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  "body": "\n## Statement\n\nLet $0 \u003c k \u003c 1$ and $k' = \\sqrt{1 - k^2}$.\nLet $K = K(k)$ and $K' = K(k')$ be [[complete elliptic integrals of the first kind|RM54CPPE12]], and let $E = E(k)$ and $E' = E(k')$ be [[complete elliptic integrals of the second kind|VS2PSX14C8]].\nThen\n$$E K' + E' K - K K' = \\frac{\\pi}{2}.$$\n\n## Remarks\n\nProof idea: differentiation under the integral sign gives $\\frac{dE}{dk} = \\frac{E - K}{k}$ and $\\frac{dK}{dk} = \\frac{E - k'^2 K}{k k'^2}$.\nWith these rules, and the same rules for $E'$ and $K'$ via $dk'/dk = -k/k'$, the derivative of the left side in $k$ is $0$, so the left side is constant.\nTo find the constant, let $k \\to 0$: then $K, E \\to \\pi/2$ and $E' \\to 1$, while $K'$ grows only like $\\log(4/k)$ and $E - K = O(k^2)$, so $(E - K)K' \\to 0$ and the left side tends to $\\pi/2$.\nExample: at $k = 1/\\sqrt{2}$ one has $k = k'$, so $K = K'$ and $E = E'$, and the relation becomes $2EK - K^2 = \\pi/2$.\nThere is a parallel form for the periods $\\omega_1, \\omega_2$ and quasi-periods $\\eta_1, \\eta_2$ of the [[Weierstrass zeta function|QATW7046PD]]: $\\eta_1\\omega_2 - \\eta_2\\omega_1 = 2\\pi i$ when $\\operatorname{Im}(\\omega_2/\\omega_1) \u003e 0$.\n\n## Explanations\n\n### L1 human\n\nThe four complete elliptic integrals $K, E, K', E'$ are transcendental functions of $k$, and none of them has a simple closed form.\nYet this particular combination of them is always exactly $\\pi/2$, whatever the modulus.\nIt is one of the few exact links between these integrals and the number $\\pi$.\n\n### L2 human\n\nLegendre's relation is the key identity behind the Gauss–Legendre algorithm for $\\pi$, also called the Brent–Salamin algorithm.\nThat algorithm combines it with the [[arithmetic–geometric mean|E0V3N93DX8]] formula for $K$ and a similar formula for $E$.\nFor an elliptic curve it is the same fact as the quasi-period relation for the Weierstrass zeta function, read for a curve with real periods.\nIt is also used to compute $E$ from $K$ at special moduli, for example at $k = 1/\\sqrt 2$, where it gives $E(1/\\sqrt2) = \\frac{\\pi/2 + K^2}{2K}$ with $K = K(1/\\sqrt2)$, a value tied to the [[lemniscate constant|V5A2S8QYQQ]].\n",
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