% 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:

$$\begin{aligned}
    \boxed{
        f(x) = \frac{c_1}{x - h_1} + \frac{c_2}{x - h_2} + ... + \frac{c_N}{x - h_N}
    }
\end{aligned}$$

Then the constant coefficients $c_n$ 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 the following 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 $c_{1,1}, ..., c_{1,m}$.