Elliptic Integrals

There are three basic forms of elliptic integrals, only two of which are implemented in this research at this time.  Elliptic integrals of the first kind have the form:

, for .

The elliptic integral of the second kind has the form:

, for .

Complete elliptic integrals of the first and second kind are represented by:

and , respectively.

Jacobi Elliptic Functions

Elliptic functions are defined as the inverses of functions developed by elliptic integrals.  The notation used for the specific functions below was developed by Jacobi.

The amplitude of Jacobi elliptic functions is defined as the inverse of the elliptic integral of the first kind.

If , then .

We then can define three basic Jacobi elliptic functions in the following way:

The remaining Jacobi elliptic functions are of the form:

,

where p and q are elements of the set {c,d,n,s} and is defined to be 1.  The period of the functions sn(x|m) and cn(x|m) is 4K(m) and the period of dn(x|m) is 2K(m).

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