Complex Analysis of Real Functions IV: Non-Integrable Real Functions

TL;DR

This paper extends the complex-analytic framework to include a broad class of non-integrable real functions, enabling a unified treatment alongside integrable functions and distributions within the same structure.

Contribution

It demonstrates that certain non-integrable real functions can be represented within the existing complex-analytic structure, expanding the scope of the framework.

Findings

Non-integrable real functions can be represented in the complex-analytic structure.

Unified treatment of integrable and non-integrable functions within the same framework.

Extension of the complex analysis approach to broader classes of functions.

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 a certain class of non-integrable real functions can be represented within that same structure. In previous papers it was shown that essentially all integrable real functions, as well as all singular Schwartz distributions, can be represented within that same complex-analytic structure. The large class of non-integrable real functions which we analyze here can therefore be represented side by side with those other real objects, thus allowing all these objects to be treated in a unified way.