With careful standardization of argument conventions, the Wolfram Language 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

JacobiZeta  ▪  EllipticLog  ▪  ArithmeticGeometricMean  ▪  InverseWeierstrassP

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 ϕ

CarlsonRF — Carlson's elliptic integral functions

CarlsonRK  ▪  CarlsonRE  ▪  CarlsonRM  ▪  CarlsonRG  ▪  CarlsonRJ  ▪  CarlsonRD  ▪  CarlsonRC