I know usual techniques in partial fractions such as the ones described in Wiki. But I don't understand how this process works in a precise manner.
I've seen in some complex analysis textbooks, discussions of partial fractions. So should I be aided by complex analysis to properly (and precisely) grasp the idea of partial fraction?
Please help!
Consider a rational function $p(z)/q(z)$ where $p(z)$ and $q(z)$ are polynomials, and the degree of $q$ is greater than that of $p$. Factor $q(z) = \prod_{j=1}^m (z - r_j)^{d_j}$, where $r_j$ are complex numbers and $d_j$ positive integers. Then the basic result is that we can write $$ \dfrac{p(z)}{q(z)} = \sum_{j=1}^m \sum_{k=1}^{d_j} \dfrac{c_{jk}}{(z - r_j)^k}$$ for some complex numbers $c_{jk}$.
The "complex analysis" proof of this goes like this: let $\sum_{k=1}^{d_j} c_{jk}/(z - r_j)^k$ be the principal part of $p(z)/q(z)$ at the pole $r_j$. Let $f(z) = p(z)/q(z) - \sum_{j=1}^m \sum_{k=1}^{d_j} c_{jk}/(z - r_j)^k$. Its singularities at the $r_j$ are removable; after removing those singularities, we have an entire function that goes to $0$ as $z \to \infty$. By Liouville's theorem it is $0$.