Elliptic rational functions

Properties

The canonical properties

• ${\displaystyle R_n^2(\xi ,x)\le 1}$ for ${\displaystyle |x|\le 1}$
• ${\displaystyle R_n^2(\xi ,x)=1}$ at ${\displaystyle |x|=1}$
• ${\displaystyle R_n^2(\xi ,-x)=R_n^2(\xi ,x)}$
• ${\displaystyle R_n^2(\xi ,x)>1}$ for ${\displaystyle x>1}$
• 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 (Lutovac, Tošić & Evans 2001, § 13.2). The following properties are derived:

Normalization

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

${\displaystyle R_n(\xi ,1)=1}$

Nesting property

The nesting property is written:

${\displaystyle R_m(R_n(\xi ,\xi ),R_n(\xi ,x))=R_{m\cdot n}(\xi ,x)}$

This is a very important property:

• If ${\displaystyle R_n}$ is known for all prime n, then nesting property gives ${\displaystyle R_n}$ for all n. In particular, since ${\displaystyle R_2}$ and ${\displaystyle R_3}$ can be expressed in closed form without explicit use of the Jacobi elliptic functions, then all ${\displaystyle R_n}$ for n of the form ${\displaystyle n=2^a{3}^b}$ can be so expressed.
• It follows that if the zeroes of ${\displaystyle R_n}$ for prime n are known, the zeros of all ${\displaystyle R_n}$ 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:

${\displaystyle L_{m\cdot n}(\xi )=L_m(L_n(\xi ))}$

Limiting values

The elliptic rational functions are related to the Chebyshev polynomials of the first kind ${\displaystyle T_n(x)}$ by:

${\displaystyle \underset{\xi =\to \mathrm{\infty }}{lim}{R}_n(\xi ,x)=T_n(x)}$

Symmetry

${\displaystyle R_n(\xi ,-x)=R_n(\xi ,x)}$ for n even

${\displaystyle R_n(\xi ,-x)=-R_n(\xi ,x)}$ for n odd

Equiripple

${\displaystyle R_n(\xi ,x)}$ has equal ripple of ${\displaystyle \pm 1}$ in the interval ${\displaystyle -1\le x\le 1}$. By the inversion relationship (see below), it follows that ${\displaystyle 1/R_n(\xi ,x)}$ has equiripple in ${\displaystyle -1/\xi \le x\le 1/\xi }$ of ${\displaystyle \pm 1/L_n(\xi )}$.

Inversion relationship

The following inversion relationship holds:

${\displaystyle R_n(\xi ,\xi /x)=\frac{R_n(\xi ,\xi )}{R_n(\xi ,x)}}$

This implies that poles and zeroes come in pairs such that

${\displaystyle x_{pi}{x}_{zi}=\xi }$

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 ${\displaystyle x_{ni}(\xi )}$ or ${\displaystyle x_{ni}}$ when ${\displaystyle \xi }$ 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 (Lutovac, Tošić & Evans 2001, § 12.6). Using the fact that for any z

${\displaystyle \mathrm{c}\mathrm{d}\left((2m-1)K\left(1/z\right),\frac{1}{z}\right)=0}$

the defining equation for the elliptic rational functions implies that

${\displaystyle n\frac{K(1/L_n)}{K(1/\xi )}{\mathrm{c}\mathrm{d}}^{-1}(x_m,1/\xi )=(2m-1)K(1/L_n)}$

so that the zeroes are given by

${\displaystyle x_m=\mathrm{c}\mathrm{d}\left(K(1/\xi )\frac{2m-1}{n},\frac{1}{\xi }\right).}$

Using the inversion relationship, the poles may then be calculated.

From the nesting property, if the zeroes of ${\displaystyle R_m}$ and ${\displaystyle R_n}$ can be algebraically expressed (i.e. without the need for calculating the Jacobi ellipse functions) then the zeroes of ${\displaystyle R_{m\cdot n}}$ can be algebraically expressed. In particular, the zeroes of elliptic rational functions of order ${\displaystyle 2^i{3}^j}$ may be algebraically expressed (Lutovac, Tošić & Evans 2001, § 12.9, 13.9). For example, we can find the zeroes of ${\displaystyle R_8(\xi ,x)}$ as follows: Define

${\displaystyle X_n\equiv R_n(\xi ,x)L_n\equiv R_n(\xi ,\xi )t_n\equiv \sqrt{1-1/L_n^2}.}$

Then, from the nesting property and knowing that

${\displaystyle R_2(\xi ,x)=\frac{(t+1)x^2-1}{(t-1)x^2+1}}$

where ${\displaystyle t\equiv \sqrt{1-1/{\xi }^2}}$ we have:

${\displaystyle L_2=\frac{1+t}{1-t},L_4=\frac{1+t_2}{1-t_2},L_8=\frac{1+t_4}{1-t_4}}$

${\displaystyle X_2=\frac{(t+1)x^2-1}{(t-1)x^2+1},X_4=\frac{(t_2+1)X_2^2-1}{(t_2-1)X_2^2+1},X_8=\frac{(t_4+1)X_4^2-1}{(t_4-1)X_4^2+1}.}$

These last three equations may be inverted:

${\displaystyle x=\frac{1}{\pm \sqrt{1+t\left(\frac{1-X_2}{1+X_2}\right)}},X_2=\frac{1}{\pm \sqrt{1+t_2\left(\frac{1-X_4}{1+X_4}\right)}},X_4=\frac{1}{\pm \sqrt{1+t_4\left(\frac{1-X_8}{1+X_8}\right)}}.}$

To calculate the zeroes of ${\displaystyle R_8(\xi ,x)}$ we set ${\displaystyle X_8=0}$ in the third equation, calculate the two values of ${\displaystyle X_4}$, then use these values of ${\displaystyle X_4}$ in the second equation to calculate four values of ${\displaystyle X_2}$ and finally, use these values in the first equation to calculate the eight zeroes of ${\displaystyle R_8(\xi ,x)}$. (The ${\displaystyle t_n}$ 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:

${\displaystyle R_1(\xi ,x)=x}$

${\displaystyle R_2(\xi ,x)=\frac{(t+1)x^2-1}{(t-1)x^2+1}}$

where

${\displaystyle t\equiv \sqrt{1-\frac{1}{{\xi }^2}}}$

${\displaystyle R_3(\xi ,x)=x\frac{(1-x_p^2)(x^2-x_z^2)}{(1-x_z^2)(x^2-x_p^2)}}$

where

${\displaystyle G\equiv \sqrt{4{\xi }^2+(4{\xi }^2({\xi }^2-1){)}^{2/3}}}$

${\displaystyle x_p^2\equiv \frac{2{\xi }^2\sqrt{G}}{\sqrt{8{\xi }^2({\xi }^2+1)+12G{\xi }^2-G^3}-\sqrt{G^3}}}$

${\displaystyle x_z^2={\xi }^2/x_p^2}$

${\displaystyle R_4(\xi ,x)=R_2(R_2(\xi ,\xi ),R_2(\xi ,x))=\frac{(1+t)(1+\sqrt{t}{)}^2{x}^4-2(1+t)(1+\sqrt{t})x^2+1}{(1+t)(1-\sqrt{t}{)}^2{x}^4-2(1+t)(1-\sqrt{t})x^2+1}}$

${\displaystyle R_6(\xi ,x)=R_3(R_2(\xi ,\xi ),R_2(\xi ,x))}$ etc.

See Lutovac, Tošić & Evans (2001, § 13) for further explicit expressions of order n=5 and ${\displaystyle n=2^i{3}^j}$.

The corresponding discrimination factors are:

${\displaystyle L_1(\xi )=\xi }$

${\displaystyle L_2(\xi )=\frac{1+t}{1-t}={\left(\xi +\sqrt{{\xi }^2-1}\right)}^2}$

${\displaystyle L_3(\xi )={\xi }^3{\left(\frac{1-x_p^2}{{\xi }^2-x_p^2}\right)}^2}$

${\displaystyle L_4(\xi )={\left(\sqrt{\xi }+({\xi }^2-1{)}^{1/4}\right)}^4{\left(\xi +\sqrt{{\xi }^2-1}\right)}^2}$

${\displaystyle L_6(\xi )=L_3(L_2(\xi ))}$ etc.

The corresponding zeroes are ${\displaystyle x_{nj}}$ where n is the order and j is the number of the zero. There will be a total of n zeroes for each order.

${\displaystyle x_{11}=0}$

${\displaystyle x_{21}=\xi \sqrt{1-t}}$

${\displaystyle x_{22}=-x_{21}}$

${\displaystyle x_{31}=x_z}$

${\displaystyle x_{32}=0}$

${\displaystyle x_{33}=-x_{31}}$

${\displaystyle x_{41}=\xi \sqrt{\left(1-\sqrt{t}\right)\left(1+t-\sqrt{t(t+1)}\right)}}$

${\displaystyle x_{42}=\xi \sqrt{\left(1-\sqrt{t}\right)\left(1+t+\sqrt{t(t+1)}\right)}}$

${\displaystyle x_{43}=-x_{42}}$

${\displaystyle x_{44}=-x_{41}}$

From the inversion relationship, the corresponding poles ${\displaystyle x_{p,ni}}$ may be found by ${\displaystyle x_{p,ni}=\xi /(x_{ni})}$