---
title: "Partial fraction decomposition"
sort_title: "Partial fraction decomposition"
date: 2021-02-22
categories:
- Mathematics
layout: "concept"
---

**Partial fraction decomposition** or **partial fraction expansion**
is a method to rewrite quotients of two polynomials $$g(x)$$ and $$h(x)$$,
where the numerator $$g(x)$$ is of lower order than $$h(x)$$,
as sums 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 $$c_{1,1}, ..., c_{1,m}$$.



## References
1.  O. Bang,
    *Applied mathematics for physicists: lecture notes*, 2019,
    unpublished.