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The bottom line of the "Analytic Continuation" page on WolframMathWorld states:

there exists a meromorphic function in the unit disk where every point on the unit circle is a limit point of the set of poles.

I must admit that I've been rather surprised by such a statement, therefore I tried to construct this example by myself and to google for it, but nothing came up. Do you have any idea on how such a function looks like?

Thanks in advance!

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Problem 53 here says that if $$B(z)=\prod_{n=1}^{\infty}{|a_n|\over a_n}{a_n-z\over1-\overline{a_n}z}$$ with $0<|a_n|<1$ and $\sum(1-|a_n|)<\infty$ then $B(z)$ is a holomorphic function on the disk with zeros precisely at the points $a_n$, $n=1,2,\dots$. The $a_n$ can be chosen so that every point of the unit circle is a limit point.

It would seem that the reciprocal would do what you want.

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  • $\begingroup$ Thanks, this does exactly what I was looking for! I'm going to do the exercise then! $\endgroup$ – Giovanni De Gaetano Jun 11 '14 at 10:04

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