rational function

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

$$R(x)=\frac{P(x)}{Q(x)},$$

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) $R(x_1,\mathrm{\dots },x_n)$ has the form

$$R(x_1,\mathrm{\dots },x_n)=\frac{P(x_1,\mathrm{\dots },x_n)}{Q(x_1,\mathrm{\dots },x_n)},$$

where $P(x_1,\mathrm{\dots },x_n)$ and $Q(x_1,\mathrm{\dots },x_n)$ are polynomials in the variables $(x_1,\mathrm{\dots },x_n)$ with coefficients in some field or ring $S$.

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

Title	rational function
Canonical name	RationalFunction
Date of creation	2013-03-22 13:38:54
Last modified on	2013-03-22 13:38:54
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	6
Author	CWoo (3771)
Entry type	Definition
Classification	msc 26C15
Synonym	rational expression
Related topic	PolynomialRing
Related topic	FractionField
Related topic	RealFunction
Related topic	PropertiesOfEntireFunctions
Related topic	IntegrationOfFractionPowerExpressions