Plot of elliptic rational functions for x between -1 and 1 for orders 1,2,3 and 4 with discrimination factor ξ=1.1. All are bounded between -1 and 1 and all have the value 1 at x=1.
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:
where
cd(u,k) is the Jacobi elliptic cosine function.
K() is a complete elliptic integral of the first kind.
is the discrimination factor, equal to the minimum value of the magnitude of for .
For many cases, in particular for orders of the form n = 2a3b 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.
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