2021/03/25 by G. P. Balakumar, Balakumar, G. P.
Mathematics · #Meromorphic and Entire Functions #Mathematics and Applications #Holomorphic and Operator Theory
paper · pdf · doi:10.48550/arxiv.2103.13986
It is a classical fact that domains of convergence of power series of several complex variables are characterized as logarithmically convex complete Reinhardt domains; let D \subsetneq ℂN be such a domain. We show that a necessary as well as sufficient condition for a power series g to have D as its domain of convergence is that it admits a certain decomposition into elementary power series; specifically, g can be expressed as a sum of a sequence of power series gn with the property that each of the logarithmic images Gn of their domains of convergence are half-spaces, all containing the logarithmic image G of D and such that the largest open subset of ℂN on which all the gn's and g converge absolutely is D. In short, every power series admits a decomposition into elementary power series. The proof of this leads to a new way of arriving at a constructive proof of the aforementioned classical fact. This proof inturn leads to another decomposition result in which the Gn's are now wedges formed by intersections of pairs of \it supporting half-spaces of G. Along the way, we also show that in each fiber of the restriction of the absolute map to the boundary of the domain of convergence of g, there exists a singular point of g.