Categories: Mathematics.

# 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 $$c_{1,1}, ..., c_{1,m}$$.