arXiv subjects

Sekar Nugraheni

Publications and source records attributed to Sekar Nugraheni.

2 recordsLinked to original sources

arXiv2026

Complex hyper-power series and generalized complex analytic functions

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.

math.FA↗

arXiv2025

Dirac delta as a generalized holomorphic function

The definition of a non-trivial space of generalized functions of a complex variable allowing to consider derivatives of continuous functions is a non-obvious task, e.g. because of Morera theorem, because distributional Cauchy-Riemann equations implies holomorphicity and of course because including Dirac delta seems incompatible with the identity theorem. Surprisingly , these results can be achieved if we consider a suitable non-Archimedean extension of the complex field, i.e. a ring where infinitesimal and infinite numbers return to be available. In this first paper, we set the definition of generalized holomorphic function and prove the extension of several classical theorems, such as Cauchy-Riemann equations, Goursat, Looman-Menchoff and Montel theorems, generalized differentiability implies smoothness, intrinsic embedding of compactly supported distributions, closure with respect to composition and hence non-linear operations on these generalized functions. The theory hence addresses several limitations of Colombeau theory of generalized holomorphic functions. The final aim of this series of papers is to prove the Cauchy-Kowalevski theorem including also distributional PDE or singular boundary conditions and nonlinear operations.

math.FA↗