| Elliptic Integrals |
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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 |
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For the amplitude 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 |
, the elliptic integrals are said to be complete.
| Jacobi's Elliptic Functions | |||||||||||||||||||
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   where
Jacobi's Elliptic Functions have the following properties:
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is the amplitude 
