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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. –  Chris Eagle May 12 '12 at 22:41
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2 Answers 2

up vote -1 down vote accepted

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. –  Jonas Meyer May 9 '13 at 0:57
    
Ah, I will fix that right away. –  Libertron May 10 '13 at 15:51
    
Fixed, sorry about that. –  Libertron May 10 '13 at 15:55
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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 $0$. –  Jonas Meyer May 10 '13 at 16:06
    
How would that issue be resolved, then? Should the points be considered on the entire complex plane? –  Libertron May 10 '13 at 22:05
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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? –  Libertron May 12 '12 at 23:12
    
@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 $\log 0$? –  Jonas Meyer May 13 '12 at 3:01
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