ISSN 2291-8639
Volume 13, Number 2 (2017), 161-169
http://www.etamaths.com
ON THE COMPOSITION AND NEUTRIX COMPOSITION OF THE DELTA FUNCTION AND THE FUNCTION cosh−1(|x|1/r+ 1)
BRIAN FISHER1, EMIN OZCAG2,∗AND FATMA AL-SIREHY3
Abstract. LetF be a distribution inD0and letfbe a locally summable function. The composition F(f(x)) ofF andfis said to exist and be equal to the distributionh(x) if the limit of the sequence
{Fn(f(x))}is equal toh(x), whereFn(x) =F(x)∗δn(x) forn= 1,2, . . . and{δn(x)}is a certain
regular sequence converging to the Dirac delta function. It is proved that the neutrix composition
δ(s)[cosh−1(x1/r
+ + 1)] exists and
δ(s)[cosh−1(x1/r+ + 1)] =− M−1
X
k=0 kr+r
X
i=0
k
i
(−1)i+krcr,s,k
(kr+r)k! δ
(k)(x),
fors=M−1, M, M+ 1, . . .andr= 1,2, . . . ,where
cr,s,k= i
X
j=0
i
j
(−1)kr+r−i(2j−i)s+1
2s+i+1 ,
M is the smallest integer for whichs−2r+ 1<2M randr≤s/(2M+ 2).
Further results are also proved.
1. Introduction
LetDbe the space of infinitely differentiable functions with compact support, letD0 be the space of
distributions defined onD.
A sequence of functions{fn} is said to be regular if
(i)fn is infinitely differentiable for all n,
(ii) the sequence{hfn, ϕi}converges to a limit hf, ϕifor everyϕ∈ D,
(iii) hf, ϕiis continuous inϕin the sense that limn→∞hfn, ϕi= 0 for each sequenceϕn →0 inD, see [24].
There are many ways to construct a sequence of regular functions which converges toδ(x).For instance letρbe a fixed infinitely differentiable function having the properties:
(i) ρ(x) = 0 for|x| ≥1, (ii) ρ(x)≥0,
(iii) ρ(x) =ρ(−x), (iv)
Z 1
−1
ρ(x)dx= 1,
putting δn(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-functionδ(x).
Further, ifF is a distribution inD0 andF
n(x) =hF(x−t), δn(x)i,then{Fn(x)}is a regular sequence of infinitely differentiable functions converging toF(x).
In the framework of the theory of distributions, no meaning can be generally given to expressions of the form F(f(x)) where F and f are arbitrary distributions. However, in elementary particle physics one finds the need to evaluate δ2(x) when calculating the transition rates of certain particle
interactions, [14]. In addition, there are terms proportional to powers of theδfunctions at the origin
Received 7thSeptember, 2016; accepted 4th November, 2016; published 1stMarch, 2017.
2010Mathematics Subject Classification. 33B10, 46F30, 46F10, 41A30.
Key words and phrases. distribution; delta function; composition of distributions; neutrix composition of distributions.
c
2017 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.
coming from the measure of path integration [10]. The composition of a distribution and an infinitely differentiable function is extended to distributions by continuity provided the derivative of the infinitely differentiable function is different from zero, [2]. The composition of a distribution and an infinitely differentiable function is extended to distributions by continuity provided the derivative of the infinitely differentiable function is different from zero, [2]. Fisher [5] defined the composition of a distribution
F and a summable function f which has a single simple root in the open interval (a, b), and it was recently generalized in [18] by allowing f to be a distribution. Antosik [1] defined the composition
g(f(x)) as the limit of the sequence{gn(fn)} providing the limit exists. By this definition he defined the compositions√δ= 0, √δ2+ 1 = 1 +δ, log(1 +δ) = 0, sinδ= 0, cosδ= 1 and 1
1+δ = 1.
For many pairs of distributions, it is not possible to define their compositions by using the defini-tion of Antosik. Using the neutrix calculus developed by van der Corput [3], Fisher gave a general principle for the discarding of unwanted infinite quantities from asymptotic expansions and this has been exploited in context of distributions, see [4,5]. The technique of neglecting appropriately defined infinite quantities was devised by Hadamard and the resulting finite value extracted from divergent integral is referred to as the Hadamard finite part, see [16]. In fact his method can be regarded as a particular applications of the neutrix calculus.
The following definition of the neutrix composition of distributions is a generalization of Gel’fand and Shilov’s definition of the composition involving the delta function [15], and was given in [5].
Definition 1.1. LetF be a distribution inD0 and letf be a locally summable function. We say that the
neutrix compositionF(f(x))exists and is equal tohon the open interval(a, b), with−∞< a < b <∞, if
N−lim n→∞
Z ∞
−∞
Fn(f(x))ϕ(x)dx=hh(x), ϕ(x)i
for allϕ inD[a, b], whereFn(x) =F(x)∗δn(x) forn= 1,2, . . . andN is the neutrix, see [3], having
domainN0 the positive and rangeN00 the real numbers, with negligible functions which are finite linear sums of the functions
nλlnr−1n, lnrn: λ >0, r= 1,2, . . .
and all functions which converge to zero in the usual sense asntends 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→∞
Z ∞
−∞
Fn(f(x))ϕ(x)dx=hh(x), ϕ(x)i
for allϕinD[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 off(n),see [4,6,7,16].
2. Main Results
By using Fisher’s definition Koh and Li give meaning toδk and (δ0)k fork= 2,3, . . . ,see [17], and the more general form (δ(r))k was considered by Kou and Fisher in [18]. The meaning has been given to the symbolδk
+ in [22] and the k-th powers ofδfor negative integers were defined in [21].
Recently, in [20] Chenkuan Li and Changpin Li used Caputo fractional derivatives and Definition 1.1 and chose the followingδ−sequence
δn(x) =n
π
e−nx2 (x∈R)
to redefine powers of the distributionsδk(x) and (δ0)k(x) for some values ofk∈R.
The following two theorems were proved in [6] and [7] respectively.
Theorem 2.1. The neutrix compositionδ(s)(sgnx|x|λ)exists and
COMPOSITIONδ (cosh (|x| + 1)) 163
fors= 0,1,2, . . .and (s+ 1)λ= 1,3, . . . and
δ(s)(sgnx|x|λ) = (−1)(s+1)(λ+1)s!
λ[(s+ 1)λ−1]!δ
((s+1)λ−1)(x)
fors= 0,1,2, . . .and (s+ 1)λ= 2,4, . . . .
Theorem 2.2. The compositions δ(2s−1)(sgnx|x|1/s)andδ(s−1)(|x|1/s)exist and
δ(2s−1)(sgnx|x|1/s) = 1 2(2s)!δ
0(x),
δ(s−1)(|x|1/s) = (−1)s−1δ(x)
fors= 1,2, . . . .
The next theorem was proved in [9].
Theorem 2.3. The neutrix compositionδ(s)(sinh−1x1+/r)exists and δ(s)(sinh−1x1+/r) =
M−1
X
k=0
kr+r−1
X
i=0
kr+r−1
i
(−1)i+kra s,k,i 2kr+rk! δ
(k)(x),
fors= 0,1,2, . . .andr= 1,2, . . . ,whereM is the smallest positive integer greater than(s−r+ 1)/r
and
ar,s,k,i=
(−1)s[(kr+r−2i)s+ (kr+r−2i−2)s]
2 .
In particular, the neutrix composition δ(sinh−1x1+/r)exists and
δ(sinh−1x1+/r) = 0,
forr= 2,3, . . . .
In the following, we define the function δ(s)[cosh−1(x1+/r+ 1)] by
δ(s)[cosh−1(x1+/r+ 1)] =
δ(s)[cosh−1(|x|1/r+ 1)], x≥0,
0, x <0
and we define the functionδ(s)[cosh−1(x1/r
− + 1)] by
δ(s)[cosh−1(x1−/r+ 1)] =
δ(s)[cosh−1(
|x|1/r+ 1)], x≤0,
0, x >0
forr= 1,2, . . .ands= 0,1,2, . . . .
We also use the following easily proved lemma.
Lemma 2.1.
Z 1
0
tiρ(s)(t)dt=
0, 0≤i < s, 1
2(−1)
ss!, i=s
fors= 0,1,2, . . . .
We now prove
Theorem 2.4. The neutrix compositionδ(s)[cosh−1(x1+/r+ 1)] exists and
δ(s)[cosh−1(x1+/r+ 1)] = M−1
X
k=0
kr+r
X
i=0
k
i
(−1)krc r,s,k (kr+r)k! δ
(k)(x) (2.1)
fors=M −1, M, M+ 1, . . .andr= 1,2, . . . ,where
cr,s,k= i
X
j=0
i
j
(−1)kr+r+s−i(2j−i)s+1
2i+1 ,
M is the smallest integer for whichs−2r+ 1<2M r andr≤s/(2M + 2).
In particular, the neutrix composition δ[cosh−1(x++ 1)]exists and
forr= 1,2, . . . and the neutrix compositionδ0[cosh−1(x++ 1)] exists and
δ0[cosh−1(x++ 1)] =
1
4δ(x). (2.3)
Proof. To prove equation (1), we first of all have to evaluate
Z 1
−1
δn(s)[cosh−1(x+1/r+ 1)]xkdx = ns+1
Z 1
−1
ρ(s)[ncosh−1(x1+/r+ 1)]xkdx
= ns+1
Z 1
0
ρ(s)[ncosh−1(x1/r+ 1)]xkdx
+ns+1
Z 0
−1
ρ(s)(0)xkdx
= I1+I2. (2.4)
It is obvious that
N−lim
n→∞ I2= Nn−→∞limn
s+1
Z 0
−1
ρ(s)(0)xkdx= 0, (2.5)
fork= 0,1,2, . . . .
Making the substitutiont=ncosh−1(x1/r+ 1), we have for large enoughn
I1 = rns
Z 1
0
[cosh(t/n)−1]kr+r−1sinh(t/n)ρ(s)(t)dt
= −rn
s+1
kr+r
Z 1
0
[cosh(t/n)−1]kr+rρ(s+1)(t)dt
= −rn
s+1
kr+r
kr+r
X
i=0
kr+r
i
(−1)kr+r−i
Z 1
0
coshi(t/n)ρ(s+1)(t)dt
= −rn
s+1
kr+r
kr+r
X
i=0
kr+r
i i X j=0 i j
(−1)kr+r−i 2i
Z 1
0
exp[(2j−i)t/n]ρ(s+1)(t)dt
= −rn
s+1
kr+r
kr+r
X
i=0
kr+r
i i X j=0 i j ∞ X m=0
(−1)kr+r−i(2j−i)m 2im!nm
Z 1
0
tmρ(s+1)(t)dt.
It follows that
N−lim n→∞ I1 =
r kr+r
kr+r
X
i=0
kr+r
i i X j=0 i j
(−1)kr+r+s−i(2j−i)s+1
2i(s+ 1)!
Z 1
0
ts+1ρ(s+1)(t)dt
= r
kr+r
kr+r
X
i=0
kr+r i i X j=0 i j
(−1)kr+r+s−i(2j−i)s+1 2i+1
= r
kr+r
kr+r
X
i=0
kr+r
i
cr,s,k, (2.6)
fork= 0,1,2, . . . .
Whenk=M, we have
|I1| ≤
rns+1 M r+r
Z 1
0
[cosh(t/n)−1]
M r+rρ(s+1)(t)
dt
≤ rns+1
Z 1
0
[(t/n)2+O(n−4)]M r+r|ρ(s+1)(t)|dt
≤ rns−2M r−2r+1
Z 1
0
[1 +O(n−4M r−4r)]|ρ(s+1)(t)|dt
COMPOSITIONδ (cosh (|x| + 1)) 165
Thus, ifψ is an arbitrary continuous function, then
lim n→∞
Z 1
0
δn(s)[cosh−1(x+1/r+ 1)]xMψ(x)dx= 0, (2.7)
sinces−2M r−2r+ 1<0.
We also have
Z 0
−1
δ(ns)[cosh−1(x+1/r+ 1)]ψ(x)dx = ns+1
Z 0
−1
ρ(s)(0)ψ(x)dx
and it follows that
N−lim n→∞
Z 0
−1
δn(s)[(sinh−1x+)1/r]ψ(x)dx= 0. (2.8)
If now ϕis an arbitrary function in D[−1,1],then by Taylor’s Theorem, we have
ϕ(x) = M−1
X
k=0
ϕ(k)(0) k! x
k+xM
M!ϕ
(M)(ξx),
where 0< ξ <1, and so
N−lim n→∞ hδ
(s)
n [cosh−1(x
1/r
+ + 1)], ϕ(x)i = N−lim
n→∞
M−1
X
k=0
ϕ(k)(0)
k!
Z 1
0
δn(s)[cosh−1(x
1/r
+ + 1)]xkdx
+ N−lim n→∞
M−1
X
k=0
ϕ(k)(0) k!
Z 0
−1
δn(s)[cosh−1(x1+/r+ 1)]xkdx
+ lim n→∞
1
M!
Z 1
0
δ(ns)[cosh−1(x+1/r+ 1)]xMϕ(M)(ξx)dx
+ lim n→∞
1
M!
Z 0
−1
δn(s)[cosh−1(x+1/r+ 1)]xMϕ(M)(ξx)dx
= M−1
X
k=0
kr+r
X
i=0
k
i
rc
r,s,kϕ(k)(0) (kr+r)k! + 0
= M−1
X
k=0
kr+r
X
i=0
k
i
(−1)krc r,s,k (kr+r)k! hδ
(k)(x), ϕ(x)
i, (2.9)
on using equations (4) to (9). This proves equation (1) on the interval (−1,1).
It is clear that δ(s)[cosh−1(x+1/r+ 1)] = 0 forx >0 and so equation (1) holds forx >0.
Now suppose thatϕis an arbitrary function inD[a, b], wherea < b <0. Then
Z b
a
δn(s)[cosh−1(x+1/r+ 1)]ϕ(x)dx=ns+1
Z b
a
ρ(s)(0)ϕ(x)dx
and so
N−lim n→∞
Z b
a
δn(s)[cosh−1(x+1/r+ 1)]ϕ(x)dx= 0.
It follows thatδ(s)[cosh−1(x1/r
+ + 1)] = 0 on the interval (a, b).Sinceaandbare arbitrary, we see that
equation (1) holds on the real line.
To prove equation (2), we note that in this case s= 0 and so M = 0 forr= 1,2, . . . .The sum in equation (1) is therefore empty and equation (2) follows.
When r = s = 1 it follows that M = 1 and equation (3) then follows from equation (1). This completes the proof of the theorem.
Corollary 2.1. The neutrix compositionδ(s)[cosh−1(x1−/r+ 1)] exists and
δ(s)[cosh−1(x1−/r+ 1)] = M−1
X
k=0
kr+r
X
i=0
k
i
rc
r,s,k (kr+r)k!δ
fors=M −1, M, M+ 1, . . .andr= 1,2, . . . ,
In particular, the neutrix composition δ[cosh−1(x1−/r+ 1)] exists and
δ[cosh−1(x1−/r+ 1)] = 0 (2.11)
forr= 1,2, . . . and the neutrix compositionδ0[cosh−1(x1/r
− + 1)] exists and
δ0[cosh−1(x1−/r+ 1)] = 1
4δ(x). (2.12)
Proof. Equations (10) to (12) follow immediately on replacing x by −x in equations (1) to (3) respectively.
Corollary 2.2. The neutrix compositionδ(s)[cosh−1(|x|1/r+ 1)]exists and
δ(s)[cosh−1(|x|1/r+ 1)] = M−1
X
k=0
kr+r
X
i=0
k
i
[1 + (−1)k]rc r,s,k (kr+r)k! δ
(k)(x) (2.13)
fors=M −1, M, M+ 1, . . .andr= 1,2, . . . ,
In particular, the neutrix composition δ[cosh−1(|x|1/r+ 1)]exists and
δ[cosh−1(|x|1/r+ 1)] = 0 (2.14)
forr= 1,2, . . . and the neutrix compositionδ0[cosh−1(|x|1/r+ 1)] exists and
δ0[cosh−1(|x|1/r+ 1)] = 1
2δ(x). (2.15)
Proof. Equation (13) follows from equations (1) and (10) on noting that
δ(s)[cosh−1(|x|1/r+ 1)] =δ(s)[cosh−1(x1/r
+ + 1)] +δ
(s)[cosh−1(x1/r
− + 1)].
Equations (14) to (15) follow similarly.
Theorem 2.5. The neutrix compositionδ(s)[cosh−1(x
++ 1)1/r]exists and
δ(s)[cosh−1(x++ 1)1/r] =
M−1
X
k=0
kr+r
X
i=0
kr+r
i
(−1)kb r,s,k
k! δ
(k)(x) (2.16)
fors=M −1, M, M+ 1, . . .andr= 1,2, . . . ,where
br,s,k = ri+r
X
j=0
ri+r
j
(−1)s+k−ir(2j−ri−r)s+1
2ri+r+1(ri+r) ,
M is the smallest integer for whichs+ 1<2M r andr≤(s+ 1)/(2M).
Proof. To prove equation (16), we first of all have to evaluate
Z 1
−1
δ(ns)[cosh−1(x++ 1)1/r]xkdx = ns+1
Z 1
−1
ρ(s)[ncosh−1(x++ 1)1/r]xkdx
= ns+1
Z 1
0
ρ(s)[ncosh−1(x++ 1)1/r]xkdx
+ns+1
Z 0
−1
ρ(s)(0)xkdx
= J1+J2. (2.17)
It is obvious that
N−lim
n→∞ J2= Nn−→∞limn
s+1Z 0
−1
ρ(s)(0)xkdx= 0, (2.18)
COMPOSITIONδ (cosh (|x| + 1)) 167
Making the substitutiont=ncosh−1(x++ 1)1/r, we have for large enoughn
J1 = rns
Z 1
0
[coshr(t/n)−1]kcoshr−1(t/n) sinh(t/n)ρ(s)(t)dt
= rns
k X i=0 k i
(−1)k−i
Z 1
0
coshri+r−1(t/n) sinh(t/n)ρ(s)(t)dt
= −rns+1
k
X
i=0
k
i
(−1)k−i
ri+r
Z 1
0
coshri+r(t/n)ρ(s+1)(t)dt
= −rns+1
k
X
i=0
k
i
ri+r X
j=0
ri+r
j
(−1)k−i 2ri+r(ri+r)
Z 1
0
exp[(2j−ri−r)t/n]ρ(s+1)(t)dt
= −rns+1
kr+r
X
i=0
kr+r
i
ri+r X
j=0
ri+r
j
∞
X
m=0
(−1)k−i(2j−ri−r)m 2ri+r(ri+r)m!nm
Z 1
0
tmρ(s+1)(t)dt.
It follows that
N−lim
n→∞ J1 = −
kr+r
X
i=0
kr+r i
ri+r X
j=0
ri+r j
(−1)k−ir(2j−ri−r)s+1 2ri+r(ri+r)(s+ 1)!
Z 1
0
ts+1ρ(s+1)(t)dt
= kr+r
X
i=0
kr+r
i
ri+r X
j=0
ri+r
j
(−1)s+k−ir(2j−ri−r)s+1
2ri+r+1(ri+r)
= kr+r
X
i=0
kr+r
i
br,s,k, (2.19)
fork= 0,1,2, . . . .
Whenk=M, we have
|J1| ≤ rns
Z 1
0
[cosh
r(t/n)−1]Mcoshr−1(t/n) sinh(t/n)ρ(s)(t)
dt
≤ rns
Z 1
0
[(t/n)
2r+O(n−4r)]Mcoshr−1(t/n) sinh(t/n)ρ(s)(t)
dt
= O(ns−2M r−1).
Thus, ifψ is an arbitrary continuous function, then
lim n→∞
Z 1
0
δn(s)[cosh−1(x++ 1)1/r]xMψ(x)dx= 0, (2.20)
sinces−2M r−1<0.
We also have
Z 0
−1
δ(ns)[cosh−1(x++ 1)1/r]ψ(x)dx = ns+1
Z 0
−1
ρ(s)(0)ψ(x)dx
and it follows that
N−lim n→∞
Z 0
−1
δn(s)[(sinh−1x+)1/r]ψ(x)dx= 0. (2.21)
If now ϕis an arbitrary function in D[−1,1],then by Taylor’s Theorem, we have
ϕ(x) = M−1
X
k=0
ϕ(k)(0)
k! x k+x
M
M!ϕ
where 0< ξ <1, and so
N−lim n→∞ hδ
(s)
n [cosh
−1(x
++ 1)1/r], ϕ(x)i=
= N−lim n→∞
M−1
X
k=0
ϕ(k)(0)
k!
Z 1
0
δn(s)[cosh−1(x++ 1)1/r]xkdx
+ N−lim n→∞
M−1
X
k=0
ϕ(k)(0)
k!
Z 0
−1
δn(s)[cosh
−1
(x++ 1)1/r]xkdx
+ lim n→∞
1
M!
Z 1
0
δ(ns)[cosh−1(x++ 1)1/r]xMϕ(M)(ξx)dx
+ lim n→∞
1
M!
Z 0
−1
δn(s)[cosh−1(x++ 1)1/r]xMϕ(M)(ξx)dx
= M−1
X
k=0
kr+r
X
i=0
kr+r i
br,s,kϕ(k)(0)
k! + 0
= M−1
X
k=0
kr+r
X
i=0
kr+r i
(−1)kbr,s,k
k! hδ
(k)(x), ϕ(x)i, (2.22)
on using equations (17) to (22). This proves equation (16) on the interval (−1,1).
Replacing xby−xin equation (16), we get
Corollary 2.3. The neutrix compositionδ(s)[cosh−1(x
−+ 1)1/r]exists and
δ(s)[cosh−1(x−+ 1)1/r] =
M−1
X
k=0
kr+r
X
i=0
kr+r
i
b
r,s,k
k! δ
(k)(x) (2.23)
fors=M −1, M, M+ 1, . . .andr= 1,2, . . . ,
Corollary 2.4. The neutrix compositionδ(s)[cosh−1(|x|+ 1)1/r]exists and
δ(s)[cosh−1(|x|+ 1)1/r] = M−1
X
k=0
kr+r
X
i=0
kr+r
i
[1 + (−1)]kb r,s,k
k! δ
(k)(x) (2.24)
fors=M −1, M, M+ 1, . . .andr= 1,2, . . . ,
Proof. Equation (24) follows from equations (16) and (23) on noting that
δ(s)[cosh−1(|x|+ 1)1/r] =δ(s)[cosh−1(x++ 1)1/r] +δ(s)[cosh−1(x−+ 1)1/r].
For further related results on the neutrix composition of distributions, see [11], [12], [13], [19] and [23].
References
[1] P. Antosik, Composition of Distributions, Technical Report no.9 University of Wisconsin, Milwaukee, (1988-89), pp.1-30.
[2] P. Antosik, J. Mikusinski and R. Sikorski, Theory of Distributions, The sequential Approach, PWN-Elsevier, Warszawa-Amsterdam (1973).
[3] J.G. van der Corput, Introduction to the neutrix calculus, J. Analyse Math., 7(1959), 291-398. [4] B. Fisher, On defining the distributionδ(r)(f(x)), Rostock. Math. Kolloq., 23(1983), 73-80.
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COMPOSITIONδ (cosh (|x| + 1)) 169
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1
Department of Mathematics and Computer Science, Leicester University, England
2
Department of Mathematics, Hacettepe University, Ankara, Turkey
3
Department of Mathematics, Jeddah, King Abdulaziz University, Jeddah, Saudi Arabia
∗
