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Elliptic Rational Functions

# Definition

We define the elliptic rational function in terms of the Jacobi elliptic functions as

where cd is one of the 12 Jacobi elliptic functions (see JacobiCD in [9]), is the inverse cd Jacobi elliptic function (see InverseJacobiCD in [9]), is the complete elliptic integral of the first kind (see EllipticK in [9]), is the order (a positive integer), is the selectivity factor (), and is the discrimination factor defined in the Introduction. It can also be defined as the value of the elliptic rational function for , that is, [4].

can be represented by using the parametric equations

where is an intermediate variable [4].

Traditionally, for known , , and , can be computed as follows.

1. Find using EllipticK.

2. Find from the inverse cd Jacobi elliptic function.

3. Determine and from the degree equation

4. Find using the cd Jacobi elliptic function.

The Chebyshev polynomial can be derived from for because , that is,

The elliptic rational function in terms of the Jacobi elliptic functions can be expanded as a rational function in terms of [4]. The explicit formulas of the first three functions are

In Mathematica, can be represented as follows.

Note that JacobiDN requires instead of .

The meaning of the Jacobi elliptic function notations follows: means is the modulus, means is the parameter. The most common notation uses the form with , but the elliptic functions are implemented in Mathematica using the form with instead.