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:
Rn(\xi,x)\equivcd\left(n
K(1/Ln(\xi)) | |
K(1/\xi) |
cd-1(x,1/\xi),1/Ln(\xi)\right)
where
Ln(\xi)=Rn(\xi,\xi)
Rn(\xi,x)
|x|\ge\xi
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.
Rn(\xi,x)=r
|
where
xi
xpi
r0
Rn(\xi,1)=1
Rn(\xi,x)=r
|
2(\xi,x)\le | |
R | |
n |
1
|x|\le1
2(\xi,x)= | |
R | |
n |
1
|x|=1
2(\xi,x) | |
R | |
n |
2(\xi,x)>1 | |
R | |
n |
x>1
The only rational function satisfying the above properties is the elliptic rational function . The following properties are derived:
The elliptic rational function is normalized to unity at x=1:
Rn(\xi,1)=1
The nesting property is written:
Rm(Rn(\xi,\xi),Rn(\xi,x))=Rm ⋅ (\xi,x)
This is a very important property:
Rn
Rn
R2
R3
Rn
n=2a3b
Rn
Rn
Lm ⋅ (\xi)=Lm(Ln(\xi))
The elliptic rational functions are related to the Chebyshev polynomials of the first kind
Tn(x)
\lim\xi= → inftyRn(\xi,x)=Tn(x)
Rn(\xi,-x)=Rn(\xi,x)
Rn(\xi,-x)=-Rn(\xi,x)
Rn(\xi,x)
\pm1
-1\lex\le1
1/Rn(\xi,x)
-1/\xi\lex\le1/\xi
\pm1/Ln(\xi)
The following inversion relationship holds:
R | ||||
|
This implies that poles and zeroes come in pairs such that
xpixzi=\xi
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
xni(\xi)
xni
\xi
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
cd\left((2m-1)K\left(1/z\right), | 1 |
z |
\right)=0
the defining equation for the elliptic rational functions implies that
n
K(1/Ln) | |
K(1/\xi) |
cd-1(xm,1/\xi)=(2m-1)K(1/Ln)
so that the zeroes are given by
x | , | ||||
|
1 | |
\xi |
\right).
Using the inversion relationship, the poles may then be calculated.
From the nesting property, if the zeroes of
Rm
Rn
Rm ⋅
2i3j
R8(\xi,x)
Xn\equivRn(\xi,x) Ln\equivRn(\xi,\xi) tn\equiv
2}. | |
\sqrt{1-1/L | |
n |
Then, from the nesting property and knowing that
R | ||||
|
where
t\equiv\sqrt{1-1/\xi2}
L | ||||
|
,
L | ||||
|
,
L | ||||
|
X | ||||
|
,
X | ||||||||||||||||||||||||||||
|
,
X | ||||||||||||||||||||||||||||
|
.
These last three equations may be inverted:
x=
1 | |||||
|
To calculate the zeroes of
R8(\xi,x)
X8=0
X4
X4
X2
R8(\xi,x)
tn
We may write the first few elliptic rational functions as:
R1(\xi,x)=x
R | ||||
|
where
t\equiv\sqrt{1-
1 | |
\xi2 |
R | ||||||||||||||||||||||||||||||||||
|
where
G\equiv\sqrt{4\xi2+(4\xi2(\xi2-1))2/3
| ||||
x | ||||
p |
2=\xi | |
x | |
z |
2 | |
p |
R4(\xi,x)=R2(R2(\xi,\xi),R
2x | ||||
|
4-2(1+t)(1+\sqrt{t})x2+1} {(1+t)(1-\sqrt{t})2x4-2(1+t)(1-\sqrt{t})x2+1}
R6(\xi,x)=R3(R2(\xi,\xi),R2(\xi,x))
See for further explicit expressions of order n=5 and
n=2i3j
The corresponding discrimination factors are:
L1(\xi)=\xi
L | ||||
|
=\left(\xi+\sqrt{\xi2-1}\right)2
| ||||||||||||||||
L | ||||||||||||||||
3(\xi)=\xi |
\right)2
2-1) | |
L | |
4(\xi)=\left(\sqrt{\xi}+(\xi |
1/4\right)4\left(\xi+\sqrt{\xi2-1}\right)2
L6(\xi)=L3(L2(\xi))
The corresponding zeroes are
xnj
x11=0
x21=\xi\sqrt{1-t}
x22=-x21
x31=xz
x32=0
x33=-x31
x41=\xi\sqrt{\left(1-\sqrt{t}\right)\left(1+t-\sqrt{t(t+1)}\right)}
x42=\xi\sqrt{\left(1-\sqrt{t}\right)\left(1+t+\sqrt{t(t+1)}\right)}
x43=-x42
x44=-x41
From the inversion relationship, the corresponding poles
xp,ni
xp,ni=\xi/(xni)