 TOPICS  # Rational Function

A quotient of two polynomials and , is called a rational function, or sometimes a rational polynomial function. More generally, if and are polynomials in multiple variables, their quotient is called a (multivariate) rational function. The term "rational polynomial" is sometimes used as a synonym for rational function. However, this usage is strongly discouraged since by analogy with complex polynomial and integer polynomial, rational polynomial should properly refer to a polynomial with rational coefficients.

A rational function has no singularities other than poles in the extended complex plane. Conversely, if a single-values function has no singularities other than poles in the extended complex plane, then it is a rational function (Knopp 1996, p. 137). In addition, a rational function can be decomposed into partial fractions (Knopp 1996, p. 139).

Abel's Curve Theorem, Closed Form, Fundamental Theorem of Symmetric Functions, Inside-Outside Theorem, Polynomial, Quotient-Difference Algorithm, Rational Integer, Rational Number, Rational Polynomial, Riemann Curve Theorem Explore this topic in the MathWorld classroom

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## References

Flajolet, P. and Sedgewick, R. "Analytic Combinatorics: Functional Equations, Rational and Algebraic Functions." http://www.inria.fr/RRRT/RR-4103.html.Knopp, K. "Rational Functions." §35 in Theory of Functions Parts I and II, Two Volumes Bound as One, Part I. New York: Dover, pp. 96 and 137-139, 1996.

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Rational Function

## Cite this as:

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