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Research Article | Open Access

Volume 2011 |Article ID 846736 | https://doi.org/10.1155/2011/846736

Brian Fisher, Adem Kılıçman, "On the Composition and Neutrix Composition of the Delta Function with the Hyperbolic Tangent and Its Inverse Functions", Journal of Applied Mathematics, vol. 2011, Article ID 846736, 13 pages, 2011. https://doi.org/10.1155/2011/846736

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

Accepted13 Jun 2011
Published01 Aug 2011

#### 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 . Other operations can be defined only for particular class of distributions; these may be called irregular, and among them are multiplication of distributions, see , and composition [3, 4], convolution products, see , further in , 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 . 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 , and was given in .

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 , 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  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  and , 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 .

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 .

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 .

#### 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.

1. H. Eltayeb, A. Kılıçman, and B. Fisher, “A new integral transform and associated distributions,” Integral Transforms and Special Functions, vol. 21, no. 5-6, pp. 367–379, 2010.
2. B. Fisher and A. Kılıçman, “A commutative neutrix product of ultradistributions,” Integral Transforms and Special Functions, vol. 4, no. 1-2, pp. 77–82, 1996.
3. B. Fisher and A. Kılıçman, “On the composition and neutrix composition of the delta function and powers of the inverse hyperbolic sine function,” Integral Transforms and Special Functions, vol. 21, no. 12, pp. 935–944, 2010.
4. B. Fisher and A. Kılıçman, “On the neutrix composition of the delta and inverse hyperbolic sine functions,” Journal of Applied Mathematics, vol. 2011, Article ID 612353, 12 pages, 2011. View at: Publisher Site | Google Scholar
5. A. Kılıçman, “On the commutative neutrix product of distributions,” The Indian Journal of Pure and Applied Mathematics, vol. 30, no. 8, pp. 753–762, 1999. View at: Google Scholar | Zentralblatt MATH
6. A. Kılıçman, “A comparison on the commutative neutrix convolution of distributions and the exchange formula,” Czechoslovak Mathematical Journal, vol. 51, no. 3, pp. 463–471, 2001.
7. A. Kılıçman, B. Fisher, and S. Pehlivan, “The neutrix convolution product of ${x}_{+}^{\lambda }$${\mathrm{ln}}^{r}{x}_{+}$ and ${x}_{-}^{\mu }{\mathrm{ln}}^{s}{x}_{-}$,” Integral Transforms and Special Functions, vol. 7, no. 3-4, pp. 237–246, 1998.
8. A. Kılıçman and H. Eltayeb, “A note on defining singular integral as distribution and partial differential equations with convolution term,” Mathematical and Computer Modelling, vol. 49, no. 1-2, pp. 327–336, 2009.
9. P. Antosik, “Composition of distributions,” Tech. Rep. 9, University of Wisconsin, 1989. View at: Google Scholar
10. B. Fisher, “On defining the change of variable in distributions,” Rostocker Mathematisches Kolloquium, no. 28, pp. 33–40, 1985. View at: Google Scholar | Zentralblatt MATH
11. B. Fisher, “On defining the distribution ${\left({x}_{+}^{r}\right)}_{-}^{-s}$,” Univerzitet u Novom Sadu. Zbornik Radova Prirodno-Matematičkog Fakulteta. Serija za Matemati, vol. 15, no. 1, pp. 119–129, 1985. View at: Google Scholar | Zentralblatt MATH
12. B. Fisher, “The composition and neutrix composition of distributions,” in Mathematical Methods in Engineering, pp. 59–69, Springer, New York, NY, USA, 2007. View at: Google Scholar | Zentralblatt MATH
13. I. M. Gel'fand and G. E. Shilov, Generalized Functions, vol. 1, Academic Press, New York, NY, USA, 1st edition, 1964.
14. J. G. van der Corput, “Introduction to the neutrix calculus,” Journal d'Analyse Mathématique, vol. 7, pp. 281–398, 1959. View at: Publisher Site | Google Scholar
15. Y. J. Ng and H. van Dam, “Neutrix calculus and finite quantum field theory,” Journal of Physics A, vol. 38, no. 18, pp. L317–L323, 2005.
16. Y. J. Ng and H. van Dam, “An application of neutrix calculus to quantum field theory,” International Journal of Modern Physics A, vol. 21, no. 2, pp. 297–312, 2006.
17. B. Fisher, “The delta function and the composition of distributions,” Demonstratio Mathematica, vol. 35, no. 1, pp. 117–123, 2002. View at: Google Scholar | Zentralblatt MATH
18. B. Fisher, T. Kraiweeradechachai, and E. Özçağ, “Results on the neutrix composition of the delta function,” Hacettepe Journal of Mathematics and Statistics, vol. 36, no. 2, pp. 147–156, 2007. View at: Google Scholar | Zentralblatt MATH
19. B. Fisher, B. Jolevska-Tuneska, and E. Özçağ, “Further results on the compositions of distributions,” Integral Transforms and Special Functions, vol. 13, no. 2, pp. 109–116, 2002.
20. B. Fisher, A. Kananthai, G. Sritanatana, and K. Nonlaopon, “The composition of the distributions ${x}_{-}^{ms}\mathrm{ln}{x}_{-}$ and ${x}_{+}^{r-p/m}$,” Integral Transforms and Special Functions, vol. 16, no. 1, pp. 13–20, 2005.
21. B. Fisher and K. Taş, “On the composition of the distributions ${x}_{+}^{-r}$ and ${x}_{+}^{\mu }$,” The Indian Journal of Pure and Applied Mathematics, vol. 36, no. 1, pp. 11–22, 2005. View at: Google Scholar | Zentralblatt MATH
22. B. Fisher and K. Taş, “On the composition of the distributions ${x}^{-1}$ln$|x|$ and ${x}_{+}^{r}$,” Integral Transforms and Special Functions, vol. 16, no. 7, pp. 533–543, 2005.

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