{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $0 \u003c k \u003c 1$, and let $\\operatorname{sn}$, $\\operatorname{cn}$, $\\operatorname{dn}$ be the [[Jacobi elliptic functions|P071B692JF]] with modulus $k$.\nFor all complex $u, v$ such that both sides are defined, with\n$$\\Delta = 1 - k^2\\operatorname{sn}^2 u \\operatorname{sn}^2 v,$$\none has\n$$\\operatorname{sn}(u + v) = \\frac{\\operatorname{sn} u \\operatorname{cn} v \\operatorname{dn} v + \\operatorname{sn} v \\operatorname{cn} u \\operatorname{dn} u}{\\Delta},$$\n$$\\operatorname{cn}(u + v) = \\frac{\\operatorname{cn} u \\operatorname{cn} v - \\operatorname{sn} u \\operatorname{sn} v \\operatorname{dn} u \\operatorname{dn} v}{\\Delta},$$\n$$\\operatorname{dn}(u + v) = \\frac{\\operatorname{dn} u \\operatorname{dn} v - k^2 \\operatorname{sn} u \\operatorname{sn} v \\operatorname{cn} u \\operatorname{cn} v}{\\Delta}.$$\n\n## Remarks\n\nProof idea: fix $c$ and set $v = c - u$; using the derivative rules $\\operatorname{sn}' = \\operatorname{cn}\\operatorname{dn}$, $\\operatorname{cn}' = -\\operatorname{sn}\\operatorname{dn}$, $\\operatorname{dn}' = -k^2\\operatorname{sn}\\operatorname{cn}$ and the relations $\\operatorname{sn}^2 + \\operatorname{cn}^2 = 1$, $\\operatorname{dn}^2 + k^2\\operatorname{sn}^2 = 1$, one checks that each right side has derivative $0$ in $u$.\nSo each right side depends only on $u + v = c$, and putting $u = 0$ (where $\\operatorname{sn} 0 = 0$, $\\operatorname{cn} 0 = \\operatorname{dn} 0 = 1$) shows it equals the left side.\nBy analytic continuation the identities hold wherever both sides are defined.\nExample: for $k \\to 0$ the formulas become $\\sin(u + v) = \\sin u\\cos v + \\sin v\\cos u$ and the cosine addition formula, with $\\operatorname{dn} = 1$ and $\\Delta = 1$.\nWith $k = 1$, where $\\operatorname{sn} = \\tanh$ and $\\operatorname{cn} = \\operatorname{dn} = \\operatorname{sech}$, the first formula becomes $\\tanh(u + v) = (\\tanh u + \\tanh v)/(1 + \\tanh u\\tanh v)$, after factoring $1 - \\tanh^2 u\\tanh^2 v = (1 - \\tanh u\\tanh v)(1 + \\tanh u\\tanh v)$.\nExample of use: with $v = K$, where $\\operatorname{sn} K = 1$, $\\operatorname{cn} K = 0$, $\\operatorname{dn} K = k'$, one gets $\\operatorname{sn}(u + K) = \\operatorname{cn} u / \\operatorname{dn} u$.\n\n## Explanations\n\n### L1 human\n\nThe sine and cosine of a sum can be written with the sines and cosines of the parts.\nThe Jacobi elliptic functions have the same property, but the formulas carry the denominator $1 - k^2\\operatorname{sn}^2 u\\operatorname{sn}^2 v$.\nWhen the modulus $k$ is $0$, the denominator is $1$ and the familiar trigonometric formulas come back.\n\n### L2 human\n\nThe addition formulas reflect the group law on the elliptic curve $y^2 = (1 - x^2)(1 - k^2x^2)$, parametrised by $x = \\operatorname{sn} u$, $y = \\operatorname{cn} u\\operatorname{dn} u$.\nThey give the shifts by quarter and half periods, such as $\\operatorname{sn}(u + 2K) = -\\operatorname{sn} u$, from the special values at $K$ and $2K$.\nThey are also the algebraic addition theorem behind Euler's addition theorem for the integral $\\int dt/\\sqrt{(1 - t^2)(1 - k^2t^2)}$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "P071B692JF"
    ],
    "kind": "statement",
    "msc": [
      "33E05"
    ],
    "name": "Addition formulas for Jacobi elliptic functions",
    "role": "theorem",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Jacobi_elliptic_functions"
      }
    ],
    "tag": "QWSRBJ1G1E",
    "uses": [
      "P071B692JF"
    ]
  },
  "kind": "statement",
  "link": "https://tamnd.github.io/math2/node/QWSRBJ1G1E/",
  "links": [
    {
      "field": "about",
      "target": "P071B692JF"
    },
    {
      "field": "uses",
      "target": "P071B692JF"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/QWSRBJ1G1E.md",
  "msc": [
    "33E05"
  ],
  "name": "Addition formulas for Jacobi elliptic functions",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/QWS/QWSRBJ1G1E.md",
  "status": null,
  "tag": "QWSRBJ1G1E",
  "uri": "m2://QWSRBJ1G1E",
  "url": "https://tamnd.github.io/math2/wiki/addition-formulas-for-jacobi-elliptic-functions/",
  "writing": {
    "tag": "QWSRBJ1G1E",
    "kind": "statement",
    "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
      }
    ]
  }
}