If $p(x)$ and $q(x)$ are polynomials such that $\deg q(x) > \deg p(x)$ then the rational function $p(x)/q(x)$ has a partial fraction expansion
$$ \frac{p(x)}{q(x)} = \frac{a_1}{(x-q_1)^{m_1}}+\frac{a_2}{(x-q_2)^{m_2}}+\cdots +\frac{a_r}{(x-q_r)^{m_r}} $$ where $q(x) = \prod_{j=1}^r (x-q_j)^{m_j}$ and the $a_j\ (\ 1\leq j \leq r \leq \deg q(x)\ )$ are constants.
I know there is a way to calculate the $a_j$ using the calculus of residues but cannot find a good reference which describes this (everything online is just at the freshman calculus level). Can anyone provide a good link or discussion of this method? In addition, although the statement that a PFE exists such that the $a_j$ are constants is intuitively clear, I don't have a good quality proof of this and I'd be interested in obtaining one.