On the Composition and Neutrix Composition of the Delta Function with the Hyperbolic Tangent and Its Inverse 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. It is proved that the neutrix composition 𝛿^{(𝑟𝑠−1)}((tanh𝑥₊)^1/𝑟) exists and 𝛿^{(𝑟𝑠−1)}((tanh𝑥₊)^1/𝑟)=∑^𝑠−1_𝑘=0∑^𝐾𝑘_𝑖=0((−1)^𝑘𝑐_{𝑠−2𝑖−1,𝑘}(𝑟𝑠)!/2𝑠𝑘!)𝛿^(𝑘)(𝑥) 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 (tanh⁻¹𝑥)^𝑘={∑^∞_𝑖=0(𝑥^2𝑖+1/(2𝑖+1))}^𝑘=∑^∞_𝑗=𝑘𝑐_𝑗,𝑘𝑥^𝑗, 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)∫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, 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 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 [14], having domain 𝑁′ the positive integers 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). 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)(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 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,….

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

Theorem 1.4. 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), (1.7) for 𝑠=0,1,2,… and 𝑟=1,2,….
In particular, the composition 𝛿(ln(1+|𝑥|)) exists and𝛿(ln[1+|𝑥|))=𝛿(𝑥). (1.8)

Theorem 1.5. 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), (1.9) 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 and𝛿^(𝑠)ln1+|𝑥|^1/𝑟=0, (1.10) for 𝑠=0,1,2,…,𝑟−2 and 𝑟=2,3,… and 𝛿^(𝑟−1)ln1+|𝑥|^1/𝑟=(−1)^𝑟−1𝑟!𝛿(𝑥), (1.11) for 𝑟=2,3,….

The following two theorems were proved in [4].

Theorem 1.6. 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), (1.12) 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 (1.13) In particular, the neutrix composition 𝛿(sinh⁻¹𝑥₊) exists and 𝛿sinh⁻¹𝑥₊=12𝛿(𝑥). (1.14)

Theorem 1.7. 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), (1.15) for 𝑠=1,2,…, where 𝑏_{𝑠,𝑘,𝑖}=(𝑘−2𝑖+1)^2𝑠−1+(𝑘−2𝑖−1)^2𝑠−1. (1.16) In particular 𝛿^′sinh⁻¹sgn𝑥⋅𝑥²=𝛿^′(𝑥)(𝑥).4.3!−2𝛿 (1.17)

2. Main Result

In the next theorem, the constants 𝑐_𝑗,𝑘 are defined by the expansion tanh⁻¹𝑥^𝑘=^∞_𝑖=0𝑥^2𝑖+12𝑖+1^𝑘=^∞_𝑗=𝑘𝑐_𝑗,𝑘𝑥^𝑗, (2.1) for 𝑘=0,1,2,….

Theorem 2.1. The neutrix composition 𝛿^{(𝑟𝑠−1)}((tanh𝑥₊)^1/𝑟) exists and 𝛿^{(𝑟𝑠−1)}tanh𝑥₊^1/𝑟=^𝑠−1𝐾_𝑘=0^𝑘_𝑖=0(−1)^𝑘𝑐_{𝑠−2𝑖−1,𝑘}(𝑟𝑠)!𝛿2𝑠𝑘!^(𝑘)(𝑥), (2.2) for 𝑟,𝑠=1,2,…, where 𝐾_𝑘 is the integer part of (𝑠−𝑘−1)/2.
In particular, the neutrix compositions 𝛿^(𝑟−1)((tanh𝑥₊)^1/𝑟) and 𝛿^(2𝑟−1)((tanh𝑥₊)^1/𝑟) exist and𝛿^(𝑟−1)tanh𝑥₊^1/𝑟=(−1)^𝑟−1𝑟!2𝛿(𝑥),𝛿^(2𝑟−1)tanh𝑥₊^1/𝑟=(2𝑟)!4𝛿(𝑥)−𝛿^′(𝑥), (2.3)(2.4) for 𝑟=1,2,….

Proof. To prove (2.2), first of all we evaluate ¹₋₁𝛿_𝑛^{(𝑟𝑠−1)}tanh𝑥₊^1/𝑟𝑥^𝑘𝑑𝑥. (2.5) We have ¹₋₁𝛿_𝑛^{(𝑟𝑠−1)}tanh𝑥₊^1/𝑟𝑥^𝑘𝑑𝑥=𝑛^𝑟𝑠1₋₁𝜌^{(𝑟𝑠−1)}𝑛tanh𝑥₊^1/𝑟𝑥^𝑘𝑑𝑥=𝑛^𝑟𝑠1₀𝜌^{(𝑟𝑠−1)}𝑛(tanh𝑥)^1/𝑟𝑥^𝑘𝑑𝑥+𝑛^𝑟𝑠0₋₁𝜌^{(𝑟𝑠−1)}(0)𝑥^𝑘𝑑𝑥=𝐼_1,𝑘+𝐼_2,𝑘. (2.6) It is obvious that 𝑁−lim_𝑛→∞𝐼_2,𝑘=𝑁−lim_𝑛→∞⁰₋₁𝛿_𝑛^{(𝑟𝑠−1)}tanh𝑥₊^1/𝑟𝑥^𝑘𝑑𝑥=0, (2.7) for 𝑘=0,1,2,….
Making the substitution 𝑡=𝑛(tanh𝑥)^1/𝑟, we have for large enough 𝑛𝐼_1,𝑘=𝑟𝑛^{𝑟𝑠−𝑟1}₀𝑡^𝑟−1tanh⁻¹𝑡𝑛^𝑟𝑘𝑡1−𝑛^2𝑟−1𝜌^{(𝑟𝑠−1)}(𝑡)𝑑𝑡=𝑟^∞∞_{𝑖=0𝑗=𝑘}¹₀𝑐_𝑗,𝑘𝑡^{𝑟𝑗+2𝑟𝑖+𝑟−1}𝑛^{𝑟𝑗+2𝑟𝑖+𝑟−𝑟𝑠}𝜌^{(𝑟𝑠−1)}(𝑡)𝑑𝑡. (2.8) It follows that 𝑁−lim_𝑛→∞𝐼_1,𝑘=𝑁−lim_𝑛→∞𝑛^𝑟𝑠1₀𝜌^{(𝑟𝑠−1)}𝑛(tanh𝑥)^1/𝑟𝑥^𝑘𝑑𝑥=𝑟^𝑘0_𝑖=0¹₀𝑐_{𝑠−2𝑖−1,𝑘}𝑡^𝑟𝑠−1𝜌^{(𝑟𝑠−1)}(𝑡)=𝑑𝑡_𝑘0𝑖=0(−1)^𝑟𝑠−1𝑟𝑐_{𝑠−2𝑖−1,𝑘}(𝑟𝑠−1)!2, (2.9)(2.10) 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 𝐼_1,0=𝑟¹₀𝑐_0,0𝑡^{2𝑟𝑖+𝑟−1}𝜌^(𝑟−1)(𝑡)𝑑𝑡, (2.11) and so ordinary limit exists lim_𝑛→∞𝐼_1,0=𝑟¹₀𝑡^𝑟−1𝜌^(𝑟−1)(𝑡)()𝑑𝑡=−1^𝑟−1𝑟!2, (2.12) 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 𝑁−lim_𝑛→∞𝐼_1,0=𝑟𝑐_1,0(2𝑟−1)!2=(2𝑟)!4,𝑁−lim_𝑛→∞𝐼_1,1=𝑟𝑐_1,1(2𝑟−1)!2=(2𝑟)!4, (2.13) for 𝑟=1,2,…, since 𝑐_1,0=0, and 𝑐_1,1=1.
When 𝑘=𝑠, we have𝐼_1,𝑠=𝑟𝑛^{𝑟𝑠−𝑟1}₀𝑡^𝑟−1tanh⁻¹𝑡𝑛^𝑟𝑠𝑡1−𝑛^2𝑟−1𝜌^{(𝑟𝑠−1)}(𝑡)𝑑𝑡≤𝑟𝑛^{𝑟𝑠−𝑟1}₀tanh⁻¹1𝑛^𝑟𝑠𝑡1−𝑛^2𝑟−1𝜌^{(𝑟𝑠−1)}(𝑡)𝑑𝑡=𝑟𝑛^{𝑟𝑠−𝑟1}₀𝑛^−𝑟𝑠+𝑂𝑛^{−𝑟𝑠−2𝑟}𝜌^{(𝑟𝑠−1)}(𝑡𝑛)𝑑𝑡=𝑂^−𝑟. (2.14) Thus, if 𝜓 is an arbitrary continuous function, then lim_𝑛→∞¹₀𝛿_𝑛^{(𝑟𝑠−1)}tanh⁻¹𝑥₊^1/𝑟𝑥^𝑠𝜓(𝑥)𝑑𝑥=0. (2.15) We also have ⁰₋₁𝛿_𝑛^{(𝑟𝑠−1)}tanh⁻¹𝑥₊^1/𝑟𝜓(𝑥)𝑑𝑥=𝑛^𝑟𝑠0₋₁𝜌^{(𝑟𝑠−1)}(0)𝜓(𝑥)𝑑𝑥, (2.16) and it follows that 𝑁−lim_𝑛→∞⁰₋₁𝛿_𝑛^{(𝑟𝑠−1)}tanh⁻¹𝑥₊^1/𝑟𝜓(𝑥)𝑑𝑥=0. (2.17) If now 𝜑 is an arbitrary function in 𝒟[−1,1], then by Taylor's Theorem, we have 𝜑(𝑥)=^𝑠−1_𝑘=0𝜑^(𝑘)(0)𝑥𝑘!^𝑘+𝑥^𝑠𝜑𝑠!^(𝑠)(𝜉𝑥), (2.18) where 0<𝜉<1, and so 𝑁−lim_𝑛→∞𝛿_𝑛^{(𝑟𝑠−1)}tanh⁻¹𝑥₊^1/𝑟,𝜑(𝑥)=𝑁−lim_𝑛→∞^𝑠−1_𝑘=0𝜑^(𝑘)(0)𝑘!¹₀𝛿_𝑛^{(𝑟𝑠−1)}tanh⁻¹𝑥₊^1/𝑟𝑥^𝑘𝑑𝑥+𝑁−lim_𝑛→∞^𝑠−1_𝑘=0𝜑^(𝑘)(0)𝑘!⁰₋₁𝛿_𝑛^{(𝑟𝑠−1)}tanh⁻¹𝑥₊^1/𝑟𝑥^𝑘𝑑𝑥+lim_𝑛→∞1𝑠!¹₀𝛿_𝑛^{(𝑟𝑠−1)}tanh⁻¹𝑥₊^1/𝑟𝑥^𝑠𝜑^(𝑠)(𝜉𝑥)𝑑𝑥+𝑁−lim_𝑛→∞1𝑠!⁰₋₁𝛿_𝑛^{(𝑟𝑠−1)}tanh⁻¹𝑥₊^1/𝑟𝑥^𝑠𝜑^(𝑠)()=𝜉𝑥𝑑𝑥_{𝑠−1𝑘𝑘=00𝑖=0}𝑟𝑐_{𝑠−2𝑖−1,𝑘}(𝑟𝑠−1)!2𝜑^(𝑘)(0)=𝑘!+0_{𝑠−1𝑘𝑘=00𝑖=0}𝑐_{𝑠−2𝑖−1,𝑘}(𝑟𝑠)!()2𝑠−1^𝑘𝛿𝑘!^(𝑘)(𝑥),𝜑(𝑥), (2.19) 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 𝛿^{(𝑟𝑠−1)}((tanh⁻¹𝑥₊)^1/𝑟)=0 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^𝑏_𝑎𝛿_𝑛^{(𝑟𝑠−1)}tanh⁻¹𝑥₊^1/𝑟𝜑(𝑥)𝑑𝑥=𝑛^𝑟𝑠𝑏_𝑎𝜌^{(𝑟𝑠−1)}(0)𝜑(𝑥)𝑑𝑥, (2.20) and so 𝑁−lim_𝑛→∞^𝑏_𝑎𝛿_𝑛^{(𝑟𝑠−1)}tanh⁻¹𝑥₊^1/𝑟𝜑(𝑥)𝑑𝑥=0. (2.21) It follows that 𝛿^{(𝑟𝑠−1)}((tanh⁻¹𝑥₊)^1/𝑟)=0 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 𝛿^{(𝑟𝑠−1)}(tanh|𝑥|)^1/𝑟=^𝑠−1𝐾_𝑘=0^𝑘_𝑖=01+(−1)^𝑘𝑟𝑐_{𝑠−2𝑖−1,𝑘}(𝑟𝑠−1)!𝛿2𝑘!^(𝑘)(𝑥), (2.22) 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 and𝛿^(𝑟−1)(tanh|𝑥|)^1/𝑟=(−1)^𝑟−1(𝑥),𝛿𝑟!𝛿_(2𝑟−1)(tanh|𝑥|)^1/𝑟=(2𝑟)!2𝛿(𝑥), (2.23) for 𝑟=1,2,….

Proof. To prove (2.22), we note that ¹₋₁𝛿_𝑛^{(𝑟𝑠−1)}tanh⁻¹|𝑥|^1/𝑟𝑥^𝑘𝑑𝑥=𝑛^𝑟𝑠1₋₁𝜌^{(𝑟𝑠−1)}𝑛tanh⁻¹|𝑥|^1/𝑟𝑥^𝑘𝑑𝑥=𝑛^𝑟𝑠1+(−1)^𝑘1₀𝜌^(𝑠)𝑛tanh⁻¹𝑥^1/𝑟𝑥^𝑘𝑑𝑥, (2.24) 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 tanh^𝑘𝑥=^∞_𝑗=𝑘𝑏_𝑗,𝑘𝑥^𝑗, (2.25) for 𝑘=0,1,2,….

Theorem 2.3. The neutrix composition 𝛿^(𝑠)(tanh⁻¹𝑥₊^1/𝑟) exists and 𝛿^(𝑠)tanh⁻¹𝑥₊^1/𝑟=^𝐾−1_𝑘=0(−1)^𝑠+𝑘𝑏_{𝑠+1,𝑘𝑟+𝑟}(𝑠+1)!𝛿2𝑘!^(𝑘)(𝑥), (2.26) 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 ¹₋₁𝛿_𝑛^(𝑠)tanh⁻¹𝑥₊^1/𝑟𝑥^𝑘𝑑𝑥. (2.27) We have ¹₋₁𝛿_𝑛^(𝑠)tanh⁻¹𝑥₊^1/𝑟𝑥^𝑘𝑑𝑥=𝑛^𝑠+11₋₁𝜌^(𝑠)𝑛tanh⁻¹𝑥₊^1/𝑟𝑥^𝑘𝑑𝑥=𝑛^𝑠+11₀𝜌^(𝑠)𝑛tanh⁻¹𝑥^1/𝑟𝑥^𝑘𝑑𝑥+𝑛^𝑠+10₋₁𝜌^(𝑠)(0)𝑥^𝑘𝑑𝑥=𝐽_1,𝑘+𝐽_2,𝑘. (2.28) It is obvious that 𝑁−lim_𝑛→∞𝐽_2,𝑘=𝑁−lim_𝑛→∞⁰₋₁𝛿_𝑛^(𝑠)tanh⁻¹𝑥₊^1/𝑟𝑥^𝑘𝑑𝑥=0. (2.29) Making the substitution 𝑡=𝑛(tanh⁻¹𝑥^1/𝑟), we have for large enough 𝑛𝐽_1,𝑘=𝑟𝑛^𝑠1₀tanh^{𝑘𝑟+𝑟−1}𝑡𝑛sech²𝑡𝑛𝜌^(𝑠)(𝑡)=𝑛𝑑𝑡_𝑠+1𝑘+1¹₀𝜌^(𝑠)(𝑡)𝑑tanh^{𝑘𝑟+𝑟}𝑡𝑛𝑛=−^𝑠+1𝑘+1¹₀tanh^{𝑘𝑟+𝑟}𝑡𝑛𝜌^(𝑠+1)(𝑡)1𝑑𝑡=−𝑘+1^∞_{𝑗=𝑘𝑟+𝑟}𝑛^{𝑠−𝑗+11}₀𝑏_{𝑗,𝑘𝑟+𝑟}𝑡^𝑗𝜌^(𝑠+1)(𝑡)𝑑𝑡, (2.30) and it follows that 𝑁−lim_𝑛→∞𝐽_1,𝑘=𝑁−lim_𝑛→∞𝑟𝑛^𝑠1₀tanh^{𝑘𝑟+𝑟−1}𝑡𝑛sech²𝑡𝑛𝜌^(𝑠)(𝑡)𝑏𝑑𝑡=−^{𝑠+1,𝑘𝑟+𝑟}𝑘+1¹₀𝑡^𝑠+1𝜌^(𝑠+1)(𝑡)=()𝑑𝑡−1^𝑠𝑏_{𝑠+1,𝑘𝑟+𝑟}(𝑠+1)!2().𝑘+1 (2.31) In particular, when 𝑠=0 and 𝑟=1, we have 𝐾=1 and then lim_𝑛→∞𝐽_1,0=𝑏_1,12=12. (2.32) When 𝑘=𝐾, we have 𝐽_1,𝑘≤1𝐾+1^∞_{𝑗=𝐾𝑟+𝑟}𝑛^{𝑠−𝑗+11}₀𝑏_{𝑗,𝐾𝑟+𝑟}𝑡^𝑗𝜌^(𝑠+1)(𝑡)𝑛𝑑𝑡=𝑂^{𝑠−𝐾𝑟−𝑟+1}. (2.33) Thus, if 𝜓 is an arbitrary continuous function, then ¹₀𝛿_𝑛^(𝑠)tanh⁻¹𝑥₊^1/𝑟𝑥^𝐾𝜓(𝑥)𝑑𝑥=𝑂𝑛^{𝑠−𝐾𝑟−𝑟+1}, (2.34) and so lim_𝑛→∞¹₀𝛿_𝑛^(𝑠)tanh⁻¹𝑥₊^1/𝑟𝑥^𝐾𝜓(𝑥)𝑑𝑥=lim_𝑛→∞𝑟𝑛^𝑠1₀tanh^{𝐾𝑟+𝑟−1}𝑡𝑛sech²𝑡𝑛𝜌^(𝑠)(𝑡)𝜓(𝑡)𝑑𝑡=0, (2.35) since 𝑠−𝐾𝑟−𝑟+1<0.
We also have⁰₋₁𝛿_𝑛^(𝑠)tanh⁻¹𝑥₊^1/𝑟𝜓(𝑥)𝑑𝑥=𝑛^𝑠+10₋₁𝜌^(𝑠)(0)𝜓(𝑥)𝑑𝑥, (2.36) and it follows that 𝑁−lim_𝑛→∞⁰₋₁𝛿_𝑛^(𝑠)tanh⁻¹𝑥₊^1/𝑟𝜓(𝑥)𝑑𝑥=0. (2.37) If now 𝜑 is an arbitrary function in 𝒟[−1,1], then by Taylor's Theorem, we have 𝜑(𝑥)=^𝐾−1_𝑘=0𝜑^(𝑘)(0)𝑥𝑘!^𝑘+𝑥^𝐾𝜑𝐾!^(𝐾)(𝜉𝑥), (2.38) where 0<𝜉<1, and so 𝑁−lim_𝑛→∞𝛿_𝑛^(𝑠)tanh⁻¹𝑥₊^1/𝑟,𝜑(𝑥)=𝑁−lim_𝑛→∞^𝐾−1_𝑘=0𝜑^(𝑘)(0)𝑘!¹₀𝛿_𝑛^(𝑠)tanh⁻¹𝑥₊^1/𝑟𝑥^𝑘𝑑𝑥+𝑁−lim_𝑛→∞^𝐾−1_𝑘=0𝜑^(𝑘)(0)𝑘!⁰₋₁𝛿_𝑛^(𝑠)tanh⁻¹𝑥₊^1/𝑟𝑥^𝑘𝑑𝑥+lim_𝑛→∞1𝐾!¹₀𝛿_𝑛^(𝑠)tanh⁻¹𝑥₊^1/𝑟𝑥^𝐾𝜑^(𝐾)(𝜉𝑥)𝑑𝑥+𝑁−lim_𝑛→∞1𝐾!⁰₋₁𝛿_𝑛^(𝑠)tanh⁻¹𝑥₊^1/𝑟𝑥^𝐾𝜑^(𝐾)()=𝜉𝑥𝑑𝑥_{𝐾−1𝑘=0}(−1)^𝑠𝑏_{𝑠+1,𝑘𝑟+𝑟}(𝑠+1)!𝜑^(𝑘)(0)=2𝑘!+0_{𝐾−1𝑘=0}(−1)^𝑠+𝑘𝑏_{𝑠+1,𝑘𝑟+𝑟}(𝑠+1)!𝛿2𝑘!^(𝑘)(𝑥),𝜑(𝑥), (2.39) on using (2.9) and (2.12). This proves (2.26) on the interval (−1,1). It is clear that 𝛿^(𝑠)(tanh⁻¹𝑥₊^1/𝑟)=0 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^𝑏_𝑎𝛿_𝑛^(𝑠)tanh⁻¹𝑥₊^1/𝑟𝜑(𝑥)𝑑𝑥=𝑛^𝑠+1𝑏_𝑎𝜌^(𝑠)(0)𝜑(𝑥)𝑑𝑥, (2.40) and so 𝑁−lim_𝑛→∞^𝑏_𝑎𝛿_𝑛^(𝑠)tanh⁻¹𝑥₊^1/𝑟𝜑(𝑥)𝑑𝑥=0. (2.41) It follows that 𝛿^(𝑠)(tanh⁻¹𝑥₊^1/𝑟)=0 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 𝛿^(𝑠)tanh⁻¹|𝑥|^1/𝑟=^𝐾−1_𝑘=0(−1)^𝑠+𝑘+(−1)^𝑠𝑏_{𝑠+1,𝑘𝑟+𝑟}(𝑠+1)!𝛿2𝑘!^(𝑘)(𝑥), (2.42) 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 𝛿tanh⁻¹|𝑥|=𝛿(𝑥). (2.43)

Proof. To prove (2.42), we note that ¹₋₁𝛿_𝑛^(𝑠)tanh⁻¹|𝑥|^1/𝑟𝑥^𝑘𝑑𝑥=𝑛^𝑠+11₋₁𝜌^(𝑠)𝑛tanh⁻¹|𝑥|^1/𝑟𝑥^𝑘𝑑𝑥=𝑛^𝑠+1(−1)^𝑠+𝑘+(−1)^𝑠1₀𝜌^(𝑠)𝑛tanh⁻¹𝑥^1/𝑟𝑥^𝑘𝑑𝑥, (2.44) 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.