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  "body": "\n## Definition\n\nConsider the system of [[ordinary differential equations|0NT64JV96A]]\n$$\\operatorname{sm}'(z) = \\operatorname{cm}(z)^2, \\qquad \\operatorname{cm}'(z) = -\\operatorname{sm}(z)^2,$$\nwith initial values $\\operatorname{sm}(0) = 0$ and $\\operatorname{cm}(0) = 1$.\nThe *Dixon elliptic functions* $\\operatorname{sm}$ and $\\operatorname{cm}$ are its unique solution near $z = 0$, continued by [[analytic continuation|0M3YQJ81NB]] to meromorphic functions on $\\C$.\n\n## Remarks\n\nThe derivative of $\\operatorname{sm}^3 + \\operatorname{cm}^3$ is $3\\operatorname{sm}^2\\operatorname{cm}^2 - 3\\operatorname{cm}^2\\operatorname{sm}^2 = 0$, so $\\operatorname{sm}^3 + \\operatorname{cm}^3 = 1$ everywhere: the point $(\\operatorname{cm} z, \\operatorname{sm} z)$ lies on the Fermat cubic $x^3 + y^3 = 1$.\nHence $\\operatorname{sm}' = (1 - \\operatorname{sm}^3)^{2/3}$ near $0$, and $\\operatorname{sm}$ is the inverse of $x \\mapsto \\int_0^x dt/(1 - t^3)^{2/3}$ near $0$.\nBoth functions are [[elliptic|FF9ZPCZ59K]], with real period\n$$\\pi_3 = 3\\int_0^1 \\frac{dt}{(1 - t^3)^{2/3}} = B(1/3, 1/3) \\approx 5.29992.$$\nLet $\\omega = e^{2\\pi i/3}$.\nExample: $f(z) = \\omega^{-1}\\operatorname{sm}(\\omega z)$ and $g(z) = \\operatorname{cm}(\\omega z)$ satisfy $f' = g^2$, $g' = -\\omega^3 f^2 = -f^2$, $f(0) = 0$, $g(0) = 1$; by uniqueness $\\operatorname{sm}(\\omega z) = \\omega\\operatorname{sm}(z)$ and $\\operatorname{cm}(\\omega z) = \\operatorname{cm}(z)$.\nSo $\\omega\\pi_3$ is also a period, and the period lattice contains $\\pi_3(\\Z + \\Z\\omega)$.\nNon-example: $\\sin$ and $\\cos$ satisfy $\\sin' = \\cos$, not $\\cos^2$, and $\\sin^3 + \\cos^3$ is not constant (it is $1$ at $0$ and $2^{-1/2}$ at $\\pi/4$).\n",
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