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.
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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,
is the amplitude, 
, and 
; 
is the parameter, 
; 
is the characteristic.
, the elliptic integrals are said to be complete.
is the amplitude 
