# Extension of holomorphic function

Let $\Omega$ be a simply-connected domain in $\mathbb{C}$, and $A$ be a closed simply-connected set such that $A\subset \Omega$ and $\Omega\setminus A$ is connected. Let $f$ be a holomorphic function in $\Omega\setminus A$. It is true that $f$ can be extended to meromorphic function $g$ in $\Omega$ ?

• Generally, no. Try to find (simple) counterexamples. – Daniel Fischer Feb 20 '15 at 15:19
• @DanielFischer, right. Assume that $g$ can have only isolated singularities (essential, pole, removable). It is still false ? – mikis Feb 20 '15 at 15:32

Take $\Omega$ to be the whole of the complex plane, and $A$ to be the complement of the unit disk. There are functions that have the unit circle as natural boundary. For example, the function $1+z+z^2+z^4+z^8+z^{16}+\cdots$ does not extend to any open set containing the unit disk.