I'm taking a complex variable class and I've recently been introduced to the Laurent series as
$$f(z)=\sum_{-\infty}^{\infty}a_n(z-z_0)^n$$
where $z_0$ is the singularity point and the expression holds for $z \in D^*= \{ z\in \mathbb{C} : |z|<r \} - \{z_0 \} $.
However, further in the subject, I've come across the following theorem:
Given $f(z)=1/Q(z)$ with $Q(z)$ a polinomial of roots $\alpha_1,...,\alpha_r$, then you can rewrite $f$ as the sum of polinomials in $1/(z-\alpha_i)$
Now, the proof starts by considering the principal parts $P_1,...,P_r$ of the function $f$ in each singularity $\alpha_i$.
My problem is that I don't fully understand what it means for a function to have different principal parts for different singularities. As far as I was concerned, the Laurent series was only developed for a single singularity.
I also fail to understand the meaning behind defining a new function as the original one minus all the principal parts, i get that this has to do with "removing singularities" but I can't see why it works (mainly because I don't really understand the mathematical expression for the Laurent series of a function with multiple singularities).
Just to make sure, it's not the theorem itself I'm having trouble with but more with the fact that I don't understand the Laurent series for a function with more than one singularity.
I'd appreciate if someone could help me with this issue.
Thanks in advance and sorry for my poor English.