Legendre form

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In mathematics, the Legendre forms of elliptic integrals, F(φ,k), E(φ,k) and Π(φ,k,n) are defined by

F(\phi,k) = \int_0^\phi \frac{1}{\sqrt{1 - k^2 \sin^2(t)}} dt,
E(\phi,k) = \int_0^\phi \sqrt{1 - k^2 \sin^2(t)}\,dt,

and

\Pi(\phi,k,n) = \int_0^\phi \frac{1}{(1 + n \sin^2(t))\sqrt{1 - k^2 \sin^2(t)}}\,dt.

The Legendre form of an elliptic curve is given by

y2 = x(x − 1)(x − λ)


[edit] See also


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