Is it possible to have an analytic function on the unit disk $\mathbb{D}$ that has infinitely many isolated zeros? What is a good example? I guess then that would make this analytic function nontrivial, correct?
Also, what is an example of a meromorphic function on the complex plane with simple poles and points log$n$, for $n \geq 0$? All I know right now is probably that the principal part of this function would be of the form $\frac{1}{z- log n}$ , but I'm not so sure about that. Any guidance would be appreciated.