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

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.

• Why do you want to define it in terms of conditional convergence? If I were defining it, I would define it using absolute convergence. Indeed, that is how I would interpret the definition you stated: if no order is specified, that means that the series converges in the more general sense of a series indexed by an arbitrary set (which means it converges regardless of the order). Feb 3, 2017 at 4:53

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).
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.