Complex Analysis of Real Functions II: Singular Schwartz Distributions

TL;DR

This paper extends the complex-analytic framework within the unit disk to include singular Schwartz distributions, enabling a unified treatment of integrable functions and singular objects in complex analysis.

Contribution

It introduces a method to represent singular Schwartz distributions within the complex-analytic structure of the unit disk, building on previous work that represented integrable functions.

Findings

Singular Schwartz distributions can be represented in the complex-analytic structure.

Unified treatment of integrable functions and singular objects in complex analysis.

Extension of the framework to include singular distributions.

Abstract

In the context of the complex-analytic structure within the unit disk centered at the origin of the complex plane, that was presented in a previous paper, we show that singular Schwartz distributions can be represented within that same structure, so long as one defines the limits involved in an appropriate way. In that previous paper it was shown that essentially all integrable real functions can be represented within the complex-analytic structure. The infinite collection of singular objects which we analyze here can thus be represented side by side with those real functions, thus allowing all these objects to be treated in a unified way.