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# Elliptic Rational Function

Elliptic rational functions are a special class of rational functions that have nice properties for approximating other functions over the interval . In particular, they are equiripple, satisfy over , are minimax approximations over , exhibit monotonic increase on , 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

 (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) (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)

Chebyshev Polynomial of the First Kind, Elliptic Function, Rational Function

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## References

Antoniou, A. Digital Filters: Analysis and Design. New York: McGraw-Hill, 1979.Daniels, R. W. Approximation Methods for Electronic Filter Design. New York: McGraw-Hill, 1974.Lutovac, M. D.; Tosic, D. V.; and Evans, B. L. Filter Design for Signal Processing Using MATLAB and Mathematica. Upper Saddle River, NJ: Prentice-Hall, 2001.

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Elliptic Rational Function

## Cite this as:

Weisstein, Eric W. "Elliptic Rational Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/EllipticRationalFunction.html