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 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:

 Derivative Sum