Elliptic rational functions 
are a special class of rational
functions that have nice properties for approximating other functions over the
interval ![x in [-1,1]](/images/equations/EllipticRationalFunction/Inline2.svg)
.
In particular, they are equiripple, satisfy 
over 
, are minimax
approximations over 
, exhibit monotonic increase on ![x in [1,xi]](/images/equations/EllipticRationalFunction/Inline6.svg)
, and have minimal order 
. Additional properties include symmetry
 |
(1)
|
normalization
 |
(2)
|
the property
 |
(3)
|
and the nesting property
 |
(4)
|
(Lutovac et al. 2001).
Letting the discrimination factor 
be the largest value of 
for 
, the elliptic rational functions can be defined by
![R_n(xi,x)=cd(n(K([L_n(xi)]^(-1)))/(K(xi^(-1)))cd^(-1)(x,xi^(-1)),[L_n(xi)]^(-1)),](/images/equations/EllipticRationalFunction/NumberedEquation5.svg) |
(5)
|
where 
is a complete elliptic integral
of the first kind, 
is a Jacobi
elliptic function, and 
is an inverse Jacobi elliptic function. For 
,
2, and 3, the functions are given by
 |  |  |
(6)
|
 |  |  |
(7)
|
 |  | ![([1+dn(2/3K(xi^_),xi^_)]^2x^2-[1+2dn(2/3K(xi^_),xi^_)]^2)/([dn^2(2/3K(xi^_),xi^_)]x^2+1)x,](/images/equations/EllipticRationalFunction/Inline23.svg) |
(8)
|
where 
.
can be expressed in closed form without using elliptic functions for 
of the form 
.
The elliptic rational functions are related to the Chebyshev polynomials of the first kind 
by
 |
(9)
|
