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.
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 are the zeroes and are the poles, and 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
for
at
for
The slope at x=1 is as large as possible
The slope at x=1 is larger than the corresponding slope of the Chebyshev polynomial of the same order.
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:
If is known for all prime n, then nesting property gives for all n. In particular, since and can be expressed in closed form without explicit use of the Jacobi elliptic functions, then all for n of the form can be so expressed.
It follows that if the zeroes of for prime n are known, the zeros of all can be found. Using the inversion relationship, the poles can also be found.
The nesting property implies the nesting property of the discrimination factor:
has equal ripple of in the interval. By the inversion relationship, 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 or when 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 and can be algebraically expressed then the zeroes of can be algebraically expressed. In particular, the zeroes of elliptic rational functions of order may be algebraically expressed. For example, we can find the zeroes of 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 we set in the third equation, calculate the two values of, then use these 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. 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. 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 may be found by
