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Journal of Applied Mathematics
Volume 2011, Article ID 612353, 12 pages
http://dx.doi.org/10.1155/2011/612353
Research Article

On the Neutrix Composition of the Delta and Inverse Hyperbolic Sine Functions

1Department of Mathematics, University of Leicester, Leicester LE1 7RH, UK
2Department of Mathematics and Institute for Mathematical Research, University Putra Malaysia, 43400 UPM, Serdang, Selangor, Malaysia

Received 24 December 2010; Accepted 8 April 2011

Academic Editor: A. A. Soliman

Copyright © 2011 Brian Fisher and Adem Kılıçman. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let be a distribution in and let be a locally summable function. The composition of and is said to exist and be equal to the distribution if the limit of the sequence is equal to , where for and is a certain regular sequence converging to the Dirac delta function. In the ordinary sense, the composition does not exists. In this study, it is proved that the neutrix composition exists and is given by , for and , where . Further results are also proved.