Allowable Changes of Variable for Recently Developed Improper Integral Definitions

TL;DR

This paper establishes sufficient conditions under which variable changes are valid in new improper integral definitions, clarifying when the existence of one integral implies the existence of the transformed integral.

Contribution

It provides the first comprehensive criteria for variable changes in recently developed improper integral definitions, extending previous understanding.

Findings

Conditions for valid variable changes are established.

Existence of one integral does not always imply the other's existence.

In certain cases, the existence of either integral guarantees the other's existence.

Abstract

Sufficient conditions for performing changes of the variable of integration when using the new definitions of improper integrals given in in "An Alternative Definition for Improper Integral with Infinite Limit" (arXiv:0805.3559v1) and "An Alternative Definition for Improper Integral with Finite Limits" (arXiv:0810.4654v1) are presented. These changes of variable are valid when both the initial and final integrals exist. In general, the existence of one of the integrals does not guarantee the existence of the other. In a restricted set of changes of variable, the existence of either integral guarantees existence of the other.