1
$\begingroup$

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 $\Omega$ ?

$\endgroup$
  • $\begingroup$ Generally, no. Try to find (simple) counterexamples. $\endgroup$ – Daniel Fischer Feb 20 '15 at 15:19
  • $\begingroup$ @DanielFischer, right. Assume that $g$ can have only isolated singularities (essential, pole, removable). It is still false ? $\endgroup$ – mikis Feb 20 '15 at 15:32
0
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.