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  "body": "\n## Summary\n\nThe Jacobi elliptic functions $\\operatorname{sn}$, $\\operatorname{cn}$ and $\\operatorname{dn}$ are [[elliptic functions|FF9ZPCZ59K]] that depend on a modulus $k$ and generalise the sine and cosine.\nThey come from inverting the incomplete [[elliptic integral|NXMCYGCB59]] of the first kind, through the [[Jacobi amplitude|WDHHJHVP7N]].\nThey satisfy $\\operatorname{sn}^2 + \\operatorname{cn}^2 = 1$ and simple differential equations, and they solve the pendulum equation exactly.\nNine further functions such as $\\operatorname{sd} = \\operatorname{sn}/\\operatorname{dn}$ are formed as quotients.\n\n## Explanations\n\n### L1 human\n\nThe sine is the inverse of the integral $\\int_0^x dt/\\sqrt{1 - t^2}$.\nIf one puts a second factor under the root and inverts $\\int_0^x dt/\\sqrt{(1 - t^2)(1 - k^2t^2)}$, one gets $\\operatorname{sn}$.\nFor $k = 0$ the functions reduce to $\\sin$, $\\cos$ and the constant $1$; for $k = 1$ they become $\\tanh$, $\\operatorname{sech}$ and $\\operatorname{sech}$.\nIn between they look like sine waves that have been flattened at the tops.\nUnlike the sine, they also repeat in an imaginary direction, so they are doubly periodic in the complex plane.\n",
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