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'partial fractions of expressions'
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| Title of object: |
partial fractions of expressions |
| Canonical Name: |
PartialFractionsOfExpressions |
| Type: |
Definition |
| Created on: |
2004-04-29 04:01:52 |
| Modified on: |
2004-04-29 04:21:48 |
| Classification: |
msc:26C15 |
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}
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%\usepackage{amsthm}
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%\usepackage{xypic}
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Content:
Let $R(z) = \frac{P(z)}{Q(z)}$ be a {\em fractional expression}, i.e., a quotient of the polynomials $P(z)$ and $Q(z)$ such that $Q(z)$ does not divide $P(z)$. Let's restrict to the case that the coefficients are real or complex numbers. If the distinct complex zeros of the denominator are $b_1, b_2, ..., b_t$ with the orders $\nu_1, \nu_2, ..., \nu_t$ ($t \ge 0$), and the numerator has not common zeros, then $R(z)$ can be decomposed uniquely as the sum
$$R(z) = H(z)+
\sum_{j=1}^t(\frac{A_{j1}}{z-b_j}+\frac{A_{j2}}{(z-b_j)^2}+...+\frac{A_{j\nu_j}}{(z-b_j)^{\nu_j}}),$$
where $H(z)$ is a polynomial and the $A_{jk}$'s are certain complex numbers.
Cf. the partial fractions of fractional numbers. |
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