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

Authors: Brian Fisher; Adem Kılıçman
Journal: Journal of Applied Mathematics
 (2011)

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## Abstract

LetF 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 {Fn(f(x))} is equal to h(x), where Fn(x)=F(x)*δn(x) for n=1,2,… and {δn(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]=∑k=0sr+r-1∑i=0k(ki)((-1)krcs,k,i/2k+1k!)δ(k)(x), for s=0,1,2,… and r=1,2,…, where cs,k,i=(-1)ss![(k-2i+1)rs-1+(k-2i-1)rs+r-1]/(2(rs+r-1)!). Further results are also proved.


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## Body

## 1. Introduction

In the following, we let𝒟 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, letρ(x) be a function in 𝒟[-1,1] having the following properties:(i)
ρ(x)≥0,(ii)
ρ(x)=ρ(-x),(iii)
∫-11ρ(x)dx=1.Puttingδn(x)=nρ(nx) for n=1,2,…, it follows that {δn(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 Fn(x)=F(x)*δn(x)=〈F(x-t),φ(t)〉, then {Fn(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.
LetF 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
(1.1)N-lim⁡n→∞∫-∞∞Fn(f(x))φ(x)dx=〈h(x),φ(x)〉,
for all φ in 𝒟[a,b], where Fn(x)=F(x)*δn(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
(1.2)nλln⁡r-1n,ln⁡rn:λ>0,r=1,2,…
and all functions which converge to zero in the usual sense as n tends to infinity.
In particular, we say that the compositionF(f(x)) exists and is equal to h on the open interval (a,b) if
(1.3)lim⁡n→∞∫-∞∞Fn(f(x))φ(x)dx=〈h(x),φ(x)〉,
for all φ in 𝒟[a,b].Note that taking the neutrix limit of a functionf(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 compositionδ(s)(sgnx|x|λ) exists and
(1.4)δ(s)(sgn⁡x|x|λ)=0,
for s=0,1,2,… and (s+1)λ=1,3,…, and
(1.5)δ(s)(sgn⁡x|x|λ)=(-1)(s+1)(λ+1)s!λ[(s+1)λ-1]!δ((s+1)λ-1)(x),
for s=0,1,2,…, and (s+1)λ=2,4,….Theorem 1.3.
The neutrix compositionsδ(2s-1)(sgnx|x|1/s) and  δ(s-1)(|x|1/s) exist and
(1.6)δ(2s-1)(sgn⁡x|x|1/s)=12(2s)!δ′(x),δ(s-1)(|x|1/s)=(-1)s-1δ(x),
for s=1,2,….Theorem 1.4.
The neutrix compositionδ(s)(sinh-1x+1/r) exists and
(1.7)δ(s)[(sinh⁡-1x+)1/r]=∑k=0(s+1)/r-1∑i=0k(ki)(-1)krcs,k,i2k+1k!δ(k)(x),
for s=0,1,2,… and r=1,2,…, where
(1.8)cr,s,k,i=(-1)ss![(k-2i+1)rs+r-1+(k-2i-1)rs+r-1]2(rs+r-1)!.The next two theorems were proved in [13].Theorem 1.5.
The neutrix compositionδ(s)[lnr(1+|x|)] exists and
(1.9)δ(s)[ln⁡r(1+|x|)]=∑k=0sr+r-1∑i=0k(ki)(-1)s-i[1+(-1)k]s!(i+1)rs+r-12r(rs+r-1)!k!δ(k)(x).
for s=0,1,2,…, and r=1,2,….
In particular, the compositionδ[ln(1+|x|)] exists and
(1.10)δ[ln⁡[1+|x|)]=δ(x).Theorem 1.6.
The neutrix compositionδ(s)[ln(1+|x1/r|)] exists and
(1.11)δ(s)[ln⁡(1+|x1/r|)]=∑k=0m-1∑i=0kr+r-1(kr+r-1i)(-1)r+s+i-1[1+(-1)k]r(i+1)s2k!δ(k)(x),
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 compositionδ(s)[ln(1+|x1/r|)] exists and
(1.12)δ(s)[ln⁡(1+|x1/r|)]=0,
for s=0,1,2,…,r-2 and r=2,3,… and
(1.13)δ(r-1)[ln⁡(1+|x1/r|)]=(-1)r-1r!δ(x),
for r=2,3,….


## 2. Main Results

We now prove the following theorem.Theorem 2.1.
The neutrix compositionδ(s)[(sinh-1x+)r] exists and
(2.1)δ(s)[(sinh⁡-1x+)r]=∑k=0sr+r-1∑i=0k(ki)(-1)krcs,k,i2k+1k!δ(k)(x),
for s=0,1,2,… and r=1,2,…, where
(2.2)cr,s,k,i=(-1)ss![(k-2i+1)rs+r-1+(k-2i-1)rs+r-1]2(rs+r-1)!.
In particular, the neutrix composition δ(sinh-1x+) exists and
(2.3)δ(sinh⁡-1x+)=12δ(x).Proof.
To prove (2.1), we first of all evaluate
(2.4)∫-11δn(s)[(sinh⁡-1x+)r]xkdx.
We have
(2.5)∫-11δn(s)[(sinh⁡-1x+)r]xkdx=ns+1∫-11ρ(s)[(nsinh⁡-1x+)r]xkdx=ns+1∫01ρ(s)[n(sinh⁡-1x)r]xkdx+ns+1∫-10ρ(s)(0)xkdx=I1+I2.
It is obvious that
(2.6)N-lim⁡n→∞I2=N-lim⁡n→∞∫-10δn(s)[(sinh⁡-1x+)r]xkdx=0,
for k=0,1,2,….
Making the substitutiont=n(sinh-1x)r, we have for large enough n(2.7)I1=ns-r+1r∫01t1/(r-1)sinh⁡k(tn)1/rcosh⁡(tn)1/rρ(s)(t)dt×∫01t1/(r-1){exp⁡[(k-2i+1)(tn)1/r]+exp⁡[(k-2i-1)(tn)1/r]}ρ(s)(t)dt,
where
(2.8)n(s-1)/(r+1)∫01t1/(r-1){exp⁡[(k-2i+1)(tn)1/r]+exp⁡[(k-2i-1)(tn)1/r]}ρ(s)(t)dt=∑p=0∞∫01[(k-2i+1)p+(k-2i-1)p]t(p/r)+(1/r)-1p!n(p/r)+(1/r)-s-1ρ(s)(t)dt.
It follows that
(2.9)N-limn→∞ns-1/r+1∫01t1/(r-1){exp⁡[(k-2i+1)(tn)1/r]+exp⁡[(k-2i-1)(tn)1/r]}ρ(s)(t)dt=(-1)ss![(k-2i+1)rs+r-1+(k-2i-1)rs+r-1]2(rs+r-1)!=cr,s,k,i,
and by applying the neutrix limit we obtain
(2.10)N-lim⁡n→∞I1=N-lim⁡n→∞∫01δn(s)[(sinh⁡-1x+)r]xkdx=12k+1r∑i=0k(ki)(-1)icr,s,k,i
for k=0,1,2,….
Whenk=sr+r, we have
(2.11)|I1|=∫01|δn(s)[(sinh⁡-1x+)r]xsr+r|dx=ns+1∫01|ρn(s)[n(sinh⁡-1x)r]xsr+r|dx≤n(s-1)/(r+1)2sr+rrexp⁡(sr+r+1)∫01|[1-exp⁡[-2(tn)1/r]sr+rρ(s)(t)]|dt=n(s-1)/(r+1)2sr+rrexp⁡(sr+r+1)∫01[2(tn)1/r+O(n-2/r)]sr+r|ρ(s)(t)|dt≤n-1/rexp⁡(sr+r+1)∫01[1+O(n-2/r)]|ρ(s)(t)|dt=O(n-1/r).
Thus, if ψ is an arbitrary continuous function, then
(2.12)lim⁡n→∞∫01δn(s)[(sinh⁡-1x+)r]xrs+rψ(x)dx=0.
We also have
(2.13)∫-10δn(s)[(sinh⁡-1x+)r]ψ(x)dx=ns+1∫-10ρ(s)(0)ψ(x)dx,
and it follows that
(2.14)N-lim⁡n→∞∫-10δn(s)[(sinh⁡-1x+)r]ψ(x)dx=0.
If now φ is an arbitrary function in 𝒟[-1,1], then by Taylor's Theorem, we have
(2.15)φ(x)=∑k=0sr+r-1φ(k)(0)k!xk+xrs+r(rs+r)!φ(rs+r)(ξx),
where 0<ξ<1, and so
(2.16)N-lim⁡n→∞〈δn(s)[(sinh⁡-1x+)1/r],φ(x)〉=N-lim⁡n→∞∑k=0sr+r-1φ(k)(0)k!∫01δn(s)[(sinh⁡-1x+)r]xkdx+N-lim⁡n→∞∑k=0sr+r-1φ(k)(0)k!∫-10δn(s)[(sinh⁡-1x+)r]xkdx+lim⁡n→∞1(sr+r)!∫01δn(s)[(sinh⁡-1x+)r]xsr+rφ(sr+r)(ξx)dx+lim⁡n→∞1(sr+r)!∫-10δn(s)[(sinh⁡-1x+)r]xsr+rφ(sr+r)(ξx)dx=∑k=0sr+r-1∑i=0k(ki)rcr,s,k,iφ(k)(0)2k+1k!+0=∑k=0sr+r-1∑i=0k(ki)(-1)krcr,s,k,i2k+1k!〈δ(k)(x),φ(x)〉,
on using (2.3) to (2.14). This proves (2.1) on the interval (-1,1).
It is clear thatδ(s)[(sinh-1x+)r]=0 for x>0 and so (2.1) holds for x>-1.
Now, suppose thatφ is an arbitrary function in 𝒟[a,b], where a<b<0. Then,
(2.17)∫abδn(s)[(sinh⁡-1x+)r]φ(x)dx=ns+1∫abρ(s)(0)φ(x)dx
and so
(2.18)N-lim⁡n→∞∫abδn(s)[(sinh⁡-1x+)r]φ(x)dx=0.
It follows thatδ(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 compositionδ(s)[(sinh-1|x|)r] exists and
(2.19)δ(s)[(sinh⁡-1|x|)r]=∑k=0sr+r-1∑i=0k(ki)[(-1)k+1]cr,s,k,i2k+1k!δ(k)(x),
for s=0,1,2,… and r=1,2,….
In particular, the compositionδ(sinh-1|x|) exists and
(2.20)δ(sinh⁡-1|x|)=12δ(x).Proof.
To prove (2.19), we note that
(2.21)∫-11δn(s)[(sinh⁡-1|x|)r]xkdx=ns+1∫-11ρ(s)[(nsinh⁡-1|x|)r]xkdx=ns+1[1+(-1)k]∫01ρ(s)[n(sinh⁡-1x)r]xkdx,
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 compositionδ(2s-1)[sinh-1(sgnx·x2)] exists and
(2.22)δ(2s-1)[sinh⁡-1(sgn⁡x⋅x2)]=∑k=02s-1∑i=0i+k+1(ki)(-1)kbs,k,i2k+1(2k+1)!δ(k)(x),
for s=1,2,…, where
(2.23)bs,k,i=(k-2i+1)2s-1+(k-2i-1)2s-1.Proof.
To prove (2.22), we now have to evaluate
(2.24)∫-11δn(2s-1)[sinh⁡-1(sgnx⋅x2)]xkdx.
We have
(2.25)∫-11δn(2s-1)[sinh⁡-1(sgn⁡x⋅x2)]xkdx=n2s∫-11ρ(2s-1)[nsinh-1(sgn⁡x⋅x2)]xkdx={2n2s∫01ρ(2s-1)[n(sinh⁡-1x2)]xkdx,kodd,0,keven.
Making the substitution t=n(sinh-1x2), we have for large enough n(2.26)∫-11δn(2s-1)[sinh⁡-1(sgnx⋅x2)]xkdx=2n2s∫01ρ(2s-1)[n(sinh⁡-1x2)]x2k+1dx=n2s-12k+1∑i=0kki(-1)i∫01{exp⁡[(k-2i+1)tn]+exp⁡[(k-2i-1)tn]}ρ(2s-1)(t)dt,
where
(2.27)n2s-1∫01{exp⁡[(k-2i+1)tn]+exp⁡[(k-2i-1)tn]}ρ(s)(t)dt=∑p=0∞∫01[(k-2i+1)p+(k-2i-1)p]tpp!np-2s+1ρ(2s-1)(t)dt.
It follows that
(2.28)N-limn→∞n2s-1∫01{exp⁡[(k-2i+1)tn]+exp⁡[(k-2i-1)tn]}ρ(s)(t)dt=N-lim⁡n→∞∑p=0∞∫01[(k-2i+1)p+(k-2i-1)p]tpp!np-2s+1ρ(2s-1)(t)dt=-(k-2i+1)2s-1+(k-2i-1)2s-12=bs,k,i2,
and so by using the neutrix limit, we have
(2.29)N-lim⁡n→∞∫-11δn(2s-1)[sinh⁡-1(sgn⁡x⋅x2)]x2k+1dx=∑i=0k(ki)(-1)i+1bs,k,i2k+1,
for k=0,1,2,….
Whenk=2s, we have
(2.30)∫-11|δn(2s-1)[sinh⁡-1(sgn⁡x⋅x2)]x4s+1|dx=n2s∫-11ρ(2s-1)[n(sinh⁡-1x2)]x4s+1dx≤n2s-12s-1exp⁡(s+1)∫-11|[1-exp⁡(-2tn)]2sρ(2s-1)(t)|dt=n2s-12s-1exp⁡(s+1)∫-11|[2tn+O(n-2)]2sρ(2s-1)(t)|dt≤22s+1n-1exp⁡(s+1)∫-11[1+O(n-2/r)]|ρ(2s-1)(t)|dt=O(n-1).
Thus, if ψ is an arbitrary continuous function, then
(2.31)lim⁡n→∞∫-11δn(2s-1)[sinh⁡-1(sgn⁡x⋅x2)]x4s+1ψ(x)dx=0.
If now φ is an arbitrary function in 𝒟[-1,1], then by Taylor's Theorem, we have
(2.32)φ(x)=∑k=04sφ(k)(0)k!xk+x4s+1(4s+1)!φ(4s+1)(ξx),
where 0<ξ<1, and so
(2.33)N-lim⁡n→∞〈δn(2s-1)[sinh⁡-1(sgn⁡x⋅x2)],φ(x)〉=N-lim⁡n→∞∑k=02s-1φ(2k+1)(0)(2k+1)!∫-11δn(2s-1)[sinh⁡-1(sgn⁡x⋅x2)]x2k+1dx+lim⁡n→∞1(4s+1)!∫-11δn(4s+1)[sinh⁡-1(sgn⁡x⋅x2)]x4s+1φ(4s+1)(ξx)dx=∑k=02s-1∑i=0k(ki)(-1)i+1bs,k,iφ(k)(0)2k+1(2k+1)!+0=∑k=02s-1∑i=0k(ki)(-1)i+k+1bs,k,i2k+1(2k+1)!〈δ(k)(x),φ(x)〉,
on using (2.25) to (2.31), proving (2.22) on the interval (-1,1). However, it is clear that δn(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 compositionδ′[sinh-1sgnx·x2)] exists and
(2.34)δ′[sinh⁡-1(sgn⁡x⋅x2)]=δ′(x)4.3!-2δ(x).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].



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