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% Partial fraction decomposition
# Partial fraction decomposition
**Partial fraction decomposition** or **expansion** is a method to rewrite a
quotient of two polynomials $g(x)$ and $h(x)$, where the numerator
$g(x)$ is of lower order than $h(x)$, as a sum of fractions with $x$ in
the denominator:
$$\begin{aligned}
f(x) = \frac{g(x)}{h(x)} = \frac{c_1}{x - h_1} + \frac{c_2}{x - h_2} + ...
\end{aligned}$$
Where $h_n$ etc. are the roots of the denominator $h(x)$. If all $N$ of
these roots are distinct, then it is sufficient to simply posit:
$$\begin{aligned}
\boxed{
f(x) = \frac{c_1}{x - h_1} + \frac{c_2}{x - h_2} + ... + \frac{c_N}{x - h_N}
}
\end{aligned}$$
The constants $c_n$ can either be found the hard way,
by multiplying the denominators around and solving a system of $N$
equations, or the easy way by using this trick:
$$\begin{aligned}
\boxed{
c_n = \lim_{x \to h_n} \big( f(x) (x - h_n) \big)
}
\end{aligned}$$
If $h_1$ is a root with multiplicity $m > 1$, then the sum takes the form of:
$$\begin{aligned}
\boxed{
f(x)
= \frac{c_{1,1}}{x - h_1} + \frac{c_{1,2}}{(x - h_1)^2} + ...
}
\end{aligned}$$
Where $c_{1,j}$ are found by putting the terms on a common denominator, e.g.
$$\begin{aligned}
\frac{c_{1,1}}{x - h_1} + \frac{c_{1,2}}{(x - h_1)^2}
= \frac{c_{1,1} (x - h_1) + c_{1,2}}{(x - h_1)^2}
\end{aligned}$$
And then, using the linear independence of $x^0, x^1, x^2, ...$, solving
a system of $m$ equations to find all $c_{1,1}, ..., c_{1,m}$.
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