Holomorphic function of several variables: what is the definition saying?

Question

Here is a typical definition of holomorphic function in several variables:

This definition is imprecise.

Note that there is nothing mentioned about absolute convergence. For small polydisks $\Delta$ about $w$, it will be the case the given series convergence absolutely, so the order of terms doesn't matter.

But, let's pretend we don't know anything about absolute convergence yet. Let $S$ be the set of bijections of $\mathbb{N}$ onto $\mathbb{N}^n$. The choice of such a bijection gives us a way to arrange the above sum. If $\phi \in S$, and we write $\phi(v) = (\phi(v)_1, ... , \phi(v)_n)$, let

$$f(z,\phi) = \sum\limits_{v=0}^{\infty} a_{\phi(v)_1, ... , \phi(v)_n} (z_1 - w_1)^{\phi(v)_1} \cdots (z_n - w_n)^{\phi(v)_n}$$

So, how should a holomorphic function be defined? If you were writing that definition and wanted to be more explicit, would you say:

There exists a $\phi \in S$ such that for all $z \in U$, $f(z,\phi)$ converges.

or would you say:

For all $z \in U$, there exists a $\phi \in S$ such that $f(z,\phi)$ converges.

Answer 1

First of all, a simpler definition can be given by requiring the function to be complex-differentiable at every point of the set. Where the definition is word-by-word analogous to the real differentiability, that is a function $f(z_1,\ldots, z_n)$ is complex-differentiable at a point, if it admits a complex-linear approximation (in the usual sense).

Furthermore, as a consequence of Hartogs' theorem, it suffices to require the function to be holomorphic separately on each complex line with all coordinates but one fixed. See the book Several Complex Variables and the Geometry of Real Hypersurfaces (Studies in Advanced Mathematics) by John P. D'Angelo for more details.

Second, you do not need any ordering of map $\phi$ (see the question) for the multi-indices in the power series, in order to define the convergence. It can be defined completely generally for any series $\sum_{a\in A} c_a$ indexed over any set $A$:

We say that $c = \sum_{a\in A} c_a$ if for any $\epsilon>0$ there exists a finite subset $A_\epsilon\subset A$ such that

$$|c-\sum_{a\in B} c_a|<\epsilon$$

holds whenever $B\subset A$ is any finite subset containing $A_\epsilon$. That, in particular, will also imply the absolute convergence.