{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "01991NCP7H"
    },
    {
      "field": "uses",
      "target": "0PN2BXRS8G"
    }
  ],
  "body": "\n## Definition\n\nLet $c$ be a [[pole|0WPQYW8MTE]] of $f$.\nThe order of the pole $c$ is the unique integer $m \\ge 1$ with the following property. On some punctured disc $0 \u003c |z - c| \u003c \\delta$,\n$$f(z) = \\frac{g(z)}{(z - c)^m},$$\nwhere $g$ is [[holomorphic|0DG0Z72CQD]] on the disc $|z - c| \u003c \\delta$ and $g(c) \\neq 0$.\nA pole of order $1$ is a simple pole.\n\n## Remarks\n\nThe order of the pole of $f$ at $c$ is the [[order of the zero|064N002TX4]] of $1/f$ at $c$.\nIn the Laurent series $\\sum a_n (z - c)^n$ of $f$ at $c$, the order is $m$ if and only if $a_{-m} \\neq 0$ and $a_n = 0$ for each $n \u003c -m$.\nMany authors give a pole of order $m$ the signed order $-m$. Then zeros have positive order and poles have negative order.\nFor example, $\\frac{1}{\\sin^2 z}$ has a pole of order $2$ at each point $k\\pi$ with $k \\in \\mathbb{Z}$.\n\n## Explanations\n\n### L1 human\n\nThe order of a pole says how fast the function goes to infinity at the point.\nNear a pole of order $m$, the function behaves like a constant times $1/(z - c)^m$.\n",
  "commit": "f591989ed15704e9d8755a7a5a6426d73de7a3ae",
  "front": {
    "defines": "073MWACF1K",
    "kind": "definition",
    "msc": [
      "30D30"
    ],
    "name": "Order of a pole",
    "sources": [
      {
        "locator": "Definitions",
        "url": "https://en.wikipedia.org/wiki/Zeros_and_poles"
      }
    ],
    "tag": "0YK27TJ4WF",
    "uses": [
      "0WPQYW8MTE",
      "064N002TX4"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/0YK27TJ4WF/",
  "links": [
    {
      "field": "defines",
      "target": "073MWACF1K"
    },
    {
      "field": "uses",
      "target": "0WPQYW8MTE"
    },
    {
      "field": "uses",
      "target": "064N002TX4"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0YK27TJ4WF.md",
  "msc": [
    "30D30"
  ],
  "name": "Order of a pole",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0YK/0YK27TJ4WF.md",
  "status": null,
  "tag": "0YK27TJ4WF",
  "uri": "m2://0YK27TJ4WF",
  "url": "https://math2.tamnd.com/wiki/order-of-a-pole-definition/",
  "writing": {
    "tag": "0YK27TJ4WF",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}