{
  "aliases": [
    "sokhotski-theorem",
    "casorati-weierstrass-sokhotski-theorem"
  ],
  "backlinks": [
    {
      "field": "uses",
      "target": "08QA8QTM71"
    }
  ],
  "body": "\n## Statement\n\nLet $c \\in \\mathbb{C}$, let $r \u003e 0$, and let $f$ be [[holomorphic|0DG0Z72CQD]] on the punctured disc $0 \u003c |z - c| \u003c r$.\nAssume that $c$ is an [[essential singularity|0YKDJBV26S]] of $f$.\nThen for each $\\delta$ with $0 \u003c \\delta \u003c r$, the image $f(\\{z : 0 \u003c |z - c| \u003c \\delta\\})$ is dense in $\\mathbb{C}$.\nThus for each $w \\in \\mathbb{C}$ there is a sequence $z_n \\to c$ with $z_n \\neq c$ and $f(z_n) \\to w$.\n\n## Remarks\n\nProof idea: if the image of a punctured disc misses a disc $|w - w_0| \u003c \\varepsilon$, then $g(z) = 1/(f(z) - w_0)$ is bounded near $c$. By [[Riemann's theorem|0BSQDZWF34]], $g$ is holomorphic at $c$, and then $c$ is removable or a pole of $f$.\n[[Picard's great theorem|08QA8QTM71]] is a much stronger statement.\nIn the Russian literature, the theorem has the name of Sokhotski.\n\n## Explanations\n\n### L1 human\n\nNear an essential singularity, the function comes as close as you like to each complex number.\nFor example, $e^{1/z}$ comes close to $5$, to $-i$ and to $0$ on each small disc around $0$.\n",
  "commit": "f591989ed15704e9d8755a7a5a6426d73de7a3ae",
  "front": {
    "aliases": [
      "sokhotski-theorem",
      "casorati-weierstrass-sokhotski-theorem"
    ],
    "kind": "statement",
    "msc": [
      "30D30"
    ],
    "name": "Casorati–Weierstrass theorem",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q2226962",
      "wikipedia": "Casorati–Weierstrass theorem"
    },
    "sources": [
      {
        "locator": "Formal statement of the theorem",
        "url": "https://en.wikipedia.org/wiki/Casorati%E2%80%93Weierstrass_theorem"
      },
      {
        "locator": "Proof of the theorem",
        "url": "https://en.wikipedia.org/wiki/Casorati%E2%80%93Weierstrass_theorem"
      }
    ],
    "tag": "02HHNPC8ED",
    "uses": [
      "0YKDJBV26S",
      "0BSQDZWF34"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/02HHNPC8ED/",
  "links": [
    {
      "field": "uses",
      "target": "0YKDJBV26S"
    },
    {
      "field": "uses",
      "target": "0BSQDZWF34"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/02HHNPC8ED.md",
  "msc": [
    "30D30"
  ],
  "name": "Casorati–Weierstrass theorem",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/02H/02HHNPC8ED.md",
  "status": null,
  "tag": "02HHNPC8ED",
  "uri": "m2://02HHNPC8ED",
  "url": "https://math2.tamnd.com/wiki/casorati-weierstrass-theorem/",
  "writing": {
    "tag": "02HHNPC8ED",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}