Editing Partial fraction decomposition (section)

=== Residue method ===
{{See also|Heaviside cover-up method}}
Over the complex numbers, suppose ''f''(''x'') is a rational proper fraction, and can be decomposed into

<math display="block">f(x) = \sum_i \left( \frac{a_{i1}}{x - x_i} + \frac{a_{i2}}{( x - x_i)^2} + \cdots + \frac{a_{i k_i}}{(x - x_i)^{k_i}} \right). </math>

Let
<math display="block"> g_{ij}(x) = (x - x_i)^{j-1}f(x),</math>
then according to the [[Laurent series#Uniqueness|uniqueness of Laurent series]], ''a''<sub>''ij''</sub> is the coefficient of the term {{math|(''x'' − ''x''<sub>''i''</sub>)<sup>−1</sup>}} in the Laurent expansion of ''g''<sub>''ij''</sub>(''x'') about the point ''x''<sub>''i''</sub>, i.e., its [[residue (complex analysis)|residue]]
<math display="block">a_{ij} = \operatorname{Res}(g_{ij},x_i).</math>

This is given directly by the formula
<math display="block">a_{ij} = \frac 1 {(k_i-j)!}\lim_{x\to x_i}\frac{d^{k_i-j}}{dx^{k_i-j}} \left((x-x_i)^{k_i} f(x)\right),</math>
or in the special case when ''x''<sub>''i''</sub> is a simple root,
<math display="block">a_{i1}=\frac{P(x_i)}{Q'(x_i)},</math>
when
<math display="block">f(x)=\frac{P(x)}{Q(x)}.</math>