# 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 |