{
  "aliases": null,
  "backlinks": [
    {
      "field": "about",
      "target": "8GSKDNZ4TK"
    },
    {
      "field": "about",
      "target": "A0V7ZAEQBW"
    },
    {
      "field": "about",
      "target": "E0V3N93DX8"
    },
    {
      "field": "uses",
      "target": "8GSKDNZ4TK"
    },
    {
      "field": "uses",
      "target": "A0V7ZAEQBW"
    },
    {
      "field": "uses",
      "target": "E0V3N93DX8"
    },
    {
      "field": "uses",
      "target": "EP681WKN00"
    },
    {
      "field": "uses",
      "target": "T8NQJ0CBJM"
    },
    {
      "field": "uses",
      "target": "TN2R1TDE3G"
    },
    {
      "field": "uses",
      "target": "V5A2S8QYQQ"
    },
    {
      "field": "uses",
      "target": "W43SYCP85W"
    },
    {
      "field": "defines",
      "target": "Z6F9V4AHHH"
    }
  ],
  "body": "\n## Summary\n\nThe complete elliptic integral of the first kind is the function $K(k) = \\int_0^{\\pi/2} d\\theta / \\sqrt{1 - k^2\\sin^2\\theta}$ of the modulus $k$.\nIt is a quarter of the real period of the [[Jacobi elliptic functions|P071B692JF]] $\\operatorname{sn}$ and $\\operatorname{cn}$.\nIt gives the exact period of a pendulum and can be computed very fast with the arithmetic–geometric mean.\n\n## Explanations\n\n### L1 human\n\nFor a pendulum with small swings, the period does not depend on the amplitude.\nFor large swings it does, and the correction is given by $K(k)$, where $k$ is the sine of half the largest angle.\nAt $k = 0$ the integrand is $1$ and $K(0) = \\pi/2$.\nAs $k$ grows toward $1$ the integral grows without bound, because the pendulum takes longer and longer to swing near the top.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "33E05"
    ],
    "name": "Complete elliptic integral of the first kind",
    "sources": [
      {
        "locator": "Complete elliptic integral of the first kind",
        "url": "https://en.wikipedia.org/wiki/Elliptic_integral"
      }
    ],
    "specializes": [
      "NXMCYGCB59"
    ],
    "tag": "RM54CPPE12"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/RM54CPPE12/",
  "links": [
    {
      "field": "specializes",
      "target": "NXMCYGCB59"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/RM54CPPE12.md",
  "msc": [
    "33E05"
  ],
  "name": "Complete elliptic integral of the first kind",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/RM5/RM54CPPE12.md",
  "status": null,
  "tag": "RM54CPPE12",
  "uri": "m2://RM54CPPE12",
  "url": "https://tamnd.github.io/math2/wiki/complete-elliptic-integral-of-the-first-kind/",
  "writing": {
    "tag": "RM54CPPE12",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}