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  "body": "\n## Definition\n\nLet $R(x, y)$ be a [[rational function|0XDK9CGYKJ]] of two variables, and let $P$ be a polynomial of degree $3$ or $4$ with no repeated roots.\nAn *elliptic integral* is an integral of the form\n$$f(x) = \\int_c^x R\\big(t, \\sqrt{P(t)}\\big) \\, dt,$$\nwhere $c$ is a constant and $\\sqrt{\\cdot}$ is a [[square root|0F7FQS6XY6]].\n\nLet $0 \\le k \u003c 1$, let $\\varphi$ be real, and let $n$ be a real parameter with $n \\sin^2\\theta \\ne 1$ for $\\theta$ between $0$ and $\\varphi$.\nThe *incomplete elliptic integrals in Legendre form* of the first, second and third kind are\n$$F(\\varphi, k) = \\int_0^\\varphi \\frac{d\\theta}{\\sqrt{1 - k^2 \\sin^2\\theta}}, \\qquad E(\\varphi, k) = \\int_0^\\varphi \\sqrt{1 - k^2 \\sin^2\\theta} \\, d\\theta,$$\n$$\\Pi(n; \\varphi, k) = \\int_0^\\varphi \\frac{d\\theta}{(1 - n \\sin^2\\theta)\\sqrt{1 - k^2 \\sin^2\\theta}}.$$\nThe number $k$ is the *modulus*.\n\n## Remarks\n\nThe substitution $t = \\sin\\theta$ turns $F(\\varphi, k)$ into $\\int_0^{\\sin\\varphi} dt / \\sqrt{(1 - t^2)(1 - k^2 t^2)}$, an elliptic integral with $P(t) = (1 - t^2)(1 - k^2 t^2)$, which has the four distinct roots $\\pm 1$ and $\\pm 1/k$ when $0 \u003c k \u003c 1$.\nExample: the ellipse $x = a\\sin\\theta$, $y = b\\cos\\theta$ with $0 \u003c b \u003c a$ has arc length element $\\sqrt{a^2\\cos^2\\theta + b^2\\sin^2\\theta}\\,d\\theta = a\\sqrt{1 - e^2\\sin^2\\theta}\\,d\\theta$, where $e^2 = 1 - b^2/a^2$; so its arc length from $\\theta = 0$ to $\\theta = \\varphi$ is $a E(\\varphi, e)$.\nNon-example: $\\int_0^x dt / \\sqrt{1 - t^2} = \\arcsin x$ is not an elliptic integral, since $P(t) = 1 - t^2$ has degree $2$.\nEvery elliptic integral is a sum of elementary functions and constant multiples of integrals of the three Legendre kinds.\nThe values at $\\varphi = \\pi/2$ are the complete integrals $K(k) = F(\\pi/2, k)$, [[of the first kind|RM54CPPE12]], and $E(k) = E(\\pi/2, k)$, [[of the second kind|VS2PSX14C8]].\n",
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