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 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.

Expression as a ratio of polynomials

For even orders, the elliptic rational functions may be expressed as a ratio of two polynomials, both of order n. where x_i are the zeroes and x_{pi} are the poles, and r_0 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:

Properties

The canonical properties

The only rational function satisfying the above properties is the elliptic rational function. The following properties are derived:

Normalization

The elliptic rational function is normalized to unity at x=1:

Nesting property

The nesting property is written: This is a very important property:

Limiting values

The elliptic rational functions are related to the Chebyshev polynomials of the first kind T_n(x) by:

Symmetry

Equiripple

R_n(\xi,x) has equal ripple of \pm 1 in the interval. By the inversion relationship (see below), it follows that has equiripple in of.

Inversion relationship

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.

Poles and Zeroes

The zeroes of the elliptic rational function of order n will be written x_{ni}(\xi) or x_{ni} when \xi is implicitly known. 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. 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 R_m and R_n can be algebraically expressed (i.e. without the need for calculating the Jacobi ellipse functions) then the zeroes of can be algebraically expressed. In particular, the zeroes of elliptic rational functions of order 2^i3^j may be algebraically expressed. For example, we can find the zeroes of R_8(\xi,x) 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 R_8(\xi,x) we set X_8=0 in the third equation, calculate the two values of X_4, then use these values of X_4 in the second equation to calculate four values of X_2 and finally, use these values in the first equation to calculate the eight zeroes of R_8(\xi,x). (The t_n are calculated by a similar recursion.) Again, using the inversion relationship, these zeroes can be used to calculate the poles.

Particular values

We may write the first few elliptic rational functions as: See for further explicit expressions of order n=5 and n=2^i,3^j. The corresponding discrimination factors are: The corresponding zeroes are x_{nj} 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 x_{p,ni} may be found by