Legendre form

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

:$y^2\; =\; x(x\; -\; 1)(x\; -\; lambda)$

ee also

* Carlson symmetric form