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  "body": "\n## Summary\n\nThe Jacobi amplitude $\\operatorname{am}(u, k)$ is the inverse function of the incomplete [[elliptic integral|NXMCYGCB59]] of the first kind $u = F(\\varphi, k)$.\nIt is the angle $\\varphi$ whose sine and cosine are the [[Jacobi elliptic functions|P071B692JF]] $\\operatorname{sn} u$ and $\\operatorname{cn} u$.\nFor modulus $k = 0$ it is the identity, and for $k = 1$ it is the Gudermannian function.\n\n## Explanations\n\n### L1 human\n\nThe integral $F(\\varphi, k)$ measures something like a distorted angle: it adds up $d\\theta / \\sqrt{1 - k^2\\sin^2\\theta}$ from $0$ to $\\varphi$.\nThe amplitude goes the other way, from the value $u$ back to the angle $\\varphi$.\nFor a pendulum, $u$ is proportional to time and the amplitude tracks the angle of an auxiliary variable that turns unevenly.\nWhen $k = 0$ the integrand is $1$, so $F(\\varphi, 0) = \\varphi$ and $\\operatorname{am}(u, 0) = u$.\n",
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