ABSTRACT
A more detailed study of power series in several complex variables exhibits some properties quite unlike the familiar properties of power series in one complex variable. To simplify notation, only power series centered at the origin will be considered in this section; the results extend to the general case merely by a change of notation. As is well known, a power series ∑ icizi in one complex variable is absolutely convergent at all points z in some open disc |z|<r and divergent at all points z in the open exterior |z| > r, but the question of its convergence on the boundary circle |z| = r is generally a rather complicated one. That suggests that when examining the set of points at which a power series in several complex variables converges, it is convenient to consider initially the largest open set in which the series converges, the domain of convergence of the power series. A point Z ∈ ℂ n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203750063/1087b710-7222-48dd-8094-26eaf896ae1b/content/eq160.tif"/> is in the domain of convergence precisely when the power series converges at all points in an open neighborhood of Z. A power series necessarily converges absolutely at each point of its domain of convergence. A convenient terminology used in the discussion of such domains of convergence is the following:
