3

Suppose that $z\in\mathbb{C}$ and $f$ is a holomorphism on $U$ such that there is no holomorphism $g$ extending $f$ on $V\supseteq U$ such that $z\in V$. Is it possible for there to be such a $g$ and a holomorphism $h$ on $W$ with $V\cap W\neq\emptyset$ such that $g|_{V\cap W}=h|_{V\cap W}$ with $z\in W$? That is, are singularities preserved by analytic continuation?

Thomas Anton
  • 2,336

1 Answers1

2

The answer to radius of convergence with Taylor series, functions with branch cuts is related, and indeed its negative answer can be easily seen to imply a negative answer to this question. The global analytic function $\frac{1}{\log(z)}$ is given as a counterexample. Since $\log(z)=0$ in only one branch, there is a singularity at $0$, but also a disappearing one.

Thomas Anton
  • 2,336
  • 1
    Glad to see my question helped, something like $\frac1{e^z-e-1/\log z}$ will have moving poles (for $k$ large enough then $e^z-e-1/(2ik\pi +\log z)$ has a zero near $z=1$) – reuns Dec 12 '20 at 13:26
  • Thanks, I was just looking at complex plots and starting to notice this. Fixed. – Thomas Anton Dec 12 '20 at 13:32