Question regarding an analytic function and a meromorphic one 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.
$$\sin\left(\frac{1}{z-1}\right)$$

$$\sum_{n=1}^\infty \frac{1}{n!(z-\log n)}$$
Better example for one of them would have been an infinite Blaschke product
$$B(z)= e^{i \theta} \prod_{n=1}^{\infty} \frac{z - a_n}{1 - \bar{a}_{n}z},$$ 
where $a_1,a_2,…$ are points in $\mathbb{C}$ (?) and $0 \leq \theta \leq 2 \pi$.