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Journal of Applied Mathematics
Volume 2011 (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.

1. Introduction

In the following, we let be the space of infinitely differentiable functions with compact support, let be the space of infinitely differentiable functions with support contained in the interval , and let be the space of distributions defined on .

Now, let be a function in having the following properties:(i),(ii),(iii).

Putting for , it follows that is a regular sequence of infinitely differentiable functions converging to the Dirac delta-function . Further, if is an arbitrary distribution in and , then is a regular sequence converging to .

Since the theory of distributions is a linear theory, thus we can extend some of the operations which are valid for ordinary functions to the space of distributions and such operations are called regular operations such as: addition, multiplication by scalars; see [1]. Other operations can be defined only for a particular class of distributions or for certain restricted subclasses of distributions; these are called irregular operations such as: multiplication of distributions, convolution products, and composition of distributions; see [24]. Thus, there have been several attempts recently to define distributions of the form in , where and are distributions in ; see for example [58]. In the following, we are going to consider an alternative approach. As a starting point, we look at the following definition which is a generalization of Gel'fand and Shilov's definition of the composition involving the delta function [9], and was given in [6].

Definition 1.1. Let be a distribution in and let be a locally summable function. We say that the neutrix composition exists and is equal to on the open interval , with , if for all in , where for and is the neutrix, see [10], having domain the positive and range the real numbers, with negligible functions which are finite linear sums of the functions and all functions which converge to zero in the usual sense as tends to infinity.
In particular, we say that the composition exists and is equal to on the open interval if for all in .

Note that taking the neutrix limit of a function is equivalent to taking the usual limit of Hadamard's finite part of . The definition of the neutrix composition of distributions was originally given in [10] but was then simply called the composition of distributions.

The following three theorems were proved in [11], [8], and [12], respectively.

Theorem 1.2. The neutrix composition exists and for and , and for , and .

Theorem 1.3. The neutrix compositions and exist and for .

Theorem 1.4. The neutrix composition exists and for and , where

The next two theorems were proved in [13].

Theorem 1.5. The neutrix composition exists and for , and .
In particular, the composition exists and

Theorem 1.6. The neutrix composition exists and for and , where is the smallest non-negative integer greater than .
In particular, the composition exists and for and and for .

2. Main Results

We now prove the following theorem.

Theorem 2.1. The neutrix composition exists and for and , where In particular, the neutrix composition exists and

Proof. To prove (2.1), we first of all evaluate We have It is obvious that for .
Making the substitution , we have for large enough where It follows that and by applying the neutrix limit we obtain for .
When , we have Thus, if is an arbitrary continuous function, then We also have and it follows that If now is an arbitrary function in , then by Taylor's Theorem, we have where , and so on using (2.3) to (2.14). This proves (2.1) on the interval .
It is clear that for and so (2.1) holds for .
Now, suppose that is an arbitrary function in , where . Then, and so
It follows that on the interval . Since and are arbitrary, we see that (2.1) holds on the real line. This completes the proof of the theorem.

Corollary 2.2. The neutrix composition exists and for and .
In particular, the composition exists and

Proof. To prove (2.19), we note that and (2.19) now follows as above.
Equation (2.20) follows on noting that in the particular case , the usual limit holds in (2.10). This completes the proof of the corollary.

Theorem 2.3. The neutrix composition exists and for , where

Proof. To prove (2.22), we now have to evaluate We have Making the substitution , we have for large enough where It follows that and so by using the neutrix limit, we have for .
When , we have Thus, if is an arbitrary continuous function, then If now is an arbitrary function in , then by Taylor's Theorem, we have where , and so on using (2.25) to (2.31), proving (2.22) on the interval . However, it is clear that for and so (2.22) holds on the real line, completing the proof of the theorem.

Corollary 2.4. The composition exists and

Proof. To prove (2.34) note that in the particular case , the usual limits hold and then (2.34) is a particular case of (2.22). This completes the proof of the corollary.

For further related results on the neutrix operation of distributions, see [1222] and [2, 3, 23].

Acknowledgments

The authors would like to thank the referee(s) for the very constructive comments and suggestions that improved the paper. The paper was prepared when B. Fisher visited University Putra Malaysia and therefore the authors gratefully acknowledge that this research was partially supported by the University Putra Malaysia under the Research University Grant Scheme no. 05-01-09-0720RU.

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