Elliptic rational functions
Encyclopedia
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 filter
s. (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
called the selectivity factor. A rational elliptic function of degree n in x with selectivity factor ξ is generally defined as:
For many cases, in particular for orders of the form
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 n even)
where
are the zeroes and 
are the poles, and 
is a normalizing constant chosen such that 
. The above form would be true for odd 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:
(for n odd)
The only rational function satisfying the above properties is the elliptic rational function . The following properties are derived:
This is a very important property:
by:
for n even
for n odd
has equal ripple of 
in the interval 
. By the inversion relationship (see below), it follows that 
has equiripple in 
of 
.
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.
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 (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 
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 
. (The 
are calculated by a similar recursion.) Again, using the inversion relationship, these zeroes can be used to calculate the poles.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
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 filter
Elliptic filter
An elliptic filter is a signal processing filter with equalized ripple behavior in both the passband and the stopband...
s. (These functions are sometimes called Chebyshev rational functions, not to be confused with certain other functions of the same name
Chebyshev rational functions
In mathematics, the Chebyshev rational functions are a sequence of functions which are both rational and orthogonal. They are named after Pafnuty Chebyshev...
).
Rational elliptic functions are identified by a positive integer order n and include a parameter
called the selectivity factor. A rational elliptic function of degree n in x with selectivity factor ξ is generally defined as:- cd is the Jacobi elliptic cosine functionJacobi's elliptic functionsIn mathematics, the Jacobi elliptic functions are a set of basic elliptic functions, and auxiliary theta functions, that have historical importance with also many features that show up important structure, and have direct relevance to some applications...
. - K is a complete elliptic integralElliptic integralIn integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse. They were first studied by Giulio Fagnano and Leonhard Euler...
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
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 polynomialsChebyshev polynomials
In mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev, are a sequence of orthogonal polynomials which are related to de Moivre's formula and which can be defined recursively. One usually distinguishes between Chebyshev polynomials of the first kind which are denoted Tn and...
: 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. (for n even)where
are the zeroes and are the poles, and is a normalizing constant chosen such that . The above form would be true for odd 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: (for n odd)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 (see below), the poles can also be found. - The nesting property implies the nesting property of the discrimination factor:
Limiting values
The elliptic rational functions are related to the Chebyshev polynomials of the first kind by:Symmetry
for n even for n oddEquiripple
has equal ripple of 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 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
Chebyshev polynomials
In mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev, are a sequence of orthogonal polynomials which are related to de Moivre's formula and which can be defined recursively. One usually distinguishes between Chebyshev polynomials of the first kind which are denoted Tn and...
. 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 (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 may be algebraically expressed . For example, we can find the zeroes of as follows: DefineThen, 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 . (The 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:
-
-
- where
- where
etc.
See for further explicit expressions of order n=5 and
.
The corresponding discrimination factors are:
etc.
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 
-