International Journal of Analysis and Applications
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. 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

{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. It is proved that the neutrix composition

δ(s)[cosh−1(x
1/r
+ + 1)] exists and

δ(s)[cosh−1(x
1/r
+ + 1)] = −

M−1∑
k=0

kr+r∑
i=0

(k
i

)(−1)i+krcr,s,k
(kr + r)k!

δ(k)(x),

for s = M − 1,M,M + 1, . . . and r = 1, 2, . . . , where

cr,s,k =
i∑

j=0

(i
j

)(−1)kr+r−i(2j − i)s+1
2s+i+1

,

M is the smallest integer for which s− 2r + 1 < 2Mr and r ≤ s/(2M + 2).
Further results are also proved.

1. Introduction

Let D be the space of infinitely differentiable functions with compact support, let D′ be the space of
distributions defined on D.

A sequence of functions {fn} is said to be regular if

(i) fn is infinitely differentiable for all n,
(ii) the sequence {〈fn,ϕ〉} converges to a limit 〈f,ϕ〉 for every ϕ ∈D,
(iii) 〈f,ϕ〉 is continuous in ϕ in the sense that limn→∞〈fn,ϕ〉 = 0 for each sequence ϕn → 0 in D,

see [24].

There are many ways to construct a sequence of regular functions which converges to δ(x). For instance
let ρ be 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)
∫ 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-function δ(x).
Further, if F is a distribution in D′ and Fn(x) = 〈F(x− t),δn(x)〉, then {Fn(x)} is a regular sequence
of infinitely differentiable functions converging to F(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 the δ functions at the origin

Received 7th September, 2016; accepted 4th November, 2016; published 1st March, 2017.
2010 Mathematics 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.

161



162 FISHER, OZCAG AND AL-SIREHY

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, sin δ = 0, cos δ = 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. 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→∞

∫ ∞
−∞

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 [3], having
domain N′ the positive and range N′′ the real numbers, with negligible functions which are finite linear
sums of the functions

nλ lnr−1 n, lnr n : 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 composition F(f(x)) exists and is equal to h on the open interval

(a,b) if

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 function f(n), is equivalent to taking the usual limit of
Hadamard’s finite part of f(n), see [4, 6, 7, 16].

2. Main Results

By using Fisher’s definition Koh and Li give meaning to δk and (δ′)k for k = 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 symbol δk+ in [22] and the k-th powers of δ for 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 following δ−sequence

δn(x) =
(n
π

)
e−nx

2

(x ∈ R)

to redefine powers of the distributions δk(x) and (δ′)k(x) for some values of k ∈ R.

The following two theorems were proved in [6] and [7] respectively.

Theorem 2.1. The neutrix composition δ(s)(sgn x|x|λ) exists and

δ(s)(sgn x|x|λ) = 0



COMPOSITION δ(s)(cosh−1(|x|1/r + 1)) 163

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

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

δ(2s−1)(sgn x|x|1/s) = 1
2
(2s)!δ′(x),

δ(s−1)(|x|1/s) = (−1)s−1δ(x)
for s = 1, 2, . . . .

The next theorem was proved in [9].

Theorem 2.3. The neutrix composition δ(s)(sinh−1 x
1/r
+ ) exists and

δ(s)(sinh−1 x
1/r
+ ) =

M−1∑
k=0

kr+r−1∑
i=0

(
kr + r − 1

i

)
(−1)i+kras,k,i

2kr+rk!
δ(k)(x),

for s = 0, 1, 2, . . . and r = 1, 2, . . . , where M 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−1 x
1/r
+ ) exists and

δ(sinh−1 x
1/r
+ ) = 0,

for r = 2, 3, . . . .

In the following, we define the function δ(s)[cosh−1(x
1/r
+ + 1)] by

δ(s)[cosh−1(x
1/r
+ + 1)] =

{
δ(s)[cosh−1(|x|1/r + 1)], x ≥ 0,

0, x < 0

and we define the function δ(s)[cosh−1(x
1/r
− + 1)] by

δ(s)[cosh−1(x
1/r
− + 1)] =

{
δ(s)[cosh−1(|x|1/r + 1)], x ≤ 0,

0, x > 0

for r = 1, 2, . . . and s = 0, 1, 2, . . . .

We also use the following easily proved lemma.

Lemma 2.1. ∫ 1
0

tiρ(s)(t) dt =

{
0, 0 ≤ i < s,

1
2
(−1)ss!, i = s

for s = 0, 1, 2, . . . .
We now prove

Theorem 2.4. The neutrix composition δ(s)[cosh−1(x
1/r
+ + 1)] exists and

δ(s)[cosh−1(x
1/r
+ + 1)] =

M−1∑
k=0

kr+r∑
i=0

(
k

i

)
(−1)krcr,s,k
(kr + r)k!

δ(k)(x) (2.1)

for s = M − 1,M,M + 1, . . . and r = 1, 2, . . . , where

cr,s,k =

i∑
j=0

(
i

j

)
(−1)kr+r+s−i(2j − i)s+1

2i+1
,

M is the smallest integer for which s− 2r + 1 < 2Mr and r ≤ s/(2M + 2).
In particular, the neutrix composition δ[cosh−1(x+ + 1)] exists and

δ[cosh−1(x+ + 1)] = 0 (2.2)



164 FISHER, OZCAG AND AL-SIREHY

for r = 1, 2, . . . and the neutrix composition δ′[cosh−1(x+ + 1)] exists and

δ′[cosh−1(x+ + 1)] =
1

4
δ(x). (2.3)

Proof. To prove equation (1), we first of all have to evaluate∫ 1
−1
δ(s)n [cosh

−1(x
1/r
+ + 1)]x

k dx = ns+1
∫ 1
−1
ρ(s)[n cosh−1(x

1/r
+ + 1)]x

k dx

= ns+1
∫ 1
0

ρ(s)[n cosh−1(x1/r + 1)]xk dx

+ns+1
∫ 0
−1
ρ(s)(0)xk dx

= I1 + I2. (2.4)

It is obvious that

N−lim
n→∞

I2 = N−lim
n→∞

ns+1
∫ 0
−1
ρ(s)(0)xk dx = 0, (2.5)

for k = 0, 1, 2, . . . .
Making the substitution t = n cosh−1(x1/r + 1), we have for large enough n

I1 = rn
s

∫ 1
0

[cosh(t/n) − 1]kr+r−1 sinh(t/n)ρ(s)(t) dt

= −
rns+1

kr + r

∫ 1
0

[cosh(t/n) − 1]kr+rρ(s+1)(t) dt

= −
rns+1

kr + r

kr+r∑
i=0

(
kr + r

i

)
(−1)kr+r−i

∫ 1
0

coshi(t/n)ρ(s+1)(t) dt

= −
rns+1

kr + r

kr+r∑
i=0

(
kr + r

i

) i∑
j=0

(
i

j

)
(−1)kr+r−i

2i

∫ 1
0

exp[(2j − i)t/n]ρ(s+1)(t) dt

= −
rns+1

kr + r

kr+r∑
i=0

(
kr + r

i

) i∑
j=0

(
i

j

) ∞∑
m=0

(−1)kr+r−i(2j − i)m

2im!nm

∫ 1
0

tmρ(s+1)(t) dt.

It follows that

N−lim
n→∞

I1 =
r

kr + r

kr+r∑
i=0

(
kr + r

i

) i∑
j=0

(
i

j

)
(−1)kr+r+s−i(2j − i)s+1

2i(s + 1)!

∫ 1
0

ts+1ρ(s+1)(t) dt

=
r

kr + r

kr+r∑
i=0

(
kr + r

i

) i∑
j=0

(
i

j

)
(−1)kr+r+s−i(2j − i)s+1

2i+1

=
r

kr + r

kr+r∑
i=0

(
kr + r

i

)
cr,s,k, (2.6)

for k = 0, 1, 2, . . . .
When k = M, we have

|I1| ≤
rns+1

Mr + r

∫ 1
0

∣∣∣[cosh(t/n) − 1]Mr+rρ(s+1)(t)∣∣∣dt
≤ rns+1

∫ 1
0

[(t/n)2 + O(n−4)]Mr+r|ρ(s+1)(t)|dt

≤ rns−2Mr−2r+1
∫ 1
0

[1 + O(n−4Mr−4r)]|ρ(s+1)(t)|dt

= O(ns−2Mr−2r+1).



COMPOSITION δ(s)(cosh−1(|x|1/r + 1)) 165

Thus, if ψ is an arbitrary continuous function, then

lim
n→∞

∫ 1
0

δ(s)n [cosh
−1(x

1/r
+ + 1)]x

Mψ(x) dx = 0, (2.7)

since s− 2Mr − 2r + 1 < 0.
We also have ∫ 0

−1
δ(s)n [cosh

−1(x
1/r
+ + 1)]ψ(x) dx = n

s+1

∫ 0
−1
ρ(s)(0)ψ(x) dx

and it follows that

N−lim
n→∞

∫ 0
−1
δ(s)n [(sinh

−1 x+)
1/r]ψ(x) dx = 0. (2.8)

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

ϕ(x) =

M−1∑
k=0

ϕ(k)(0)

k!
xk +

xM

M!
ϕ(M)(ξx),

where 0 < ξ < 1, and so

N−lim
n→∞

〈δ(s)n [cosh
−1(x

1/r
+ + 1)],ϕ(x)〉 = N−lim

n→∞

M−1∑
k=0

ϕ(k)(0)

k!

∫ 1
0

δ(s)n [cosh
−1(x

1/r
+ + 1)]x

k dx

+ N−lim
n→∞

M−1∑
k=0

ϕ(k)(0)

k!

∫ 0
−1
δ(s)n [cosh

−1(x
1/r
+ + 1)]x

k dx

+ lim
n→∞

1

M!

∫ 1
0

δ(s)n [cosh
−1(x

1/r
+ + 1)]x

Mϕ(M)(ξx) dx

+ lim
n→∞

1

M!

∫ 0
−1
δ(s)n [cosh

−1(x
1/r
+ + 1)]x

Mϕ(M)(ξx) dx

=

M−1∑
k=0

kr+r∑
i=0

(
k

i

)
rcr,s,kϕ

(k)(0)

(kr + r)k!
+ 0

=

M−1∑
k=0

kr+r∑
i=0

(
k

i

)
(−1)krcr,s,k
(kr + r)k!

〈δ(k)(x),ϕ(x)〉, (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 for x > 0 and so equation (1) holds for x > 0.

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

δ(s)n [cosh
−1(x

1/r
+ + 1)]ϕ(x) dx = n

s+1

∫ b
a

ρ(s)(0)ϕ(x) dx

and so

N−lim
n→∞

∫ b
a

δ(s)n [cosh
−1(x

1/r
+ + 1)]ϕ(x) dx = 0.

It follows that δ(s)[cosh−1(x
1/r
+ + 1)] = 0 on the interval (a,b). Since a and b are 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 for r = 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 composition δ(s)[cosh−1(x
1/r
− + 1)] exists and

δ(s)[cosh−1(x
1/r
− + 1)] =

M−1∑
k=0

kr+r∑
i=0

(
k

i

)
rcr,s,k

(kr + r)k!
δ(k)(x) (2.10)



166 FISHER, OZCAG AND AL-SIREHY

for s = M − 1,M,M + 1, . . . and r = 1, 2, . . . ,
In particular, the neutrix composition δ[cosh−1(x

1/r
− + 1)] exists and

δ[cosh−1(x
1/r
− + 1)] = 0 (2.11)

for r = 1, 2, . . . and the neutrix composition δ′[cosh−1(x
1/r
− + 1)] exists and

δ′[cosh−1(x
1/r
− + 1)] =

1

4
δ(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 composition δ(s)[cosh−1(|x|1/r + 1)] exists and

δ(s)[cosh−1(|x|1/r + 1)] =
M−1∑
k=0

kr+r∑
i=0

(
k

i

)
[1 + (−1)k]rcr,s,k

(kr + r)k!
δ(k)(x) (2.13)

for s = M − 1,M,M + 1, . . . and r = 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)

for r = 1, 2, . . . and the neutrix composition δ′[cosh−1(|x|1/r + 1)] exists and

δ′[cosh−1(|x|1/r + 1)] =
1

2
δ(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(x

1/r
− + 1)].

Equations (14) to (15) follow similarly.

Theorem 2.5. The neutrix composition δ(s)[cosh−1(x+ + 1)
1/r] exists and

δ(s)[cosh−1(x+ + 1)
1/r] =

M−1∑
k=0

kr+r∑
i=0

(
kr + r

i

)
(−1)kbr,s,k

k!
δ(k)(x) (2.16)

for s = M − 1,M,M + 1, . . . and r = 1, 2, . . . , where

br,s,k =

ri+r∑
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 which s + 1 < 2Mr and r ≤ (s + 1)/(2M).

Proof. To prove equation (16), we first of all have to evaluate∫ 1
−1
δ(s)n [cosh

−1(x+ + 1)
1/r]xk dx = ns+1

∫ 1
−1
ρ(s)[n cosh−1(x+ + 1)

1/r]xk dx

= ns+1
∫ 1
0

ρ(s)[n cosh−1(x+ + 1)
1/r]xk dx

+ns+1
∫ 0
−1
ρ(s)(0)xk dx

= J1 + J2. (2.17)

It is obvious that

N−lim
n→∞

J2 = N−lim
n→∞

ns+1
∫ 0
−1
ρ(s)(0)xk dx = 0, (2.18)

for k = 0, 1, 2, . . . .



COMPOSITION δ(s)(cosh−1(|x|1/r + 1)) 167

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

J1 = rn
s

∫ 1
0

[coshr(t/n) − 1]k coshr−1(t/n) sinh(t/n)ρ(s)(t) dt

= rns
k∑
i=0

(
k

i

)
(−1)k−i

∫ 1
0

coshri+r−1(t/n) sinh(t/n)ρ(s)(t) dt

= −rns+1
k∑
i=0

(
k

i

)
(−1)k−i

ri + r

∫ 1
0

coshri+r(t/n)ρ(s+1)(t) dt

= −rns+1
k∑
i=0

(
k

i

)ri+r∑
j=0

(
ri + r

j

)
(−1)k−i

2ri+r(ri + r)

∫ 1
0

exp[(2j −ri−r)t/n]ρ(s+1)(t) dt

= −rns+1
kr+r∑
i=0

(
kr + r

i

)ri+r∑
j=0

(
ri + r

j

) ∞∑
m=0

(−1)k−i(2j −ri−r)m

2ri+r(ri + r)m!nm

∫ 1
0

tmρ(s+1)(t) dt.

It follows that

N−lim
n→∞

J1 = −
kr+r∑
i=0

(
kr + r

i

)ri+r∑
j=0

(
ri + r

j

)
(−1)k−ir(2j −ri−r)s+1

2ri+r(ri + r)(s + 1)!

∫ 1
0

ts+1ρ(s+1)(t) dt

=

kr+r∑
i=0

(
kr + r

i

)ri+r∑
j=0

(
ri + r

j

)
(−1)s+k−ir(2j −ri−r)s+1

2ri+r+1(ri + r)

=

kr+r∑
i=0

(
kr + r

i

)
br,s,k, (2.19)

for k = 0, 1, 2, . . . .
When k = M, we have

|J1| ≤ rns
∫ 1
0

∣∣∣[coshr(t/n) − 1]M coshr−1(t/n) sinh(t/n)ρ(s)(t)∣∣∣ dt
≤ rns

∫ 1
0

∣∣∣[(t/n)2r + O(n−4r)]M coshr−1(t/n) sinh(t/n)ρ(s)(t)∣∣∣ dt
= O(ns−2Mr−1).

Thus, if ψ is an arbitrary continuous function, then

lim
n→∞

∫ 1
0

δ(s)n [cosh
−1(x+ + 1)

1/r]xMψ(x) dx = 0, (2.20)

since s− 2Mr − 1 < 0.
We also have ∫ 0

−1
δ(s)n [cosh

−1(x+ + 1)
1/r]ψ(x) dx = ns+1

∫ 0
−1
ρ(s)(0)ψ(x) dx

and it follows that

N−lim
n→∞

∫ 0
−1
δ(s)n [(sinh

−1 x+)
1/r]ψ(x) dx = 0. (2.21)

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

ϕ(x) =

M−1∑
k=0

ϕ(k)(0)

k!
xk +

xM

M!
ϕ(M)(ξx),



168 FISHER, OZCAG AND AL-SIREHY

where 0 < ξ < 1, and so

N−lim
n→∞

〈δ(s)n [cosh
−1(x+ + 1)

1/r],ϕ(x)〉 =

= N−lim
n→∞

M−1∑
k=0

ϕ(k)(0)

k!

∫ 1
0

δ(s)n [cosh
−1(x+ + 1)

1/r]xk dx

+ N−lim
n→∞

M−1∑
k=0

ϕ(k)(0)

k!

∫ 0
−1
δ(s)n [cosh

−1(x+ + 1)
1/r]xk dx

+ lim
n→∞

1

M!

∫ 1
0

δ(s)n [cosh
−1(x+ + 1)

1/r]xMϕ(M)(ξx) dx

+ lim
n→∞

1

M!

∫ 0
−1
δ(s)n [cosh

−1(x+ + 1)
1/r]xMϕ(M)(ξx) dx

=

M−1∑
k=0

kr+r∑
i=0

(
kr + r

i

)
br,s,kϕ

(k)(0)

k!
+ 0

=
M−1∑
k=0

kr+r∑
i=0

(
kr + r

i

)
(−1)kbr,s,k

k!
〈δ(k)(x),ϕ(x)〉, (2.22)

on using equations (17) to (22). This proves equation (16) on the interval (−1, 1).

Replacing x by −x in equation (16), we get

Corollary 2.3. The neutrix composition δ(s)[cosh−1(x− + 1)
1/r] exists and

δ(s)[cosh−1(x− + 1)
1/r] =

M−1∑
k=0

kr+r∑
i=0

(
kr + r

i

)
br,s,k
k!

δ(k)(x) (2.23)

for s = M − 1,M,M + 1, . . . and r = 1, 2, . . . ,

Corollary 2.4. The neutrix composition δ(s)[cosh−1(|x| + 1)1/r] exists and

δ(s)[cosh−1(|x| + 1)1/r] =
M−1∑
k=0

kr+r∑
i=0

(
kr + r

i

)
[1 + (−1)]kbr,s,k

k!
δ(k)(x) (2.24)

for s = M − 1,M,M + 1, . . . and r = 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 distribution δ(r)(f(x)), Rostock. Math. Kolloq., 23(1983), 73-80.

[5] B. Fisher, On defining the change of variable in distributions, Rostock. Math. Kolloq., 28(1985), 33-40.

[6] B. Fisher, The delta function and the composition of distributions, Dem. Math., 35(1)(2002), 117-123.
[7] B. Fisher, The composition and neutrix composition of distributions, in: Kenan Taş et al. (Eds.), Mathematical

Methods in Engineering, Springer, Dordrecht, 2007, pp. 59-69.
[8] B. Fisher and B. Jolevska-Tuneska, Two results on the composition of distributions, Thai. J. Math., 3(1)(2005),

17-26.

[9] B. Fisher and A. Kılıçman, On the composition and neutrix composition of the delta function and powers of the
inverse hyperbolic sine function, Integral Transforms Spec. Funct. 21(12)(2010), 935-944.

[10] H. Kleinert, A. Chervyakov, Rules for integrals over products of distributions from coordinate independence of path
integrals, Eur. Phys. J. C Part. Fields 19(4)(2001), 743-747.



COMPOSITION δ(s)(cosh−1(|x|1/r + 1)) 169

[11] B. Fisher, A. Kananthai, G. Sritanatana and K. Nonlaopon, The composition of the distributions xms− ln x− and

x
r−p/m
+ , Integral Transforms Spec. Funct., 16(1)(2005), 13-20.

[12] B. Fisher and E. Ozcag, Some results on the neutrix composition of the delta function, Filomat, 26(6)(2012),

1247-1256.

[13] B. Fisher and K. Taş, On the composition of the distributions x−1 ln |x| and xr+, Integral Transforms Spec. Funct.,
16(7)(2005), 533-543.

[14] S. Gasiorowics, Elementery Particle Physics, John Wiley and Sons, New York, 1966.

[15] I.M. Gel’fand and G.E. Shilov, Generalized Functions, Volume I, Academic Press, New York and London, 1st
edition, 1964.

[16] D. S. Jones, Hadamard’s Finite Part., Math. Methods Appl. Sci. 19(13)(1996), 1017-1052.

[17] E. L. Koh and C. K. Li, On distributions δk and (δ′)k, Math. Nachr., 157(1992), 243-248.
[18] H. Kou and B. Fisher, On Composition of Distributions, Publ. Math. Debrecen, 40(3-4)(1992), 279-290.

[19] L. Lazarova, B. Jolevska-Tuneska, I. Akturk and E. Ozcag Note on the Distribution Composition (x
µ
+)
λ. Bull.

Malaysian Math. Soc., (2016). doi:10.1007/s40840-016-0342-2.

[20] C. K. Li and C. Li, On defining the distributions δk and (δ′)k by fractional derivatives, Appl. Math. Compt.,
246(2014), 502-513.

[21] E. Ozcag, Defining the k-th powers of the Dirac-delta distribution for negative integers, Appl. Math. Letters,

14(2001), 419-423.

[22] E. Ozcag, U. Gulen and B. Fisher, On the distribution δk+, Integral Transforms Spec. Funct., 9(2000), 57-64.

[23] E. Ozcag, L. Lazarova and B. Jolevska-Tuneska, Defining Compositions of x
µ
+, |x|

µ,x−s and x−s ln |x| as Neutrix
Limit of Regular Sequences, Commun. Math. Stat., 4(1)(2016), 63-80.

[24] G. Temple, The Theory of generalized Functions, Proc. Roy. Soc. ser. A 28(1955), 175-190.

1Department of Mathematics and Computer Science, Leicester University, England

2Department of Mathematics, Hacettepe University, Ankara, Turkey

3Department of Mathematics, Jeddah, King Abdulaziz University, Jeddah, Saudi Arabia

∗Corresponding author: ozcag1@hacettepe.edu.tr


	1. Introduction
	2.  Main Results
	References