An Alternative Definition for Improper Integral with Finite Limits

Mathematics – Classical Analysis and ODEs

Scientific paper

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Scientific paper

An alternative definition of one-dimensional improper integrals with finite limits 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 "An Alternative Definition for Improper Integral with Infinite Limit" (arXiv:0805.3559v1) via a particular change of variable of integration. For a more general class of change of variable, the infinite limit and finite limit definitions give the same values when both exist. 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.

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