Generalized improper integral definition for infinite limit

TL;DR

This paper introduces a generalized definition for one-dimensional improper integrals with infinite limits, expanding the class of integrable functions while maintaining key properties like linearity and uniqueness.

Contribution

It proposes a new integral definition that includes previously non-integrable functions and establishes criteria for interchanging limits, derivatives, and integrals under this new framework.

Findings

The new definition preserves linearity and uniqueness.

Integrals valid under the conventional definition remain unchanged.

Criteria for interchanging limits, derivatives, and integrals are established.

Abstract

A generalization of the definition of a one-dimensional improper integral with an infinite 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 preserve 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 of integration with a second integration, are determined. Examples are provided. A restriction on changing the variable of integration using integration by substitution with the new definition is demonstrated.