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).

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.