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:

${\displaystyle R_{n}(\xi ,x)\equiv \mathrm {cd} \left(n{\frac {K(1/L_{n}(\xi ))}{K(1/\xi )}}\,\mathrm {cd} ^{-1}(x,1/\xi ),1/L_{n}(\xi )\right)}$

where

• cd(u,k) is the Jacobi elliptic cosine function.
• K() is a complete elliptic integral of the first kind.
• ${\displaystyle L_{n}(\xi )=R_{n}(\xi ,\xi )}$ is the discrimination factor, equal to the minimum value of the magnitude of ${\displaystyle R_{n}(\xi ,x)}$ for ${\displaystyle |x|\geq \xi }$.

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.