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'rational function'
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| Title of object: |
rational function |
| Canonical Name: |
RationalFunction |
| Type: |
Definition |
| Created on: |
2003-05-26 03:46:51 |
| Modified on: |
2004-03-08 00:08:41 |
| Classification: |
msc:26C15 |
Revision comment (for changes between this and next version):
| Changes for correction #5909 ('Functions or not?'). |
Preamble:
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here
\def\sse{\subseteq}
\def\bigtimes{\mathop{\mbox{\Huge $\times$}}}
\def\impl{\Rightarrow} |
Content:
A real function $R(x)$ of a single variable $x$ is called
\emph{\PMlinkescapetext{rational}} 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.
In general, a rational function $R(x_1,\ldots,x_n)$ has the form
\[ R(x_1,\ldots,x_n) = \frac{P(x_1,\ldots,x_n)}{Q(x_1,\ldots,x_n)}, \]
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 $S[x_1,\ldots,x_n]$. |
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