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rational function (Definition)

A real function $ R(x)$ of a single variable $ x$ is called rational if it can be written as a quotient

$\displaystyle R(x) = \frac{P(x)}{Q(x)}, $
where $ P(x)$ and $ Q(x)$ are polynomials in $ x$ with real coefficients. When one is only interested in algebraic properties of $ R(x)$ or $ P(x)$ and $ Q(x)$, it is convenient to forget that they define functions and simply treat them as algebraic expressions in $ x$. In this case $ R(x)$ is referred to as a rational expression.

In general, a rational function (expression) $ R(x_1,\ldots,x_n)$ has the form

$\displaystyle R(x_1,\ldots,x_n) = \frac{P(x_1,\ldots,x_n)}{Q(x_1,\ldots,x_n)}, $
where $ P(x_1,\ldots,x_n)$ and $ Q(x_1,\ldots,x_n)$ are polynomials in the variables $ (x_1,\ldots,x_n)$ with coefficients in some field or ring $ S$.

In this sense, $ R(x_1,\ldots,x_n)$ can be regarded as an element of the fraction field $ S(x_1,\ldots,x_n)$ of the polynomial ring $ S[x_1,\ldots,x_n]$.



"rational function" is owned by CWoo. [ owner history (1) ]
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See Also: polynomial ring, fraction field, real function, properties of entire functions, integration of fraction power expressions

Other names:  rational expression

Attachments:
partial fractions of expressions (Definition) by pahio
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Cross-references: polynomial ring, fraction field, ring, field, expressions, functions, properties, algebraic, coefficients, real, polynomials, quotient, variable, real function
There are 45 references to this entry.

This is version 3 of rational function, born on 2003-05-26, modified 2005-03-27.
Object id is 4300, canonical name is RationalFunction.
Accessed 7423 times total.

Classification:
AMS MSC: 26C15 (Real functions :: Polynomials, rational functions :: Rational functions)
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