Theorem 2.4.5 of Hormander's book Hi I was trying to understand Theoerem 2.4.5 of Hormander's book "An Introduction to Complex Analysis in Several Variables". It shows the existence and uniqueness of power series expansion of any holomorphic function in a connect Reinhardt domain containing the origin. However I don't have a big picture of how to prove it, like why we construct $\Omega_\epsilon$. Below are some of my questions so far:

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*how to show $\Omega_\epsilon$ is open? I was trying to use definition and inverse triangle inequality to prove it but it was unsuccessful.


*Why do we need the component of $\Omega_\epsilon$, denoted by $\Omega_\epsilon^\prime$. Furthermore, why we need to note that $\Omega = \cup_{\epsilon > 0} \Omega_\epsilon^\prime$ and then when $z \in \Omega_\epsilon^\prime$, we define a such integral, called $g(z)$?


*How to show $g(z)$ is holomorphic in $\Omega_\epsilon$? I know if we can show it, then we can apply the principle of analytic continuation to derive $f = g$.
Any help will be appreciated!

 A: A Reinhardt domain $D$ is a region that can be characterized  by its schematic diagram in absolute space (the space of absolute values of the coordinate entries). For example, in two complex variables, if $D$ is Reinhardt, then $(z_1, z_2)\in D \iff (|z_1|,|z_2|)\in D $ Consequently, to visualize $D$  we first sketch the diagram in absolute space comprising  all ordered pairs of non-negative real values $ (r_1, r_2)$ in $D$.  Denote this absolute region in the first quadrant of the real plane $[0,\infty] \times [0, \infty]$ by $|D|$.
For each pair of positive such radii  in $|D|$ , a point pair $p=(r_1, r_2)$, we can construct a  multivariable Cauchy integral formula $I(p)$ that we suspect  ought to properly represent a given analytic function  $f(z)$ inside the associated poly-disk region $R(p)= \{(z_1, z_2) :|z_1|<r_1, |z_2|<r_2\}$.  To prove this representation $I(p)$ for $f(z)$ is actually independent of $p \in |D|$, we use two ideas: (i) under a small perturbation to nearby point $p'\in|D|$ the integral representation $I(p')$ agrees with that of $I(p)$ on the intersection of the two associated poly-disks $R(p)\cap R(p')$ (a region that contains the origin).  (Hormander uses dilations to perturb $p$ to $p'$. Thus one dilated poly-disk is a subregion of the other.) Also (ii) note that if we deform the point $p$ to be sufficiently near the origin we are indeed recovering $f(z)$.
Suggestion.Try drawing an open region $|D|$  in absolute space that worms around in the plane  starting from an open nbd of the origin to see how the global property is pieced together from local information.
(Broadly speaking this has the flavor of the typical path deformation principle in one complex variable for passing from local to global uniqueness of a representation of an analytic function.)
