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.

For the amplitude , the elliptic integrals are said to be complete.

Complete Elliptic Integral of the First Kind: See plot.

Complete Elliptic Integral of the Second Kind: See plot.

Complete Elliptic Integral of the Third Kind: See plots.

where is the parameter, ; is the characteristic.

where is the amplitude defined in the elliptic integral of the first kind. In addition,