# Can a holomorphic function be globally represented by a power series on an open connected set?

I have a theorem in my book (Stein) which says:

Suppose $$f$$ is holomorphic in an open set $$\Omega$$. If $$D$$ is a disc centered at $$z_0$$ and whose closure is contained in $$\Omega$$, then $$f$$ has a power series expansion at $$z_0$$

$$f(z) = \sum_{n=0}^{\infty} a_n(z-z_0)^n$$

for all $$z \in D$$.

Let's say our open set $$\Omega$$ contains $$0$$; let $$P(z)$$ be the power series of $$f$$ centered at $$0$$. Obviously, if $$\Omega$$ is not connected, $$P(z)$$ does not have to represent $$f$$ at all points of $$\Omega$$. However, what about the case where $$\Omega$$ is connected? Are there any counterexamples?

The set in which a power series converges is always an open disk together with some subset of the boundary of the disk. So, if a function $$f$$ can be represented by a single power series centered at $$0$$ on an open set $$\Omega$$, then $$\Omega$$ must be contained in an open disk $$D$$ (possibly of infinite radius) around $$0$$ such that $$f$$ extends holomorphically to $$D$$ (and conversely, since if $$f$$ is holomorphic on such a disk $$D$$ then its Taylor series at $$0$$ converges to it on the whole disk).
This gives lots of counterexamples. For instance, if $$f(z)=\frac{1}{z-1}$$, then $$f$$ is holomorphic on $$\Omega=\mathbb{C}\setminus\{1\}$$ but cannot be represented by a power series centered at $$0$$ on this domain since otherwise $$f$$ would need to extend holomorphically to all of $$\mathbb{C}$$ since $$\mathbb{C}$$ is the only disk centered at $$0$$ that contains $$\Omega$$.
A power series $$f(z) = \sum_{n\geq 0} a_n(z-z_0)^n$$ converges on a disk centered on $$z_0$$. So assuming that $$\Omega$$ is not a disk implies that there is a function which cannot be given by a single power series.
Pick a fairly general $$\Omega$$ and let $$\{z_j\}$$ be a sequence of points in $$\Omega$$ acummulating on every boundary point of $$\Omega$$. By a Theorem of Weierstrass there is a holomorphic function $$h\colon \Omega \to \mathbb{C}$$ vanishing on each $$z_j$$ and nowhere else.
Since $$\Omega$$ is not a disk, the only way $$h$$ could be given globally by a unique power series would be $$h$$ being holomorphic on a disk $$D$$ containing $$\Omega$$. This would lead us to a contradiction since $$D$$ would contain some boundary point of $$\Omega$$ which is also an acummulation point for the zeros of $$h$$ and the identity principle would tell us that $$h\equiv 0$$.