arXiv subjects

Michael A. Blischke

Publications and source records attributed to Michael A. Blischke.

3 recordsLinked to original sources

arXiv2012

Generalized improper integral definition for finite limit

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.

math.CA↗

arXiv2012

Generalized improper integral definition for infinite limit

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.

math.CA↗

arXiv2009

Allowable Changes of Variable for Recently Developed Improper Integral Definitions

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.

math.CA↗