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  "body": "\n## Definition\n\nLet $0 \u003c k \u003c 1$, and let $\\operatorname{am}(u) = \\operatorname{am}(u, k)$ be the [[Jacobi amplitude|WDHHJHVP7N]].\nFor real $u$, the *Jacobi elliptic functions* with modulus $k$ are\n$$\\operatorname{sn} u = \\sin\\operatorname{am} u, \\qquad \\operatorname{cn} u = \\cos\\operatorname{am} u, \\qquad \\operatorname{dn} u = \\sqrt{1 - k^2\\sin^2\\operatorname{am} u}.$$\nEach of them extends by [[analytic continuation|0M3YQJ81NB]] to a meromorphic function on $\\C$, still written $\\operatorname{sn}$, $\\operatorname{cn}$, $\\operatorname{dn}$.\n\n## Remarks\n\nLet $K = K(k)$ and $K' = K(\\sqrt{1 - k^2})$ be [[complete elliptic integrals of the first kind|RM54CPPE12]].\nThe extended functions are [[elliptic functions|FF9ZPCZ59K]] with these periods:\n$\\operatorname{sn}$ has periods $4K$ and $2iK'$; $\\operatorname{cn}$ has periods $4K$ and $2K + 2iK'$; $\\operatorname{dn}$ has periods $2K$ and $4iK'$.\nThey satisfy\n$$\\operatorname{sn}^2 u + \\operatorname{cn}^2 u = 1, \\qquad \\operatorname{dn}^2 u + k^2\\operatorname{sn}^2 u = 1,$$\n$$\\frac{d}{du}\\operatorname{sn} u = \\operatorname{cn} u \\operatorname{dn} u, \\quad \\frac{d}{du}\\operatorname{cn} u = -\\operatorname{sn} u \\operatorname{dn} u, \\quad \\frac{d}{du}\\operatorname{dn} u = -k^2\\operatorname{sn} u \\operatorname{cn} u.$$\nExample: at $u = K$ one has $\\operatorname{am} K = \\pi/2$, so $\\operatorname{sn} K = 1$, $\\operatorname{cn} K = 0$ and $\\operatorname{dn} K = k'$, where $k' = \\sqrt{1 - k^2}$.\nCheck of the first derivative rule: $\\frac{d}{du}\\sin\\operatorname{am} u = \\cos(\\operatorname{am} u)\\cdot\\frac{d\\operatorname{am}}{du} = \\operatorname{cn} u \\operatorname{dn} u$.\nNon-example: $\\sin$ is not one of these functions for $0 \u003c k \u003c 1$, since $\\sin$ is entire while $\\operatorname{sn}$ has poles, at $u = iK'$ for example.\nLimits: for $k \\to 0$ they become $\\sin$, $\\cos$, $1$, and the formulas with $k = 1$ give $\\tanh$, $\\operatorname{sech}$, $\\operatorname{sech}$.\n",
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