Differential Calculus and Integration of Generalized Functions over Membranes

TL;DR

This paper advances the calculus of generalized functions by introducing membranes, extending integrals, and proving key theorems like Cauchy's formula and Goursat's theorem, with applications to solving generalized transport equations.

Contribution

It introduces the concept of membranes and extends integral definitions, generalizing classical complex analysis results to a broader setting.

Findings

Proved a generalized Cauchy formula.

Established Goursat's theorem for generalized holomorphic functions.

Provided explicit solutions to generalized transport equations.

Abstract

In this paper we continue the development of the differential calculus started by Aragona-Ferandez-Juriaans. Guided by the topology introduced recently by those authors we introduce the notion of membranes and extend the definition of integrals given in [2] to integrals defined on membranes. We use this to prove a generalized version of teh Cauchy formula and to obtain the Goursat Theorem for generalized holomorphic functions. We also show that the generalized transport equation can be solved giving an explicit solution