Generalized improper integral definition for finite limit
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.
Keywords
Cite
@article{arxiv.0810.4654,
title = {Generalized improper integral definition for finite limit},
author = {Michael A. Blischke},
journal= {arXiv preprint arXiv:0810.4654},
year = {2012}
}
Comments
An error in the initial version of this paper was discovered, affecting the last section of the body. That section has been removed from this version. Also, substantial modifications were made to the style of the paper