Dirac delta as a generalized holomorphic function

TL;DR

This paper introduces a new framework for generalized holomorphic functions using non-Archimedean fields, extending classical complex analysis theorems to include Dirac delta and distributions.

Contribution

It defines generalized holomorphic functions in a non-Archimedean setting and extends key classical theorems, addressing limitations of existing theories like Colombeau's.

Findings

Classical theorems extended to generalized functions

Embedding of distributions into the new framework

Closure under composition and nonlinear operations

Abstract

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.