I was curious about the following question:
Let $U$ be a domain (open, connected subset) in $\mathbb C$. We say that a function $f: U \rightarrow \mathbb C$ satisfies property $P$ is for every $a \in U$, if $r$ is the largest radius of the open ball around $a$ contained in $U$, then there exists a holomorphic function $F_a$ on this ball which satisfies $F_a'(z) = f(z)$ for every $z$ in this ball.
What are some of the minimal conditions required on $U$ to ensure that a primitive of $f$ exists for every $f$ satisfying property $P$?
Certainly $U$ being simply connected is enough. But I feel like this should work for more general $U$.
Best Answer
Let $f \in h(U)$. If $D$ is an open disk contained in $U$ then $f$ has primitive in $D$ because $D$ is simply connected. So if $U$ satisfies your property then every $f \in H(U)$ has a primitvie. This is equivalent to the fact that $U$ is simply connected.