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  "body": "\n## Definition\n\nLet $0 \\le k \u003c 1$.\nFor real $\\varphi$ let $F(\\varphi, k) = \\int_0^\\varphi d\\theta / \\sqrt{1 - k^2\\sin^2\\theta}$ be the incomplete [[elliptic integral|NXMCYGCB59]] of the first kind.\nThe map $\\varphi \\mapsto F(\\varphi, k)$ is a strictly increasing bijection from $\\R$ onto $\\R$.\nThe *Jacobi amplitude* $\\operatorname{am}(\\cdot, k) \\colon \\R \\to \\R$ is its inverse function: $\\varphi = \\operatorname{am}(u, k)$ exactly when $u = F(\\varphi, k)$.\n\n## Remarks\n\nThe map is strictly increasing because its derivative $1/\\sqrt{1 - k^2\\sin^2\\varphi}$ lies between $1$ and $1/k'$, where $k' = \\sqrt{1 - k^2}$; the lower bound $1$ makes it unbounded in both directions.\nSo $\\operatorname{am}(\\cdot, k)$ is smooth, odd, and $\\frac{d}{du}\\operatorname{am}(u, k) = \\sqrt{1 - k^2\\sin^2\\operatorname{am}(u, k)}$.\nSince the integrand has period $\\pi$, $F(\\varphi + \\pi, k) = F(\\varphi, k) + 2K$, with $K = K(k)$ the [[complete elliptic integral of the first kind|RM54CPPE12]]; hence $\\operatorname{am}(u + 2K, k) = \\operatorname{am}(u, k) + \\pi$.\nExample: for $k = 0$, $F(\\varphi, 0) = \\varphi$, so $\\operatorname{am}(u, 0) = u$; and $\\operatorname{am}(K, k) = \\pi/2$ for every $k$.\nNon-example: for $k = 1$ the integral $F(\\varphi, 1) = \\int_0^\\varphi d\\theta/\\cos\\theta$ only makes sense for $\\abs{\\varphi} \u003c \\pi/2$, so the definition above does not apply.\nThe inverse there is the Gudermannian function $\\operatorname{am}(u, 1) = 2\\arctan(e^u) - \\pi/2$, a bijection from $\\R$ onto $(-\\pi/2, \\pi/2)$.\nThe Jacobi elliptic functions are $\\operatorname{sn} u = \\sin\\operatorname{am} u$ and $\\operatorname{cn} u = \\cos\\operatorname{am} u$.\n",
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