## Domain of Holomorphy

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?

Thank you!

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 You can use the same arguments as for a Hartogs' figure to show that any holomophic function on $D$ extends to $\mathbb{C}^2$. Also beware that in general a smallest domain of holomorphy $S\subset \mathbb{C}^2$ containing $D$ need not exist. – Jérôme Poineau Apr 19 2012 at 9:14