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  "body": "\n## Definition\n\nLet $k$ be a real number with $0 \\le k \u003c 1$.\nThe *complete elliptic integral of the first kind* is\n$$K(k) = \\int_0^{\\pi/2} \\frac{d\\theta}{\\sqrt{1 - k^2 \\sin^2\\theta}} = \\int_0^1 \\frac{dt}{\\sqrt{(1 - t^2)(1 - k^2 t^2)}}.$$\nIt is the value at $\\varphi = \\pi/2$ of the incomplete [[elliptic integral|NXMCYGCB59]] $F(\\varphi, k)$ of the first kind.\nThe *complementary modulus* is $k' = \\sqrt{1 - k^2}$, and one writes $K'(k) = K(k')$.\n\n## Remarks\n\nThe second integral comes from the first by $t = \\sin\\theta$; it is an [[improper integral|05KQ3F99BQ]] at $t = 1$ that converges.\nExample: $K(0) = \\int_0^{\\pi/2} d\\theta = \\pi/2$.\nExpanding $(1 - k^2\\sin^2\\theta)^{-1/2}$ by the binomial series and integrating term by term gives\n$$K(k) = \\frac{\\pi}{2} \\sum_{n=0}^{\\infty} \\left( \\frac{(2n)!}{2^{2n} (n!)^2} \\right)^2 k^{2n}.$$\nNon-example: the formula does not define a finite number at $k = 1$, since $\\int_0^{\\pi/2} d\\theta / \\cos\\theta$ diverges; indeed $K(k) \\to \\infty$ as $k \\to 1^-$.\nSome authors use the parameter $m = k^2$ and write $K(m)$; the value is the same function written in another variable.\nThe [[arithmetic–geometric mean|E0V3N93DX8]] gives $K(k) = \\pi / (2 M(1, k'))$.\n",
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