97 Partial fractions

Let $f(s)$ be a complex rational function. Then there exists a complex polynomial $q(s)$, integers $n_j>0$ and poles ${\lambda }_j\in 𝐂$ and $a_j\in 𝐂$, all uniquely determined, such that

$$f(s)=q(s)+\sum _{j=1}^N\frac{a_j}{{(s-{\lambda }_j)}^{n_j}}.$$

Proof of existence. Starting with $f(s)=g(s)/h(s)$, we use the Euclidean algorithm to write

$$g(s)=q(s)h(s)+r(s)$$

where $q(s)$ and $r(s)$ are polynomials, and either $r(s)=0$ or the degree of $r(s)$ is strictly less than the degree of $h(s)$; hence