In the single variable case, the power series
$$\sum_{n=0}^\infty a_n z^n $$
defines an entire function, provided that
$$R^{-1}:=\limsup_{n \to \infty} |a_n|^{1/n}=0. $$
Moreover, if $R^{-1} >0$, the series converges for $|z|<R$, and has some singularity on $|z|=R$.
I'd like to know what happens in the case of $n \geq 2$ variables: Using multi-index notation, a power series has the form
$$\sum_{|\alpha| \geq 0} a_\alpha z^\alpha .$$
Is there a way to know, based on the coefficients $\{a_\alpha\}$, if the series defines an entire function? Is there any way to gain insights on the region of convergence at all?
Thank you!
Best Answer
This answer cites passages from Lecture notes on several complex variables by H.P. Boas. The author provides information about the region of convergence of multivariate power series.
In the following we want to characterize the domains of convergence. But before that we have to introduce a little bit of notation.
We consider the normed vector space $\mathbb{C}^n$ with the Euclidian norm: $$\|(z_1,\ldots,z_n)\|_2=\sqrt{|z_1|^2+\cdots+|z_n|^2}.$$ A point $(z_1,\ldots,z_n)$ in $\mathbb{C}^n$ is denoted by a single letter $z$.
If $\alpha$ is an $n$-dimensional vector all of whose coordinates are nonnegative integers, then $z^\alpha$ means the product $z_1^{\alpha_1}\cdots z_n^{\alpha_n}$; the notation $\alpha!$ abbreviates the product $\alpha_1!\cdots\alpha_n!$; and $|\alpha|$ means $\alpha_1+\cdots+\alpha_n$. In this multi-index notation, a multivariable power series can be written in the form $\sum_{\alpha}c_{\alpha}z^{\alpha}$ as abbreviation for $\sum_{\alpha_1=0}^\infty\cdots\sum_{a_n=0}^\infty c_{\alpha_1\ldots\alpha_n}z^{\alpha_1}\cdots z^{\alpha_n}$.
Since the value of a series depends in general on the order of summation, and there is no canonical ordering of $n$-tuples on nonnegative integers when $n>1$, we restrict attention to absolute convergence. The terms of an absolutely convergent series can be reordered arbitrarily without changing the value of the sum (or the convergence of the sum).
The preceding discussion shows that a convergence domain is necessarily a complete, logarithmically convex Reinhardt domain. The following theorem of Hartogs says that this geometric property characterizes domains of convergence of power series.
The relationship of holomorphy of multi-variate functions with Reinhard domains of convergence of multi-variate power series is elaborated in the following pages. The next theorem from section 2.5 clarifies the existence of power series and covers also entire multivariate power series.
Hint: Chapter I: Entire functions in Several Complex Variables, vol. III by L.I. Ronkin might be useful for further reading. The introductory section provides a lot of additional references.