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  "body": "\n## Definition\n\nLet $x, y, z$ be nonnegative real numbers, at most one of which is $0$, and let $p \u003e 0$.\nThe *Carlson symmetric forms* are [[improper integrals|05KQ3F99BQ]].\nThey are\n$$R_F(x, y, z) = \\frac{1}{2} \\int_0^\\infty \\frac{dt}{\\sqrt{(t + x)(t + y)(t + z)}},$$\n$$R_J(x, y, z, p) = \\frac{3}{2} \\int_0^\\infty \\frac{dt}{(t + p)\\sqrt{(t + x)(t + y)(t + z)}},$$\ntogether with the special cases\n$$R_C(x, y) = R_F(x, y, y), \\qquad R_D(x, y, z) = R_J(x, y, z, z),$$\nwhere for $R_C$ one takes $x \\ge 0$ and $y \u003e 0$, and for $R_D$ one takes $z \u003e 0$.\n\n## Remarks\n\nThe integrals converge: near $t = \\infty$ the integrand of $R_F$ behaves like $t^{-3/2}$, and near $t = 0$ at most one factor vanishes, which gives at worst $t^{-1/2}$.\n$R_F$ is symmetric in $x, y, z$, $R_J$ in $x, y, z$, and $R_D$ in $x, y$.\nExample: $R_F(x, x, x) = \\frac12\\int_0^\\infty (t + x)^{-3/2}\\,dt = x^{-1/2}$ for $x \u003e 0$.\nNon-example: $R_F(0, 0, 1)$ is not defined, since two arguments are $0$ and the integrand $1/(t\\sqrt{t + 1})$ is not integrable at $t = 0$.\nWith $k' = \\sqrt{1 - k^2}$ and $0 \\le k \u003c 1$, the complete [[elliptic integrals|NXMCYGCB59]] are\n$$K(k) = R_F(0, k'^2, 1), \\qquad E(k) = R_F(0, k'^2, 1) - \\frac{k^2}{3} R_D(0, k'^2, 1).$$\n$R_C$ is elementary; for $0 \\le x \u003c y$ it equals $\\arccos(\\sqrt{x/y}) / \\sqrt{y - x}$.\n",
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