Elliptic rational functions

In mathematics the elliptic rational functions are a sequence of rational functions with real coefficients. Elliptic rational functions are extensively used in the design of elliptic electronic filters. (These functions are sometimes called Chebyshev rational functions, not to be confused with certain other functions of the same name).

Rational elliptic functions are identified by a positive integer order n and include a parameter ξ ≥ 1 called the selectivity factor. A rational elliptic function of degree n in x with selectivity factor ξ is generally defined as:

For many cases, in particular for orders of the form n = 2^a3^b where a and b are integers, the elliptic rational functions can be expressed using algebraic functions alone. Elliptic rational functions are closely related to the Chebyshev polynomials: Just as the circular trigonometric functions are special cases of the Jacobi elliptic functions, so the Chebyshev polynomials are special cases of the elliptic rational functions.

For even orders, the elliptic rational functions may be expressed as a ratio of two polynomials, both of order n.

where ${\displaystyle x_{i}}$ are the zeroes and ${\displaystyle x_{pi}}$ are the poles, and ${\displaystyle r_{0}}$ is a normalizing constant chosen such that ${\displaystyle R_{n}(\xi ,1)=1}$. The above form would be true for even orders as well except that for odd orders, there will be a pole at x=∞ and a zero at x=0 so that the above form must be modified to read:

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