Generalized improper integral definition for finite limit

TL;DR

This paper introduces a new generalized definition for one-dimensional improper integrals with finite limits, expanding the class of integrable functions while maintaining key properties like linearity and equivalence to traditional definitions.

Contribution

It presents a novel generalized integral definition for finite limits, extending integrability to previously non-integrable functions and establishing its equivalence to the infinite limit case.

Findings

The new definition preserves linearity and uniqueness.

It is equivalent to the infinite limit integral definition via a change of variables.

Criteria for interchanging integration and differentiation are derived.

Abstract

A generalization of the definition of a one-dimensional improper integral with a finite limit is presented. The new definition extends the range of valid integrals to include integrals which were previously considered to not be integrable. This definition is shown to be equivalent to the infinite limit definition presented in "Generalized improper integral definition for infinite limit" (arXiv:0805.3559) via a particular change of variable of integration. The definition preserves linearity and uniqueness. Integrals which are valid under the conventional definition have the same value under the new definition. Criteria for interchanging the order of integration and differentiation, and for interchanging the order with a second integration, are obtained. Examples are provided.