1. Motivation 2.11.1 - partial fraction decomposition

let $f (z)$ be a rational function. Then there exists polynomials $h_{i} (z)$ of the form:

h_{i} (z) = \frac{A (z)}{(z - a)^{n}}

For $A \in C [x]$ of smaller degree than $n$ , and $a$ a pole of $f$ such that:

f (z) - \sum_{i = 1}^{n} h_{i} (z) \in C [x]

In general, if $a$ is a pole of $f$ of order $n$ , we might have multiple $h_{i}$ with denominators $(z - a)^{m}$ for 1;; $m \leq n$ .

Remark

This can be applied more generally to polynomial rings over a field, where the denominators are now allowed to be powers of irreducible polynomials. More information on wikipedia