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Elliptic Integrals
With careful standardization of argument conventions, Mathematica provides full coverage of elliptic integrals, with arbitrary-precision numerical evaluation for complex values of all parameters, as well as extensive symbolic transformations and simplifications.
EllipticK complete elliptic integral of the first kind
EllipticF incomplete elliptic integral of the first kind
EllipticE complete and incomplete elliptic integral of the second kind
EllipticPi complete and incomplete elliptic integral of the third kind
    
EllipticNomeQ convert from parameter m to nome q
InverseEllipticNomeQ convert from nome q to parameter m
JacobiAmplitude convert from argument u and parameter m to amplitude
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