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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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    This is not an example of anything asked for in the question. However, if you instead used infinite Blaschke products, you would get examples for the first part of the question.2013-05-09
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    Ah, I will fix that right away.2013-05-10
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    Fixed, sorry about that.2013-05-10
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    The points $a_1,a_2,\ldots$ cannot be chosen arbitrarily from the unit disk. Many choices will result in the product being identically $0$.2013-05-10
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    How would that issue be resolved, then? Should the points be considered on the entire complex plane?2013-05-10
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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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    For the first example, could you please elaborate a little? You might want to be careful with the second example since we are considering $n \geq 0$, or am I wrong?2012-05-12
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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