Complex hyper-power series and generalized complex analytic functions

TL;DR

This paper introduces hyperpower series within the framework of Robinson-Colombeau generalized numbers to extend classical complex analysis theorems to generalized holomorphic functions.

Contribution

It develops the theory of hyperpower series and generalizes key complex analysis theorems for generalized complex analytic functions in non-Archimedean settings.

Findings

Hyperpower series are defined using hyperfinite natural numbers.

Fundamental theorems like Goursat's, Liouville's, and the identity theorem are extended.

Convergence properties of hyperpower series are established.

Abstract

This paper studies the equivalence between generalized holomorphic functions (GHF) and complex analytic functions in the framework of Robinson-Colombeau generalized numbers. In every non-Archimedean ring, the use of ordinary series is severely restricted by the topological property that a series converges (in a topology of infinitesimal neighborhoods) if and only if its general term is infinitesimal. Consequently, classical Taylor series representations for generalized functions are limited to infinitesimal neighborhoods. To overcome this drawback, we introduce and develop the theory of hyperpower series, defined by summation over the set of hyperfinite natural numbers. We establish the foundational algebraic and topological properties of hyperpower series, including their radii of convergence and sets of convergence. Building on this, we define generalized complex analytic functions and extend several fundamental theorems of complex analysis to the GHF setting, specifically, providing generalizations of Goursat's theorem, Lioville's theorem, the identity theorem, and a Paley-Wiener type theorem.