{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $k$ be a real number with $0 \\le k \\le 1$.\nThe *complete elliptic integral of the second kind* is\n$$E(k) = \\int_0^{\\pi/2} \\sqrt{1 - k^2 \\sin^2\\theta} \\, d\\theta = \\int_0^1 \\frac{\\sqrt{1 - k^2 t^2}}{\\sqrt{1 - t^2}} \\, dt.$$\nIt is the value at $\\varphi = \\pi/2$ of the incomplete [[elliptic integral|NXMCYGCB59]] $E(\\varphi, k)$ of the second kind.\nWith $k' = \\sqrt{1 - k^2}$ one writes $E'(k) = E(k')$.\n\n## Remarks\n\nExample: $E(0) = \\pi/2$, and $E(1) = \\int_0^{\\pi/2} \\cos\\theta \\, d\\theta = 1$.\nThe ellipse $x = a\\sin\\theta$, $y = b\\cos\\theta$ with $0 \u003c b \\le a$ has perimeter $4aE(e)$, where $e = \\sqrt{1 - b^2/a^2}$ is its eccentricity, because its arc length element is $a\\sqrt{1 - e^2\\sin^2\\theta}\\,d\\theta$.\nNon-example: for real $k \u003e 1$ the integrand $\\sqrt{1 - k^2\\sin^2\\theta}$ is not real for $\\theta$ near $\\pi/2$, so the formula does not define a real number there.\nSeries: $E(k) = \\frac{\\pi}{2}\\left(1 - \\sum_{n=1}^{\\infty} \\left(\\frac{(2n)!}{2^{2n}(n!)^2}\\right)^2 \\frac{k^{2n}}{2n - 1}\\right)$.\nSome authors write $E(m)$ with the parameter $m = k^2$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "VS2PSX14C8",
    "kind": "definition",
    "msc": [
      "33E05"
    ],
    "name": "Complete elliptic integral of the second kind",
    "sources": [
      {
        "locator": "Complete elliptic integral of the second kind",
        "url": "https://en.wikipedia.org/wiki/Elliptic_integral"
      }
    ],
    "tag": "CA49GFN9ZA",
    "uses": [
      "NXMCYGCB59",
      "0W29TBTY12"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/CA49GFN9ZA/",
  "links": [
    {
      "field": "defines",
      "target": "VS2PSX14C8"
    },
    {
      "field": "uses",
      "target": "NXMCYGCB59"
    },
    {
      "field": "uses",
      "target": "0W29TBTY12"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/CA49GFN9ZA.md",
  "msc": [
    "33E05"
  ],
  "name": "Complete elliptic integral of the second kind",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/CA4/CA49GFN9ZA.md",
  "status": null,
  "tag": "CA49GFN9ZA",
  "uri": "m2://CA49GFN9ZA",
  "url": "https://tamnd.github.io/math2/wiki/complete-elliptic-integral-of-the-second-kind-definition/",
  "writing": {
    "tag": "CA49GFN9ZA",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}