Are there examples for $E$ so that $D$ is a domain of holomorphy?

Let $$E$$ be a compact set in $$\mathbb{C^n}$$. Let $$D=\mathbb{C^n}-E$$.

a. Find an example for an $$E$$ so that $$D$$ is not a domain of holomorphy.

b. Are there examples for $$E$$ so that $$D$$ is a domain of holomorphy?

So, if $$E$$ is a compact set so that $$D$$ is connected, then by Hartogs Lemma, $$D$$ is not a domain of holomorphy. So an example for $$E$$ would be closed unit ball.

For part (b) if there are such examples then $$D$$ should be disconnected. I assume being disconnected is not suffices to decide that it is not a domain of holomorphy even though we use the term "domain". I tried few examples like several variable annulus etc. but everytime I got a component of $$D$$ which is not a domain of holomorphy. So far I was unable to prove that no such examples too.

Any help is appreciated.

• You can always apply Hartogs' theorem to the unbounded component of $D$. Jul 6 '17 at 14:29
• Ok so then $D$ is never a domain of holomorphy right? Jul 6 '17 at 14:33
• Not with standard definitions of domain of holomorphy. Jul 6 '17 at 14:37
• What do you mean? I am confused. Jul 6 '17 at 14:41
• For standard definitions of DoH, $D$ never is a DoH. Jul 6 '17 at 14:43

If $E$ is a nonempty compact set and $f \in \mathscr{O}(D \backslash E)$, then there exists a holomorphic function $\widetilde{f}$ such that $f(z) = \widetilde{f}(z)$ for all $z \in D \backslash E$ and $\widetilde{f}$ is holomorphic for all $z \in D$. That is, compact sets are removable. This forms the content of Hartog's extension theorem, see page 172 of Shabat's Introduction to Complex Analysis -- Part II.
So unless $E$ is empty, $D$ cannot be a domain of holomorphy.