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.

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 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.

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

is a normalizing constant chosen such that

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: The only rational function satisfying the above properties is the elliptic rational function (Lutovac, Tošić & Evans 2001, § 13.2).

The following properties are derived: The elliptic rational function is normalized to unity at x=1: The nesting property is written: This is a very important property: The elliptic rational functions are related to the Chebyshev polynomials of the first kind

The following inversion relationship holds: This implies that poles and zeroes come in pairs such that Odd order functions will have a zero at x=0 and a corresponding pole at infinity.

The zeroes of the elliptic rational function of order n will be written

The zeroes of the elliptic rational function will be the zeroes of the polynomial in the numerator of the function.

The following derivation of the zeroes of the elliptic rational function is analogous to that of determining the zeroes of the Chebyshev polynomials (Lutovac, Tošić & Evans 2001, § 12.6).

Using the fact that for any z the defining equation for the elliptic rational functions implies that so that the zeroes are given by Using the inversion relationship, the poles may then be calculated.

From the nesting property, if the zeroes of

can be algebraically expressed (i.e. without the need for calculating the Jacobi ellipse functions) then the zeroes of

In particular, the zeroes of elliptic rational functions of order

may be algebraically expressed (Lutovac, Tošić & Evans 2001, § 12.9, 13.9).

as follows: Define Then, from the nesting property and knowing that where

we have: These last three equations may be inverted: To calculate the zeroes of

in the third equation, calculate the two values of

in the second equation to calculate four values of

and finally, use these values in the first equation to calculate the eight zeroes of

are calculated by a similar recursion.)

Again, using the inversion relationship, these zeroes can be used to calculate the poles.

We may write the first few elliptic rational functions as: See Lutovac, Tošić & Evans (2001, § 13) for further explicit expressions of order n=5 and

The corresponding discrimination factors are: The corresponding zeroes are

where n is the order and j is the number of the zero.

There will be a total of n zeroes for each order.

From the inversion relationship, the corresponding poles

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 .
Plot of the absolute value of the third order elliptic rational function with ξ=1.4. There is a zero at x=0 and the pole at infinity. Since the function is antisymmetric, it is seen there are three zeroes and three poles. Between the zeroes, the function rises to a value of 1 and, between the poles, the function drops to the value of the discrimination factor L n
Plot of the absolute value of the fourth order elliptic rational function with ξ=1.4. Since the function is symmetric, it is seen that there are four zeroes and four poles. Between the zeroes, the function rises to a value of 1 and, between the poles, the function drops to the value of the discrimination factor L n
Plot of the effect of the selectivity factor ξ. The fourth order elliptic rational function is shown with values of ξ varying from nearly unity to infinity. The black curve, corresponding to ξ=∞ is the Chebyshev polynomial of order 4. The closer the selectivity factor is to unity, the steeper will be the slope in the transition region between x=1 and x=ξ.