Categories: Mathematics.

Partial fraction decomposition

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

O. Bang, Applied mathematics for physicists: lecture notes, 2019, unpublished.