Elliptic Integrals 
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. 
Complete Elliptic Integrals 
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. 
Jacobi's Elliptic Functions  
where is the amplitude defined in the elliptic integral of the first kind. In addition,
Jacobi's Elliptic Functions have the following properties:
