Categories: Mathematics.

Partial fraction decomposition

Partial fraction decomposition or partial fraction expansion is a method to rewrite quotients of two polynomials g(x)g(x) and h(x)h(x), where the numerator g(x)g(x) is of lower order than h(x)h(x), as sums of fractions with xx in the denominator:

f(x)=g(x)h(x)=c1xh1+c2xh2+...\begin{aligned} f(x) = \frac{g(x)}{h(x)} = \frac{c_1}{x - h_1} + \frac{c_2}{x - h_2} + ... \end{aligned}

Where hnh_n etc. are the roots of the denominator h(x)h(x). If all NN of these roots are distinct, then it is sufficient to simply posit:

f(x)=c1xh1+c2xh2+...+cNxhN\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 cnc_n can either be found the hard way, by multiplying the denominators around and solving a system of NN equations, or the easy way by using this trick:

cn=limxhn(f(x)(xhn))\begin{aligned} \boxed{ c_n = \lim_{x \to h_n} \big( f(x) (x - h_n) \big) } \end{aligned}

If h1h_1 is a root with multiplicity m>1m > 1, then the sum takes the form of:

f(x)=c1,1xh1+c1,2(xh1)2+...\begin{aligned} \boxed{ f(x) = \frac{c_{1,1}}{x - h_1} + \frac{c_{1,2}}{(x - h_1)^2} + ... } \end{aligned}

Where c1,jc_{1,j} are found by putting the terms on a common denominator, e.g.

c1,1xh1+c1,2(xh1)2=c1,1(xh1)+c1,2(xh1)2\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 x0,x1,x2,...x^0, x^1, x^2, ..., solving a system of mm equations to find all c1,1,...,c1,mc_{1,1}, ..., c_{1,m}.


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 O. Bang, Applied mathematics for physicists: lecture notes, 2019, unpublished.