# Example of a meromorphic function with no analytic continuation outside the unit disc

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?

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.