBT]], a polydisc with a piece cut out, the envelope is the full polydisc.\nFor a ball with a smaller closed ball removed, the envelope is the whole ball.\nIn one variable nothing new appears: every domain is its own envelope.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "32D10",
      "32D26"
    ],
    "name": "Envelope of holomorphy",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Domain_of_holomorphy"
      }
    ],
    "tag": "X71JE6A3ZJ"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/X71JE6A3ZJ/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/X71JE6A3ZJ.md",
  "msc": [
    "32D10",
    "32D26"
  ],
  "name": "Envelope of holomorphy",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/X71/X71JE6A3ZJ.md",
  "status": null,
  "tag": "X71JE6A3ZJ",
  "uri": "m2://X71JE6A3ZJ",
  "url": "https://tamnd.github.io/math2/wiki/envelope-of-holomorphy/",
  "writing": {
    "tag": "X71JE6A3ZJ",
    "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
      }
    ]
  }
}