Domain of holomorphy

Remarks

Example: every convex open set $D \subsetneq \C^n$ is a domain of holomorphy. For a boundary point $p$ there is a complex linear function $\ell$ with $\operatorname{Re} \ell(z) < \operatorname{Re} \ell(p)$ for all $z \in D$, and $f_p(z) = 1/(\ell(z) - \ell(p))$ is holomorphic on $D$ and unbounded near $p$. Suppose $U_1, U_2$ satisfied the three clauses, and let $W$ be a component of $D \cap U_2$ that meets $U_1$. Since $U_2$ is connected and not contained in $D$, it contains a boundary point $p$ of $W$, and $p$ lies on the boundary of $D$. The extension $g$ of $f_p$ equals $f_p$ on $W$ by the identity theorem, and $g$ is continuous at $p$. So $f_p$ is bounded near $p$ on $W$, which is a contradiction.

Non-example: $D = \set{z \in \C^2 : 0 < \abs{z} < 1}$. Take $U_2 = \set{\abs{z} < 1}$ and $U_1 = D$. Clauses 1 and 2 hold, and clause 3 holds by Hartogs's extension theorem with $K = \set{0}$.

Every open subset of $\C$ is a domain of holomorphy.