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 \(c_{1,1}, ..., c_{1,m}\).

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