Legendre form

In mathematics, the Legendre forms of elliptic integrals, F(φ,k), E(φ,k) and P(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

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

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