Journal of Applied MathematicsVolume 2011 (2011), Article ID 846736, 13 pagesdoi:10.1155/2011/846736

Research Article

On the Composition and Neutrix Composition of the Delta Function with the Hyperbolic Tangent and Its Inverse Functions

Brian Fisher¹ and Adem Kılıçman²

¹Department of Mathematics, University of Leicester, Leicester LE1 7RH, UK²Department of Mathematics and Institute for Mathematical Research, University Putra Malaysia, Serdang, 43400 Selangor, Malaysia

Received 20 April 2011; Accepted 13 June 2011

Academic Editor: Andrew Pickering

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. It is proved that the neutrix composition exists and for , where is the integer part of and the constants are defined by the expansion , for Further results are also proved.

1. Introduction

In the following, we let be the space of infinitely differentiable functions with compact support and let be the space of infinitely differentiable functions with support contained in the interval . A distribution is a continuous linear functional defined on . The set of all distributions defined on is denoted by and the set of all distributions defined on is denoted by .

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, we can extend some operations which are valid for ordinary functions to space of distributions; such operations may be called regular operations, and among them are addition and multiplication by scalars, see [1]. Other operations can be defined only for particular class of distributions; these may be called irregular, and among them are multiplication of distributions, see [2], and composition [3, 4], convolution products, see [5–7], further in [8], where some singular integrals were defined as distributions. Note that it is a difficult task to give a meaning to the expression , if and are singular distributions.

Thus there have been several attempts recently to define distributions of the form in , where and are distributions in , see for example [9–12]. 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 [13], and was given in [10].

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 [14], having domain the positive integers 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 . If are two distributions then in the ordinary sense the composition does not necessarily exist. Thus the definition of the neutrix composition of distributions was originally given in [10] but was then simply called the composition of distributions.

We also note that, Ng and van Dam applied the neutrix calculus, in conjunction with the Hadamard integral, developed by van der Corput, to the quantum field theories, in particular, to obtain finite results for the coefficients in the perturbation series. They also applied neutrix calculus to quantum field theory, and obtained finite renormalization in the loop calculations, see [15, 16].

The following two theorems involving derivatives of the Dirac-delta function were proved in [17] and [12], respectively.

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

Theorem 1.3. The compositions and exist and for .

The following two theorems were also proved in [18].

Theorem 1.4. The neutrix composition exists and for and . In particular, the composition exists and

Theorem 1.5. The neutrix composition exists and for and , where is the smallest nonnegative integer greater than . In particular, the composition exists and for and and for .

The following two theorems were proved in [4].

Theorem 1.6. The neutrix composition exists and for and , where In particular, the neutrix composition exists and

Theorem 1.7. The neutrix composition exists and for , where In particular

2. Main Result

In the next theorem, the constants are defined by the expansion for .

Theorem 2.1. The neutrix composition exists and for , where is the integer part of . In particular, the neutrix compositions and exist and for .

Proof. To prove (2.2), first of all we evaluate We have It is obvious that for .Making the substitution , we have for large enough It follows that for , where denotes the integer part of for .In particular, when , we have . It follows from (2.9) that and so ordinary limit exists for , since .Further, when , we have and . It follows from (2.10) that for , since , and .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 (2.6), (2.7), (2.10), (2.15), and (2.17). This proves (2.2) on the interval . It is also clear that for and so (2.2) 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 (2.2) holds on the real line.Equations (2.3) and (2.4) are just particular cases of (2.2). Equation (2.3) follows on using (2.12) and (2.4) follows on using (2.13). This completes the proof of the theorem.

Corollary 2.2. The neutrix composition exists and for , where is the integer part of . In particular, the composition exists and the neutrix composition exists and for .

Proof. To prove (2.22), we note that and (2.22) now follow as above. Further, (2.23) are particular cases of (2.22) and so follows immediately. Note that in the particular case , the ordinary limit exists in (2.12) and so the composition exists in this case. This completes the proof of the corollary.

In the next theorem, the constants are defined by the following expansion for .

Theorem 2.3. The neutrix composition exists and for and , where is the smallest integer for which .

Proof. To prove (2.26), first of all we evaluate We have It is obvious that Making the substitution , we have for large enough and it follows that In particular, when and , we have and then When , we have Thus, if is an arbitrary continuous function, then and so since .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 (2.9) and (2.12). This proves (2.26) on the interval . It is clear that for and so (2.2) 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 (2.26) holds on the real line.

Corollary 2.4. The neutrix composition exists and for and , where is the smallest integer for which . In particular, the composition exists and

Proof. To prove (2.42), we note that and (2.42) now follows as above. Equation (2.43) is a particular case of (2.42) and so follows immediately. Note that in the particular case and , the ordinary limit exists in (2.12), and so the composition exists in this case. For some related results on the neutrix composition of distributions, see [19–22].

Acknowledgments

The paper was prepared when the first author visited University Putra Malaysia, therefore the authors gratefully acknowledge that this research was partially supported by the University Putra Malaysia under the Research University Grant Scheme 05-01-09-0720RU. The authors would also like to thank the referees for valuable remarks and suggestions on the previous version of the paper.

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