How to show that $D=\{ |z_1|<1\} \cup \{ |z_2|<1\} \subset \mathbb{C}^2$ is not a domain of holomorphy. What is the smallest domain of holomorphy $ S\supset D $ ?
I think we cannot just add the distinguished boundary (the torus $|z_1|=1, |z_2|=1$) to $D$ because it would violate analiticity. Should we use logarithmically convexity argument to extend our domain? Can anyone help?
Obrigado.