I am going through Theorem $1.1$ from the book Holomorphic Functions and Integral Representations in Several Complex Variables by Michael Range (Page no. $43$) which states the following $:$
Theorem $1.1 :$ Let $n \geq 2$ and suppose that $0 \lt r_j \lt 1$ for $1 \leq j \leq n.$ Then every function $f$ holomorphic on the domain $$H(r) : = \{z \in \mathbb C^n\ : |z_j| \lt 1\ \text{for}\ j \lt n,\ r_n \lt |z_n| \lt 1 \} \cup \{z \in \mathbb C^n\ :\ |z_j| \lt r_j\ \text{for}\ j \lt n,\ |z_n| \lt 1 \}$$ has a unique holomorphic extension $\widehat {f}$ to the polydisk $P^n (0,1).$
In the proof $\widehat {f}$ on the polydisk $P^n (0, (1',\delta))$ (where $1' = (1, \cdots, 1) \in \mathbb C^{n-1}$ and $r_n \lt \delta \lt 1$) is defined in the following way $$\tag{1.1}\widehat {f} (z', z_n) = \frac {1} {2 \pi i} \int_{|\zeta| = \delta} \frac {f(z', \zeta)} {\zeta - z_n}\ d\zeta$$ where $z' \in P^{n-1} (0, 1')$ and $|z_n| \lt \delta.$ It is clear that $\widehat {f}$ is continuous on $P^n (0,(1', \delta))$ by dominated convergence theorem. But it is not clear as to why $\widehat {f}$ is holomorphic in each variable which is claimed by the author. Here's the proof $:$
I have somehow managed to understand why $\widehat {f}$ is holomorphic with respect to the $n$-th variable because if we fix $z' \in P^{n-1} (0, 1')$ the integrand on the right of the equation $(1.1)$ becomes continuous on $|\zeta| = \delta$ and hence by Cauchy's integral formula in one complex variable it follows that the integral on the right has a power series expansion on $|z_n| \lt \delta$ and hence for any $z' \in P^{n-1} (0, 1')$ the function $\widehat {f} (z', \cdot)$ is holomorphic on $|z_n| \lt \delta,$ which is what we wanted to show. But I am unable to show why $\widehat {f}$ is holomorphic with respect to $j$-th coordinate for $1 \leq j \lt n.$ Any help in this regard would be warmly appreciated.
Thanks for your time.
EDIT $:$ We need to show that Cauchy-Riemann equations hold i.e. $$\frac {\partial \widehat {f}} {\partial \overline {z_j}} = 0$$ for all $1 \leq j \lt n.$ But how to justify the interchange between limit and the integrals? I think that I need to somehow use DCT to be able to conclude that. Any idea?