Elliptic Function
engineering fundamentals Elliptic Functions
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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 phi is the amplitude, phi=am u, and x=sin(phi); k is the parameter, 0<k<1; n is the characteristic.

Complete Elliptic Integrals

For the amplitude phi=pi/2, 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 k is the parameter, 0<k<1; n is the characteristic.

Jacobi's Elliptic Functions

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


Jacobi's Elliptic Functions have the following properties:

DerivativeSum
 

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