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  "body": "\n## Summary\n\nThe Carlson symmetric forms $R_F$, $R_J$, $R_C$ and $R_D$ are a standard set of [[elliptic integrals|NXMCYGCB59]] that are symmetric in their main arguments.\nEvery elliptic integral can be written with them, and in particular the Legendre integrals $F$, $E$ and $\\Pi$.\nTheir symmetry gives a duplication theorem that leads to fast and stable numerical algorithms, which is why software libraries use them.\n\n## Explanations\n\n### L1 human\n\nLegendre's forms of the elliptic integrals depend on an angle and a modulus, and the formulas that transform them are lopsided.\nCarlson's forms put the three roots of the cubic under the square root on an equal footing, so permuting $x$, $y$, $z$ does not change the value.\nFor example $R_F(x, y, z) = \\frac{1}{2}\\int_0^\\infty dt / \\sqrt{(t + x)(t + y)(t + z)}$, and $R_F(x, x, x) = 1/\\sqrt{x}$.\nThe complete integral of the first kind is $K(k) = R_F(0, 1 - k^2, 1)$.\n",
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