Mmm so I'm given the problem;
If $f_1, f_2$ are functions with simple poles (degree of 1) at $z_0$, then show $f_1f_2$ has a pole of degree 2 at $z_0$ and provide an expression for $Res(f_1f_2; z_0)$. I.e, the residue of $f_1f_2$ at $z_0$.
This problem is trivial for rational functions, but I'm unsure how to approach this in general. Idea's I've had are to use the Laurent expansions for $f_1, f_2$, but I want to avoid this if possible. The idea is pretty simple, I just don't know of a short, concise way of proving it in general. Other ideas involve showing absolute convergence of the product of the Taylor series expansions of the denominators to rewrite the denominator as a single series where $z_0$ is a pole of degree 2. Not sure if that'd work though..
Thanks for any help!