An elliptic integral is an integral involving a rational function which contains square roots of cubic or quartic polynomials. Generally, the elliptic integrals CANNOT be expressed in terms of elementary functions.
Elliptic Integral of the First Kind: See plots.
Elliptic Integral of the Second Kind: See Plots.
Elliptic Integral of the Third Kind:
where is the amplitude, , and ; is the parameter, ; is the characteristic.