I am solving this past year qualifying exam question.
Consider the function $f(z) = \frac{1}{z}$ on the annulus $A=\{z\in \mathbb{C}: \frac{1}{2}<|z|<2\}$. I have to show that there is a sequence $\{r_n(z)\}$ of rational functions whose poles are contained in $\mathbb{C} - A$ and which approximates $f(z)$ uniformly on compact subsets of $A$.
I am not sure how to proceed to solve this problem. Also, I am not sure what topics should I cover to solve problems like these.
Thank you very much for the help.