Theorem 2.4.5 of Hormander's book

Question

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:

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

2. 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 $g(z)$?

3. 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!

Answer 1

Overview. 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 $|D|$.

Suggestion. Work through Hormander's proof in a 2D special case. Try drawing an open region $|D|$ in 2D absolute space that worms around in the plane starting from an open nbd of the origin.

In Hormander's presentation, the idea is to use a single dilation parameter $\epsilon$ that parametrizes a family of smaller Reinhardt domains that blossom out to fill the full set $\Omega$.

Answers to your numbered questions.

1. The set $\Omega(\epsilon)$ is open because it can be expressed as the set of points where the continuous function $D(z)= \frac{|z|}{dist( z, \Omega^c)}$ satisfies an open inequality (e.g $D<1/\epsilon$). (P.S. The choice of this $D(z)$ is motivated by the desire to ensure that we have control on the effect of dilations in the next step.)

The nested family of open subsets $\Omega(\epsilon) \subset \Omega$ allow us to sneak up on the full set $\Omega$ with smaller Reinhardt subdomains of $\Omega$ each of which can be dilated up slightly and still fit in $\Omega$. That wiggle room allows us to write a multivariate Cauchy integral representation for $f(z)$ valid in each subdomain. The wiggle room ensures that the Cauchy integral can be evaluated by expanding the integrand as multivariate power series term-wise that converges on the subdomain. (Multivariate geometric series expansion. ) Hormander does this at the end of the proof.

2. The open subdomain $\Omega (\epsilon)$ is a countable union of open connected components, one of which contains the origin, which is the center point of all of our power series expansions. As we vary $\epsilon$ this special component balloons up to become all of $\Omega$. It is the only component that we really care about.

3. As stated in proof, " differentiation under the integral sign shows that it is holomorphic".