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

Abstract

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. In the ordinary sense, the composition 𝛿^(𝑠)[(sinh⁻¹𝑥₊)^𝑟] does not exists. In this study, it is proved that the neutrix composition 𝛿^(𝑠)[(sinh⁻¹𝑥₊)^𝑟] exists and is given by 𝛿^(𝑠)[(sinh⁻¹𝑥₊)^𝑟]=∑^{𝑠𝑟+𝑟−1}_𝑘=0∑^𝑘_𝑖=0^k_i((−1)^𝑘𝑟𝑐_{𝑠,𝑘,𝑖}/2^𝑘+1𝑘!)𝛿^(𝑘)(𝑥), for 𝑠=0,1,2,… and 𝑟=1,2,…, where 𝑐_{𝑠,𝑘,𝑖}=(−1)^𝑠𝑠![(𝑘−2𝑖+1)^𝑟𝑠−1+(𝑘−2𝑖−1)^{𝑟𝑠+𝑟−1}]/(2(𝑟𝑠+𝑟−1)!). Further results are also proved.

1. Introduction

In the following, we let 𝒟 be the space of infinitely differentiable functions with compact support, let 𝒟[𝑎,𝑏] be the space of infinitely differentiable functions with support contained in the interval [𝑎,𝑏], and let 𝒟^′ be the space of distributions defined on 𝒟.

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)∫¹₋₁𝜌(𝑥)𝑑𝑥=1.

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, 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 [2–4]. 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 [5–8]. 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 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 𝑁−lim_𝑛→∞^∞_−∞𝐹_𝑛𝑓(𝑥)𝜑(𝑥)𝑑𝑥=ℎ(𝑥),𝜑(𝑥), (1.1) 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 [10], having domain 𝑁^′ the positive and range 𝑁^′′ the real numbers, with negligible functions which are finite linear sums of the functions 𝑛^𝜆ln^𝑟−1𝑛,ln^𝑟𝑛∶𝜆>0,𝑟=1,2,… (1.2) 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 lim_𝑛→∞^∞_−∞𝐹_𝑛𝑓(𝑥)𝜑(𝑥)𝑑𝑥=ℎ(𝑥),𝜑(𝑥), (1.3) 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). 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 𝛿^(𝑠)(sgn𝑥|𝑥|^𝜆) exists and 𝛿^(𝑠)sgn𝑥|𝑥|^𝜆=0, (1.4) for 𝑠=0,1,2,… and (𝑠+1)𝜆=1,3,…, and 𝛿^(𝑠)sgn𝑥|𝑥|^𝜆=(−1)^{(𝑠+1)(𝜆+1)}𝑠!𝜆)]!𝛿[(𝑠+1𝜆−1^{((𝑠+1)𝜆−1)}(𝑥), (1.5) for 𝑠=0,1,2,…, and (𝑠+1)𝜆=2,4,….

Theorem 1.3. The neutrix compositions 𝛿^(2𝑠−1)(sgn𝑥|𝑥|^1/𝑠) and 𝛿^(𝑠−1)(|𝑥|^1/𝑠) exist and 𝛿^(2𝑠−1)sgn𝑥|𝑥|^1/𝑠=12(2𝑠)!𝛿^′(𝑥),𝛿^(𝑠−1)|𝑥|^1/𝑠=(−1)^𝑠−1𝛿(𝑥), (1.6) for 𝑠=1,2,….

Theorem 1.4. The neutrix composition 𝛿^(𝑠)(sinh⁻¹𝑥₊^1/𝑟) exists and 𝛿^(𝑠)sinh⁻¹𝑥₊^1/𝑟=^{(𝑠+1)/𝑟−1}_𝑘=0^𝑘_𝑖=0𝑘𝑖(−1)^𝑘𝑟𝑐_{𝑠,𝑘,𝑖}2^𝑘+1𝛿𝑘!^(𝑘)(𝑥), (1.7) for 𝑠=0,1,2,… and 𝑟=1,2,…, where 𝑐_{𝑟,𝑠,𝑘,𝑖}=(−1)^𝑠𝑠!(𝑘−2𝑖+1)^{𝑟𝑠+𝑟−1}+(𝑘−2𝑖−1)^{𝑟𝑠+𝑟−1}2()!.𝑟𝑠+𝑟−1 (1.8)

The next two theorems were proved in [13].

Theorem 1.5. The neutrix composition 𝛿^(𝑠)[ln^𝑟(1+|𝑥|)] exists and 𝛿^(𝑠)ln^𝑟(1+|𝑥|)=^{𝑠𝑟+𝑟−1𝑘}_𝑘=0𝑖=0𝑘𝑖(−1)^𝑠−𝑖1+(−1)^𝑘𝑠!(𝑖+1)^{𝑟𝑠+𝑟−1}()𝛿2𝑟𝑟𝑠+𝑟−1!𝑘!^(𝑘)(𝑥). (1.9) for 𝑠=0,1,2,…, and 𝑟=1,2,….
In particular, the composition 𝛿[ln(1+|𝑥|)] exists and 𝛿[ln[1+|𝑥|)]=𝛿(𝑥). (1.10)

Theorem 1.6. The neutrix composition 𝛿^(𝑠)[ln(1+|𝑥^1/𝑟|)] exists and 𝛿^(𝑠)ln1+𝑥^1/𝑟=^𝑚−1_𝑘=0^{𝑘𝑟+𝑟−1}_𝑖=0𝑖𝑘𝑟+𝑟−1(−1)^{𝑟+𝑠+𝑖−1}1+(−1)^𝑘𝑟(𝑖+1)^𝑠𝛿2𝑘!^(𝑘)(𝑥), (1.11) for 𝑠=0,1,2,… and 𝑟=2,3,…, where 𝑚 is the smallest non-negative integer greater than (𝑠−𝑟+1)𝑟⁻¹.
In particular, the composition 𝛿^(𝑠)[ln(1+|𝑥^1/𝑟|)] exists and 𝛿^(𝑠)ln1+𝑥^1/𝑟=0, (1.12) for 𝑠=0,1,2,…,𝑟−2 and 𝑟=2,3,… and 𝛿^(𝑟−1)ln1+𝑥^1/𝑟=(−1)^𝑟−1𝑟!𝛿(𝑥), (1.13) for 𝑟=2,3,….

2. Main Results

We now prove the following theorem.

Theorem 2.1. The neutrix composition 𝛿^(𝑠)[(sinh⁻¹𝑥₊)^𝑟] exists and 𝛿^(𝑠)sinh⁻¹𝑥₊^𝑟=^{𝑠𝑟+𝑟−1𝑘}_𝑘=0𝑖=0𝑘𝑖(−1)^𝑘𝑟𝑐_{𝑠,𝑘,𝑖}2^𝑘+1𝛿𝑘!^(𝑘)(𝑥), (2.1) for 𝑠=0,1,2,… and 𝑟=1,2,…, where 𝑐_{𝑟,𝑠,𝑘,𝑖}=(−1)^𝑠𝑠!(𝑘−2𝑖+1)^{𝑟𝑠+𝑟−1}+(𝑘−2𝑖−1)^{𝑟𝑠+𝑟−1}2()!.𝑟𝑠+𝑟−1 (2.2) In particular, the neutrix composition 𝛿(sinh⁻¹𝑥₊) exists and 𝛿sinh⁻¹𝑥₊=12𝛿(𝑥). (2.3)

Proof. To prove (2.1), we first of all evaluate ¹₋₁𝛿_𝑛^(𝑠)sinh⁻¹𝑥₊^𝑟𝑥^𝑘𝑑𝑥. (2.4) We have ¹₋₁𝛿_𝑛^(𝑠)sinh⁻¹𝑥₊^𝑟𝑥^𝑘𝑑𝑥=𝑛^𝑠+11₋₁𝜌^(𝑠)𝑛sinh⁻¹𝑥₊^𝑟𝑥^𝑘𝑑𝑥=𝑛^𝑠+11₀𝜌^(𝑠)𝑛sinh⁻¹𝑥^𝑟𝑥^𝑘𝑑𝑥+𝑛^𝑠+10₋₁𝜌^(𝑠)(0)𝑥^𝑘𝑑𝑥=𝐼₁+𝐼₂. (2.5) It is obvious that 𝑁−lim_𝑛→∞𝐼₂=𝑁−lim_𝑛→∞⁰₋₁𝛿_𝑛^(𝑠)sinh⁻¹𝑥₊^𝑟𝑥^𝑘𝑑𝑥=0, (2.6) for 𝑘=0,1,2,….
Making the substitution 𝑡=𝑛(sinh⁻¹𝑥)^𝑟, we have for large enough 𝑛𝐼₁=𝑛^{𝑠−𝑟+1}𝑟¹₀𝑡^1/(𝑟−1)sinh^𝑘𝑡𝑛^1/𝑟𝑡cosh𝑛^1/𝑟𝜌^(𝑠)(𝑡)×𝑑𝑡₁₀𝑡^1/(𝑟−1))𝑡exp(𝑘−2𝑖+1𝑛^1/𝑟)𝑡+exp(𝑘−2𝑖−1𝑛^1/𝑟𝜌^(𝑠)(𝑡)𝑑𝑡, (2.7) where 𝑛^{(𝑠−1)/(𝑟+1)1}₀𝑡^1/(𝑟−1)𝑡exp(𝑘−2𝑖+1)𝑛^1/𝑟𝑡+exp(𝑘−2𝑖−1)𝑛^1/𝑟𝜌^(𝑠)(𝑡)=𝑑𝑡_∞𝑝=010(𝑘−2𝑖+1)^𝑝+(𝑘−2𝑖−1)^𝑝𝑡^{(𝑝/𝑟)+(1/𝑟)−1}𝑝!𝑛^{(𝑝/𝑟)+(1/𝑟)−𝑠−1}𝜌^(𝑠)(𝑡)𝑑𝑡. (2.8) It follows that 𝑁−lim_𝑛→∞𝑛^{𝑠−1/𝑟+11}₀𝑡^1/(𝑟−1)𝑡exp(𝑘−2𝑖+1)𝑛^1/𝑟𝑡+exp(𝑘−2𝑖−1)𝑛^1/𝑟𝜌^(𝑠)(𝑡)=()𝑑𝑡−1^𝑠𝑠!(𝑘−2𝑖+1)^{𝑟𝑠+𝑟−1}+(𝑘−2𝑖−1)^{𝑟𝑠+𝑟−1}2(𝑟𝑠+𝑟−1)!=𝑐_{𝑟,𝑠,𝑘,𝑖}, (2.9) and by applying the neutrix limit we obtain 𝑁−lim_𝑛→∞𝐼₁=𝑁−lim_𝑛→∞¹₀𝛿_𝑛^(𝑠)sinh⁻¹𝑥₊^𝑟𝑥^𝑘1𝑑𝑥=2^𝑘+1𝑟^𝑘_𝑖=0𝑘𝑖(−1)^𝑖𝑐_{𝑟,𝑠,k,𝑖} (2.10) for 𝑘=0,1,2,….
When 𝑘=𝑠𝑟+𝑟, we have 𝐼₁=¹₀𝛿_𝑛^(𝑠)sinh⁻¹𝑥₊^𝑟𝑥^{𝑠𝑟+𝑟}𝑑𝑥=𝑛^𝑠+11₀𝜌_𝑛^(𝑠)𝑛sinh⁻¹𝑥^𝑟𝑥^{𝑠𝑟+𝑟}≤𝑛𝑑𝑥_{(𝑠−1)/(𝑟+1)}2^{𝑠𝑟+𝑟}𝑟exp(𝑠𝑟+𝑟+1)¹₀𝑡1−exp−2𝑛^{1/𝑟𝑠𝑟+𝑟}𝜌^(𝑠)(𝑡=𝑛)𝑑𝑡_{(𝑠−1)/(𝑟+1)}2^{𝑠𝑟+𝑟}𝑟exp(𝑠𝑟+𝑟+1)¹₀2𝑡𝑛^1/𝑟+𝑂𝑛^{−2/𝑟𝑠𝑟+𝑟}𝜌^(𝑠)(𝑡)𝑑𝑡≤𝑛^−1/𝑟exp(𝑠𝑟+𝑟+1)¹₀1+𝑂𝑛^−2/𝑟𝜌^(𝑠)(𝑡𝑛)𝑑𝑡=𝑂^−1/𝑟. (2.11) Thus, if 𝜓 is an arbitrary continuous function, then lim_𝑛→∞¹₀𝛿_𝑛^(𝑠)sinh⁻¹𝑥₊^𝑟𝑥^{𝑟𝑠+𝑟}𝜓(𝑥)𝑑𝑥=0. (2.12) We also have ⁰₋₁𝛿_𝑛^(𝑠)sinh⁻¹𝑥₊^𝑟𝜓(𝑥)𝑑𝑥=𝑛^𝑠+10₋₁𝜌^(𝑠)(0)𝜓(𝑥)𝑑𝑥, (2.13) and it follows that 𝑁−lim_𝑛→∞⁰₋₁𝛿_𝑛^(𝑠)sinh⁻¹𝑥₊^𝑟𝜓(𝑥)𝑑𝑥=0. (2.14) If now 𝜑 is an arbitrary function in 𝒟[−1,1], then by Taylor's Theorem, we have 𝜑(𝑥)=^{𝑠𝑟+𝑟−1}_𝑘=0𝜑^(𝑘)(0)𝑥𝑘!^𝑘+𝑥^{𝑟𝑠+𝑟}()!𝜑𝑟𝑠+𝑟^{(𝑟𝑠+𝑟)}(𝜉𝑥), (2.15) where 0<𝜉<1, and so 𝑁−lim_𝑛→∞𝛿_𝑛^(𝑠)sinh⁻¹𝑥₊^1/𝑟,𝜑(𝑥)=𝑁−lim_𝑛→∞^{𝑠𝑟+𝑟−1}_𝑘=0𝜑^(𝑘)(0)𝑘!¹₀𝛿_𝑛^(𝑠)sinh⁻¹𝑥₊^𝑟𝑥^𝑘𝑑𝑥+𝑁−lim_𝑛→∞^{𝑠𝑟+𝑟−1}_𝑘=0𝜑^(𝑘)(0)𝑘!⁰₋₁𝛿_𝑛^(𝑠)sinh⁻¹𝑥₊^𝑟𝑥^𝑘𝑑𝑥+lim_𝑛→∞1(𝑠𝑟+𝑟)!¹₀𝛿_𝑛^(𝑠)sinh⁻¹𝑥₊^𝑟𝑥^{𝑠𝑟+𝑟}𝜑^{(𝑠𝑟+𝑟)}(𝜉𝑥)𝑑𝑥+lim_𝑛→∞1(𝑠𝑟+𝑟)!⁰₋₁𝛿_𝑛^(𝑠)sinh⁻¹𝑥₊^𝑟𝑥^{𝑠𝑟+𝑟}𝜑^{(𝑠𝑟+𝑟)}()=𝜉𝑥𝑑𝑥_{𝑠𝑟+𝑟−1𝑘𝑘=0𝑖=0}𝑘𝑖𝑟𝑐_{𝑟,𝑠,𝑘,𝑖}𝜑^(𝑘)(0)2^𝑘+1=𝑘!+0_{𝑠𝑟+𝑟−1𝑘𝑘=0𝑖=0}𝑘𝑖(−1)^𝑘𝑟𝑐_{𝑟,𝑠,𝑘,𝑖}2^𝑘+1𝛿𝑘!^(𝑘)(𝑥),𝜑(𝑥), (2.16) on using (2.3) to (2.14). This proves (2.1) on the interval (−1,1).
It is clear that 𝛿^(𝑠)[(sinh⁻¹𝑥₊)^𝑟]=0 for 𝑥>0 and so (2.1) holds for 𝑥>−1.
Now, suppose that 𝜑 is an arbitrary function in 𝒟[𝑎,𝑏], where 𝑎<𝑏<0. Then, ^𝑏_𝑎𝛿_𝑛^(𝑠)sinh⁻¹𝑥₊^𝑟𝜑(𝑥)𝑑𝑥=𝑛^𝑠+1𝑏_𝑎𝜌^(𝑠)(0)𝜑(𝑥)𝑑𝑥 (2.17) and so N−lim_𝑛→∞^𝑏_𝑎𝛿_𝑛^(𝑠)sinh⁻¹𝑥₊^𝑟𝜑(𝑥)𝑑𝑥=0. (2.18)
It follows that 𝛿^(𝑠)[(sinh⁻¹𝑥₊)^𝑟]=0 on the interval (𝑎,𝑏). Since 𝑎 and 𝑏 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 𝛿^(𝑠)[(sinh⁻¹|𝑥|)^𝑟] exists and 𝛿^(𝑠)sinh⁻¹|𝑥|^𝑟=^{𝑠𝑟+𝑟−1𝑘}_𝑘=0𝑖=0𝑘𝑖(−1)^𝑘+1𝑐_{𝑟,𝑠,𝑘,𝑖}2^𝑘+1𝛿𝑘!^(𝑘)(𝑥), (2.19) for 𝑠=0,1,2,… and 𝑟=1,2,….
In particular, the composition 𝛿(sinh⁻¹|𝑥|) exists and 𝛿sinh⁻¹|𝑥=1|2𝛿(𝑥). (2.20)

Proof. To prove (2.19), we note that ¹₋₁𝛿_𝑛^(𝑠)sinh⁻¹|𝑥|^𝑟𝑥^𝑘𝑑𝑥=𝑛^𝑠+11₋₁𝜌^(𝑠)𝑛sinh⁻¹|𝑥|^𝑟𝑥^𝑘𝑑𝑥=𝑛^𝑠+11+(−1)^𝑘1₀𝜌^(𝑠)𝑛sinh⁻¹𝑥^𝑟𝑥^𝑘𝑑𝑥, (2.21) and (2.19) now follows as above.
Equation (2.20) follows on noting that in the particular case 𝑠=0, the usual limit holds in (2.10). This completes the proof of the corollary.

Theorem 2.3. The neutrix composition 𝛿^(2𝑠−1)[sinh⁻¹(sgn𝑥⋅𝑥²)] exists and 𝛿^(2𝑠−1)sinh⁻¹sgn𝑥⋅𝑥²=^2𝑠−1_𝑘=0^𝑖+𝑘+1_𝑖=0𝑘𝑖(−1)^𝑘𝑏_{𝑠,𝑘,𝑖}2^𝑘+1()!𝛿2𝑘+1^(𝑘)(𝑥), (2.22) for 𝑠=1,2,…, where 𝑏_{𝑠,𝑘,𝑖}=(𝑘−2𝑖+1)^2𝑠−1+(𝑘−2𝑖−1)^2𝑠−1. (2.23)

Proof. To prove (2.22), we now have to evaluate ¹₋₁𝛿_𝑛^(2𝑠−1)sinh⁻¹s𝑔𝑛𝑥⋅𝑥²𝑥^𝑘𝑑𝑥. (2.24) We have ¹₋₁𝛿_𝑛^(2𝑠−1)sinh⁻¹sgn𝑥⋅𝑥²𝑥^𝑘𝑑𝑥=𝑛^2𝑠1₋₁𝜌^(2𝑠−1)𝑛sinh⁻¹sgn𝑥⋅𝑥²𝑥^k=𝑑𝑥2𝑛^2𝑠1₀𝜌^(2𝑠−1)𝑛sinh⁻¹𝑥²𝑥^𝑘𝑑𝑥,𝑘odd,0,𝑘even. (2.25) Making the substitution 𝑡=𝑛(sinh⁻¹𝑥²), we have for large enough 𝑛¹₋₁𝛿_𝑛^(2𝑠−1)sinh⁻¹s𝑔𝑛𝑥⋅𝑥²𝑥^𝑘𝑑𝑥=2𝑛^2𝑠1₀𝜌^(2𝑠−1)𝑛sinh⁻¹𝑥²𝑥^2𝑘+1=𝑛𝑑𝑥_2𝑠−12^𝑘𝑘+1_𝑖=0𝑘𝑖(−1)^𝑖1₀()𝑡exp𝑘−2𝑖+1𝑛()𝑡+exp𝑘−2𝑖−1𝑛𝜌^(2𝑠−1)(𝑡)𝑑𝑡, (2.26) where 𝑛^2𝑠−11₀()𝑡exp𝑘−2𝑖+1𝑛()𝑡+exp𝑘−2𝑖−1𝑛𝜌^(𝑠)(𝑡)=𝑑𝑡_∞𝑝=010(𝑘−2𝑖+1)^𝑝+(𝑘−2𝑖−1)^𝑝𝑡^𝑝𝑝!𝑛^{𝑝−2𝑠+1}𝜌^(2𝑠−1)(𝑡)𝑑𝑡. (2.27) It follows that 𝑁−lim_𝑛→∞𝑛^2𝑠−11₀()𝑡exp𝑘−2𝑖+1𝑛()𝑡+exp𝑘−2𝑖−1𝑛𝜌^(𝑠)(𝑡)𝑑𝑡=𝑁−lim^∞_{𝑛→∞𝑝=0}¹₀(𝑘−2i+1)^𝑝+(𝑘−2𝑖−1)^𝑝𝑡^𝑝𝑝!𝑛^{𝑝−2𝑠+1}𝜌^(2𝑠−1)(𝑡)=−()𝑑𝑡𝑘−2𝑖+1^2𝑠−1+(𝑘−2𝑖−1)^2𝑠−12=𝑏_{𝑠,𝑘,𝑖}2, (2.28) and so by using the neutrix limit, we have 𝑁−lim_𝑛→∞¹₋₁𝛿_𝑛^(2𝑠−1)sinh⁻¹sgn𝑥⋅𝑥²𝑥^2𝑘+1𝑑𝑥=^𝑘_𝑖=0𝑘𝑖(−1)^𝑖+1𝑏_{𝑠,𝑘,𝑖}2^𝑘+1, (2.29) for 𝑘=0,1,2,….
When 𝑘=2𝑠, we have ¹₋₁𝛿_𝑛^(2𝑠−1)sinh⁻¹sgn𝑥⋅𝑥²𝑥^4𝑠+1𝑑𝑥=𝑛^2𝑠1₋₁𝜌^(2𝑠−1)𝑛sinh⁻¹𝑥²𝑥^4𝑠+1≤𝑛𝑑𝑥_2𝑠−12^𝑠−1exp(𝑠+1)¹₋₁1−exp−2𝑡𝑛^2𝑠𝜌^(2𝑠−1)(𝑡=𝑛)𝑑𝑡_2𝑠−12^𝑠−1exp(𝑠+1)¹₋₁2𝑡𝑛+𝑂𝑛^−22𝑠𝜌^(2𝑠−1)(𝑡)𝑑𝑡≤2^2𝑠+1𝑛⁻¹exp(𝑠+1)¹₋₁1+𝑂𝑛^−2/𝑟𝜌^(2𝑠−1)(𝑡𝑛)𝑑𝑡=𝑂⁻¹. (2.30) Thus, if 𝜓 is an arbitrary continuous function, then lim_𝑛→∞¹₋₁𝛿_𝑛^(2𝑠−1)sinh⁻¹sgn𝑥⋅𝑥²𝑥^4𝑠+1𝜓(𝑥)𝑑𝑥=0. (2.31) If now 𝜑 is an arbitrary function in 𝒟[−1,1], then by Taylor's Theorem, we have 𝜑(𝑥)=^4𝑠_𝑘=0𝜑^(𝑘)(0)𝑥𝑘!^𝑘+𝑥^4𝑠+1()!𝜑4𝑠+1^(4𝑠+1)(𝜉𝑥), (2.32) where 0<𝜉<1, and so 𝑁−lim_𝑛→∞𝛿_𝑛^(2𝑠−1)sinh⁻¹sgn𝑥⋅𝑥²,𝜑(𝑥)=𝑁−lim_𝑛→∞^2𝑠−1_𝑘=0𝜑^(2𝑘+1)(0)(2𝑘+1)!¹₋₁𝛿_𝑛^(2𝑠−1)sinh⁻¹sgn𝑥⋅𝑥²𝑥^2𝑘+1𝑑𝑥+lim_𝑛→∞1(4𝑠+1)!¹₋₁𝛿_𝑛^(4𝑠+1)sinh⁻¹sgn𝑥⋅𝑥²𝑥^4𝑠+1𝜑^(4𝑠+1)()=𝜉𝑥𝑑𝑥_{2𝑠−1𝑘𝑘=0𝑖=0}𝑘𝑖(−1)^𝑖+1𝑏_{𝑠,𝑘,𝑖}𝜑^(𝑘)(0)2^𝑘+1()!=2𝑘+1+0_{2𝑠−1𝑘𝑘=0𝑖=0}𝑘𝑖(−1)^𝑖+𝑘+1𝑏_{𝑠,𝑘,𝑖}2^𝑘+1()!𝛿2𝑘+1^(𝑘)(𝑥),𝜑(𝑥), (2.33) on using (2.25) to (2.31), proving (2.22) on the interval (−1,1). However, it is clear that 𝛿_𝑛^(2𝑠−1)[sinh⁻¹(sgn𝑥⋅𝑥²)]=0 for |𝑥|>0 and so (2.22) holds on the real line, completing the proof of the theorem.

Corollary 2.4. The composition 𝛿^′[sinh⁻¹sgn𝑥⋅𝑥²)] exists and 𝛿^′sinh⁻¹sgn𝑥⋅𝑥²=𝛿^′(𝑥)(𝑥).4.3!−2𝛿 (2.34)

Proof. To prove (2.34) note that in the particular case 𝑠=1, 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 [12–22] 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.