Show that this holomorphic function can be extended to $D_{2}((0,0) ;(2,2))$

Question

consider a domain in $C^{2}$:$\Omega=D_{2}((0,0) ;(1,2)) \cup\left\{(z, w) \in \mathbb{C}^{2}:|z|<2 \text { and } 1<|w|<2\right\}$ and $f \in \operatorname{Hol}(\Omega)$, I want to show that f can be extended to $D_{2}((0,0) ;(2,2))$ and I also want to show that $\Omega$ is not a domain of holomorphy， I want to use Cauchy integral formula to formulate the extension but I don't know how to do it, could you please help me? thank you

Answer 1

Here are some hints:

1. Write $$ \varphi(z,w) = \frac{1}{2\pi} \int_{\gamma} \frac{f(z,\xi)}{\xi-w} \, d\xi $$ For $\gamma$ being some circle, say of radius 1.5. Note that this is a well-defined holomorphic function on a fairly large set (figure out what set).

2. Note that when $z$ is small, you can apply Cauchy's formula to $w \mapsto f(z,w)$.

3. Apply identity theorem to piece things together.