Let F be a distribution in 𝒟′ and let f be a locally summable function. The composition F(f(x)) of F and f is said to exist and be equal to the distribution h(x) if the limit of the sequence {F_n(f(x))} is equal to h(x), where F_n(x) = F(x)*δ_n(x) for n = 1,2, … and {δ_n(x)} is a certain regular sequence converging to the Dirac delta function. It is proved that the neutrix composition δ^(rs−1)((tanhx₊) ^1/r) exists and for r, s = 1,2, …, where K_k is the integer part of (s − k − 1)/2 and the constants c_j,k are defined by the expansion , for k = 0,1, 2, …. Further results are also proved.

1. Introduction

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

Now let ρ(x) be a function in 𝒟[−1,1] having the following properties:

(i)
ρ(x) ≥ 0,
(ii)
ρ(x) = ρ(−x),
(iii)
.

Putting δ_n(x) = nρ(nx) for n = 1,2, …, it follows that {δ_n(x)} is a regular sequence of infinitely differentiable functions converging to the Dirac delta-function δ(x). Further, if F is an arbitrary distribution in 𝒟′ and F_n(x) = F(x)*δ_n(x) = 〈F(x − t), φ(t)〉, then {F_n(x)} is a regular sequence converging to F(x).

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 F(f(x)), if F and f are singular distributions.

Thus there have been several attempts recently to define distributions of the form F(f(x)) in 𝒟′, where F and f are distributions in 𝒟′, 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 F be a distribution in 𝒟′ and let f be a locally summable function. We say that the neutrix composition F(f(x)) exists and is equal to h on the open interval (a, b), with −∞ < a < b < ∞, if

()

for all φ in 𝒟[a, b], where F_n(x) = F(x)*δ_n(x) for n = 1,2, … and N is the neutrix, see [14], having domain N′ the positive integers and range N^′′ 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 n tends to infinity.

In particular, we say that the composition F(f(x)) exists and is equal to h on the open interval (a, b) if

()

for all φ in 𝒟[a, b].

Note that taking the neutrix limit of a function f(n) is equivalent to taking the usual limit of Hadamard′s finite part of f(n). If f, g are two distributions then in the ordinary sense the composition f(g) 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 δ^(s)(sgn x|x|^λ) exists and

()

for s = 0,1, 2, … and (s + 1)λ = 1,3, … and

()

for s = 0,1, 2, … and (s + 1)λ = 2,4, ….

Theorem 1.3. The compositions δ^(2s−1)(sgn x|x|^1/s) and δ^(s−1)(|x|^1/s) exist and

()

for s = 1,2, ….

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

Theorem 1.4. The neutrix composition δ^(s)(ln ^r(1 + |x|)) exists and

()

for s = 0,1, 2, … and r = 1,2, ….

In particular, the composition δ(ln (1 + |x|)) exists and

()

Theorem 1.5. The neutrix composition δ^(s)(ln (1 + |x|^1/r)) exists and

()

for s = 0,1, 2, … and r = 2,3, …, where m is the smallest nonnegative integer greater than (s − r + 1)r⁻¹.

In particular, the composition δ^(s)(ln (1 + |x|^1/r)) exists and

()

for s = 0,1, 2, …, r − 2 and r = 2,3, … and

()

for r = 2,3, ….

The following two theorems were proved in [4].

Theorem 1.6. The neutrix composition exists and

()

for s = 0,1, 2, … and r = 1,2, …, where

()

In particular, the neutrix composition δ(sinh ⁻¹x₊) exists and

()

Theorem 1.7. The neutrix composition δ^(2s−1)(sinh ⁻¹(sgn x · x²)) exists and

()

for s = 1,2, …, where

()

In particular

()

2. Main Result

In the next theorem, the constants c_j,k are defined by the expansion

()

for k = 0,1, 2, ….

Theorem 2.1. The neutrix composition exists and

()

for r, s = 1,2, …, where K_k is the integer part of (s − k − 1)/2.

In particular, the neutrix compositions and exist and

()

for r = 1,2, ….

Proof. To prove (2.2), first of all we evaluate

()

We have

()

It is obvious that

()

for k = 0,1, 2, ….

Making the substitution t = n(tanhx)^1/r, we have for large enough n

()

It follows that

()

for k = 0,1, 2, …, s − 1, where k₀ denotes the integer part of (s − k − 1)/2 for k = 0,1, 2, ….

In particular, when s = 1, we have k = 0 = k₀. It follows from (2.9) that

()

and so ordinary limit exists

()

for r = 1,2, …, since c_0,0 = 1.

Further, when s = 2, we have k = 0,1 and k₀ = 0. It follows from (2.10) that

()

for r = 1,2, …, since c_1,0 = 0, and c_1,1 = 1.

When k = s, we have

()

Thus, if ψ is an arbitrary continuous function, then

()

We also have

()

and it follows that

()

If now φ is an arbitrary function in 𝒟[−1,1], then by Taylor′s Theorem, we have

()

where 0 < ξ < 1, and so

()

on using (2.6), (2.7), (2.10), (2.15), and (2.17). This proves (2.2) on the interval (−1,1).

It is also clear that for x > 0 and so (2.2) holds for x > −1.

Now suppose that φ is an arbitrary function in 𝒟[a, b], where a < b < 0. Then

()

and so

()

It follows that

on the interval (a, b). Since a and b 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 δ^(rs−1)((tanh|x|)^1/r) exists and

()

for r, s = 1,2, …, where K_k is the integer part of (s − k − 1)/2.

In particular, the composition δ^(r−1)((tanh|x|)^1/r) exists and the neutrix composition δ^(2r−1)((tanh|x|)^1/r) exists and

()

for r = 1,2, ….

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 s = 1, the ordinary limit exists in (2.12) and so the composition δ^(r−1)((tanh|x|)^1/r) exists in this case. This completes the proof of the corollary.

In the next theorem, the constants b_j,k are defined by the following expansion

()

for k = 0,1, 2, ….

Theorem 2.3. The neutrix composition exists and

()

for s = 0,1, 2, … and r = 1,2, …, where K is the smallest integer for which s < Kr + r − 1.

Proof. To prove (2.26), first of all we evaluate

()

We have

()

It is obvious that

()

Making the substitution t = n(tanh⁻¹x^1/r), we have for large enough n

()

and it follows that

()

In particular, when s = 0 and r = 1, we have K = 1 and then

()

When k = K, we have

()

Thus, if ψ is an arbitrary continuous function, then

()

and so

()

since s − Kr − r + 1 < 0.

We also have

()

and it follows that

()

If now φ is an arbitrary function in 𝒟[−1,1], then by Taylor′s Theorem, we have

()

where 0 < ξ < 1, and so

()

on using (2.9) and (2.12). This proves (2.26) on the interval (−1,1). It is clear that

for x > 0 and so (2.2) holds for x > −1.

Now suppose that φ is an arbitrary function in [a, b], where a < b < 0. Then

()

and so

()

It follows that

on the interval (a, b). Since a and b are arbitrary, we see that (2.26) holds on the real line.

Corollary 2.4. The neutrix composition δ^(s)(tanh⁻¹|x|^1/r) exists and

()

for s = 0,1, 2, … and r = 1,2, …, where K is the smallest integer for which s < Kr + r − 1. In particular, the composition δ(tanh⁻¹ | x|) 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 s = 0 and r = 1, the ordinary limit exists in (2.12), and so the composition δ(tanh | x|) 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.

References

1 Eltayeb H., Kılıçman A., and Fisher B., A new integral transform and associated distributions, Integral Transforms and Special Functions. (2010) 21, no. 5-6, 367–379, https://doi.org/10.1080/10652460903335061, 2666504, ZBL1191.35017.
10.1080/10652460903335061
Web of Science® Google Scholar
2 Fisher B. and Kılıçman A., A commutative neutrix product of ultradistributions, Integral Transforms and Special Functions. (1996) 4, no. 1-2, 77–82, https://doi.org/10.1080/10652469608819095, 1601266, ZBL0856.46023.
10.1080/10652469608819095
Web of Science® Google Scholar
3 Fisher B. and Kılıçman A., On the composition and neutrix composition of the delta function and powers of the inverse hyperbolic sine function, Integral Transforms and Special Functions. (2010) 21, no. 12, 935–944, https://doi.org/10.1080/10652469.2010.497271, 2739388, ZBL1210.33001.
10.1080/10652469.2010.497271
Web of Science® Google Scholar
4 Fisher B. and Kılıçman A., On the neutrix composition of the delta and inverse hyperbolic sine functions, Journal of Applied Mathematics. (2011) 2011, 12, 612353, https://doi.org/10.1155/2011/612353.
10.1155/2011/612353
Google Scholar
5 Kılıçman A., On the commutative neutrix product of distributions, The Indian Journal of Pure and Applied Mathematics. (1999) 30, no. 8, 753–762, 1704484, ZBL0930.46038.
Web of Science® Google Scholar
6 Kılıçman A., A comparison on the commutative neutrix convolution of distributions and the exchange formula, Czechoslovak Mathematical Journal. (2001) 51, no. 3, 463–471, https://doi.org/10.1023/A:1013719619356, 1851540, ZBL1079.46514.
10.1023/A:1013719619356
Web of Science® Google Scholar
7 Kılıçman A., Fisher B., and Pehlivan S., The neutrix convolution product of ln^rx₊ and , Integral Transforms and Special Functions. (1998) 7, no. 3-4, 237–246, https://doi.org/10.1080/10652469808819202, 1775830, ZBL1052.33500.
10.1080/10652469808819202
Web of Science® Google Scholar
8 Kılıçman A. and Eltayeb H., A note on defining singular integral as distribution and partial differential equations with convolution term, Mathematical and Computer Modelling. (2009) 49, no. 1-2, 327–336, https://doi.org/10.1016/j.mcm.2008.05.048, 2480055, ZBL1165.45307.
10.1016/j.mcm.2008.05.048
Google Scholar
9 Antosik P., Composition of distributions, 1989, no. 9, University of Wisconsin.
Google Scholar
10 Fisher B., On defining the change of variable in distributions, Rostocker Mathematisches Kolloquium. (1985) no. 28, 33–40, 837182, ZBL0624.46021.
Google Scholar
11 Fisher B., On defining the distribution , Univerzitet u Novom Sadu. Zbornik Radova Prirodno-Matematičkog Fakulteta. Serija za Matemati. (1985) 15, no. 1, 119–129, 849759, ZBL0624.46022.
Google Scholar
12 Fisher B., The composition and neutrix composition of distributions, Mathematical Methods in Engineering, 2007, Springer, New York, NY, USA, 59–69, ZBL1130.46306.
10.1007/978-1-4020-5678-9_4
Google Scholar
13 Gel′fand I. M. and Shilov G. E., Generalized Functions, 1964, 1, 1st edition, Academic Press, New York, NY, USA, 0166596.
Google Scholar
14 van der Corput J. G., Introduction to the neutrix calculus, Journal d′Analyse Mathématique. (1959) 7, 281–398, 0124678, https://doi.org/10.1007/BF02787689.
10.1007/BF02787689
Google Scholar
15 Ng Y. J. and van Dam H., Neutrix calculus and finite quantum field theory, Journal of Physics A. (2005) 38, no. 18, L317–L323, https://doi.org/10.1088/0305-4470/38/18/L03, 2144521, ZBL1065.81097.
10.1088/0305-4470/38/18/L03
Google Scholar
16 Ng Y. J. and van Dam H., An application of neutrix calculus to quantum field theory, International Journal of Modern Physics A. (2006) 21, no. 2, 297–312, https://doi.org/10.1142/S0217751X06025225, 2205496, ZBL1091.81058.
10.1142/S0217751X06025225
Web of Science® Google Scholar
17 Fisher B., The delta function and the composition of distributions, Demonstratio Mathematica. (2002) 35, no. 1, 117–123, 1883949, ZBL1016.46028.
10.1515/dema-2002-0114
Google Scholar
18 Fisher B., Kraiweeradechachai T., and Özçağ E., Results on the neutrix composition of the delta function, Hacettepe Journal of Mathematics and Statistics. (2007) 36, no. 2, 147–156, 2411633, ZBL1151.46310.
Web of Science® Google Scholar
19 Fisher B., Jolevska-Tuneska B., and Özçağ E., Further results on the compositions of distributions, Integral Transforms and Special Functions. (2002) 13, no. 2, 109–116, https://doi.org/10.1080/10652460212897, 1915507, ZBL1031.46044.
10.1080/10652460212897
Web of Science® Google Scholar
20 Fisher B., Kananthai A., Sritanatana G., and Nonlaopon K., The composition of the distributions and , Integral Transforms and Special Functions. (2005) 16, no. 1, 13–20, https://doi.org/10.1080/10652460412331270661, 2110411, ZBL1105.46024.
10.1080/10652460412331270661
Google Scholar
21 Fisher B. and Taş K., On the composition of the distributions and , The Indian Journal of Pure and Applied Mathematics. (2005) 36, no. 1, 11–22, 2155099, ZBL1105.46024.
Web of Science® Google Scholar
22 Fisher B. and Taş K., On the composition of the distributions x⁻¹ln|x| and , Integral Transforms and Special Functions. (2005) 16, no. 7, 533–543, https://doi.org/10.1080/10652460500105438, 2173582, ZBL1105.46024.
10.1080/10652460500105438
Web of Science® Google Scholar

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

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

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