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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.

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    Hint: $\sin (\pi z)$ has zeros at every integer. This can easily be adapted to answer both your questions.2012-05-12

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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 $0 \leq \theta \leq 2 \pi$.

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    I don't know what you mean about the entire complex plane. If |a_n|>1, then $\dfrac{z-a_n}{1-\overline{a_n}z}$ has a pole in the disk. The part of the question I guess your answer is directed toward is asking for a function analytic in the disk as opposed to meromorphic. But, there is a condition on the zeros, more or less saying that they tend to the boundary fast enough, which ensures that $B(z)$ is nontrivial. You can look it up.2013-05-10
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$\sin\left(\frac{1}{z-1}\right)$

$\sum_{n=1}^\infty \frac{1}{n!(z-\log n)}$

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    @Sachin: For the first example, note that $\sin$ is defined everywhere in the plane, so $\sin(1/(z-1))$ is defined everywhere except $z=1$. What are the zeros of $\sin$? Where is $1/(z-1)$ equal to those zeros? The answers should show you infinitely many isolated zeros in the disk. For the second example, I deliberately left out $0$. What is $\log 0$?2012-05-13