Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2011, Article ID 612353, 12 pages
doi:10.1155/2011/612353
Research Article
On the Neutrix Composition of the Delta and
Inverse Hyperbolic Sine Functions
Brian Fisher1 and Adem Kılıçman2
1
2
Department of Mathematics, University of Leicester, Leicester LE1 7RH, UK
Department of Mathematics and Institute for Mathematical Research, University Putra Malaysia,
43400 UPM, Serdang, Selangor, Malaysia
Correspondence should be addressed to Adem Kılıçman, akilicman@putra.upm.edu.my
Received 24 December 2010; Accepted 8 April 2011
Academic Editor: A. A. Soliman
Copyright q 2011 B. Fisher and A. Kılıçman. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
Let F be a distribution in D and let f be a locally summable function. The composition Ff x of F
and f is said to exist and be equal to the distribution hx if the limit of the sequence {Fn f x} is
equal to hx, where Fn x Fx ∗ 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 δs sinh−1 x r does not exists. In this study, it is proved that the neutrix composition δs sinh−1 x r exists and
k k k
k1
k!δk x, for s 0, 1, 2, . . . and
is given by δs sinh−1 x r srr−1
i0 i −1 rcs,k,i /2
k0
r 1, 2, . . ., where cs,k,i −1s s!k − 2i 1rs−1 k − 2i − 1rsr−1 /2rs 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 Da, 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 ρx be a function in D−1, 1 having the following properties:
i ρx ≥ 0,
ii ρx ρ−x,
1
iii −1 ρx dx 1.
Putting δn x nρnx for n 1, 2, . . ., it follows that {δn x} is a regular sequence of
infinitely differentiable functions converging to the Dirac delta-function δx. Further, if F is
2
Journal of Applied Mathematics
an arbitrary distribution in D and Fn x Fx ∗ δn x Fx − t, ϕt, then {Fn x} is a
regular sequence converging to Fx.
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
Ffx 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 Ffx exists and is equal to h on the open interval a, b, with
−∞ < a < b < ∞, if
N − lim
∞
n→∞
−∞
Fn fx ϕxdx hx, ϕx ,
1.1
for all ϕ in Da, b, where Fn x Fx ∗ 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
nλ lnr−1 n,
lnr n : 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 Ffx exists and is equal to h on the open
interval a, b if
∞
lim
n→∞
−∞
Fn fx ϕxdx hx, ϕx ,
1.3
for all ϕ in Da, b.
Note that taking the neutrix limit of a function fn is equivalent to taking the
usual limit of Hadamard’s finite part of fn. 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 δs sgn x|x|λ exists and
δs sgn x|x|λ 0,
1.4
Journal of Applied Mathematics
3
for s 0, 1, 2, . . . and s 1λ 1, 3, . . ., and
−1s1λ1 s! s1λ−1
δ
x,
λs 1λ − 1!
δs sgn x|x|λ 1.5
for s 0, 1, 2, . . ., and s 1λ 2, 4, . . ..
Theorem 1.3. The neutrix compositions δ2s−1 sgn x|x|1/s and δs−1 |x|1/s exist and
δ2s−1 sgn x|x|1/s δ
s−1
|x|
1/s
1
2s!δ x,
2
s−1
−1
1.6
δx,
for s 1, 2, . . ..
Theorem 1.4. The neutrix composition δs sinh−1 x1/r exists and
δ
s
−1
sinh x
1/r
s1/r−1 k
k0
i0
k −1k rcs,k,i
i
2k1 k!
δk x,
1.7
for s 0, 1, 2, . . . and r 1, 2, . . ., where
cr,s,k,i −1s s! k − 2i 1rsr−1 k − 2i − 1rsr−1
2rs r − 1!
1.8
.
The next two theorems were proved in 13.
Theorem 1.5. The neutrix composition δs lnr 1 |x| exists and
srr−1
δs lnr 1 |x| k
k0 i0
s−i
k
rsr−1
k −1 1 −1 s!i 1
i
2rrs r − 1!k!
δk x.
1.9
for s 0, 1, 2, . . ., and r 1, 2, . . ..
In particular, the composition δln1 |x| exists and
δln1 |x| δx.
1.10
Theorem 1.6. The neutrix composition δs ln1 |x1/r | exists and
δs
m−1krr−1 kr r − 1 −1rsi−1 1 −1k ri 1s
δk x,
ln 1 x1/r 2k!
i
k0 i0
1.11
for s 0, 1, 2, . . . and r 2, 3, . . ., where m is the smallest non-negative integer greater than s−r 1r −1.
4
Journal of Applied Mathematics
In particular, the composition δs ln1 |x1/r | exists and
δs ln 1 x1/r 0,
1.12
for s 0, 1, 2, . . . , r − 2 and r 2, 3, . . . and
δr−1 ln 1 x1/r −1r−1 r!δx,
1.13
for r 2, 3, . . ..
2. Main Results
We now prove the following theorem.
r
Theorem 2.1. The neutrix composition δs sinh−1 x exists and
δ
s
r
−1
sinh x
srr−1 k
k −1k rcs,k,i
2k1 k!
i
k0 i0
δk x,
2.1
for s 0, 1, 2, . . . and r 1, 2, . . ., where
cr,s,k,i −1s s! k − 2i 1rsr−1 k − 2i − 1rsr−1
.
2rs r − 1!
2.2
In particular, the neutrix composition δsinh−1 x exists and
1
δx.
2
δ sinh−1 x 2.3
Proof. To prove 2.1, we first of all evaluate
1
s
−1
δn
r
sinh−1 x
2.4
xk dx.
We have
1
−1
s
δn
sinh−1 x
r
xk dx ns1
n
1
−1
1
s1
ρs
n sinh−1 x
0
I1 I2 .
xk dx
r
ρs n sinh−1 x xk dx
0
ns1
r
−1
ρs 0xk dx
2.5
Journal of Applied Mathematics
5
It is obvious that
N − lim I2 N − lim
n→∞
0
n→∞
−1
s
δn
sinh−1 x
r
xk dx 0,
2.6
for k 0, 1, 2, . . ..
r
Making the substitution t nsinh−1 x , we have for large enough n
I1 ns−r1
r
×
1
t1/r−1 sinhk
0
1/r
1/r
t
t
cosh
ρs tdt
n
n
t1/r−1
1/r 1/r t
t
exp k − 2i 1
exp k − 2i − 1
ρs tdt,
n
n
1
1
0
2.7
where
s−1/r1
n
t
1/r−1
0
∞
1/r 1/r t
t
exp k − 2i 1
exp k − 2i − 1
ρs tdt
n
n
1 k − 2i 1p k − 2i − 1p tp/r1/r−1
p0
p!np/r1/r−s−1
0
2.8
ρs tdt.
It follows that
N − lim n
n→∞
1
s−1/r1
t
1/r−1
0
1/r 1/r t
t
exp k − 2i 1
exp k − 2i − 1
ρs t dt
n
n
−1s s! k − 2i 1rsr−1 k − 2i − 1rsr−1
2rs r − 1!
cr,s,k,i,
2.9
and by applying the neutrix limit we obtain
N − limI1 N − lim
n→∞
for k 0, 1, 2, . . ..
n→∞
1
0
s
δn
−1
sinh x
k
x dx k1
−1i cr,s,k,i
2 r i0 i
r
k
1
k
2.10
6
Journal of Applied Mathematics
When k sr r, we have
|I1 | 1
s
sinh−1 x
δn
xsrr dx
r
0
ns1
1
r
s
ρn n sinh−1 x xsrr dx
0
ns−1/r1
expsr r 1
≤
2srr r
ns−1/r1
2srr r
1/r srr
1 t
s
ρ t dt
1 − exp −2
n
0
srr
1 1/r
t
s −2/r
2
expsr r 1
O n
ρ tdt
n
0
≤ n−1/r expsr r 1
2.11
1
1 O n−2/r ρs tdt
0
O n−1/r .
Thus, if ψ is an arbitrary continuous function, then
1
lim
n→∞
s
δn
sinh−1 x
r
xrsr ψxdx 0.
2.12
0
We also have
0
−1
s
δn
sinh−1 x
r
ψxdx ns1
0
−1
ρs 0ψxdx,
2.13
and it follows that
N − lim
n→∞
0
s
−1
δn
sinh−1 x
r
ψxdx 0.
2.14
If now ϕ is an arbitrary function in D−1, 1, then by Taylor’s Theorem, we have
ϕx srr−1
k0
ϕk 0 k
xrsr
x ϕrsr ξx,
k!
rs r!
2.15
Journal of Applied Mathematics
7
where 0 < ξ < 1, and so
s
N − lim δn
n→∞
srr−1
N − lim
n→∞
k0
ϕk 0
k!
n→∞
1
k0
1
1
lim
n → ∞ sr r!
sinh−1 x
0
−1
s
δn
xk dx
xsrr ϕsrr ξx dx
0
k −1k rcr,s,k,i k0 i0
r
r
sinh−1 x
0
1
s
sinh−1 x
δn
n → ∞ sr r! −1
srr−1 k
k rcr,s,k,iϕk 0
0
2k1 k!
i
k0 i0
xk dx
sinh−1 x
lim
srr−1 k
r
0
s
δn
s
δn
ϕk 0
k!
srr−1
N − lim
, ϕx
1/r
sinh−1 x
2k1 k!
i
2.16
r
xsrr ϕsrr ξx dx
δk x, ϕx ,
on using 2.3 to 2.14. This proves 2.1 on the interval −1, 1.
r
It is clear that δs sinh−1 x 0 for x > 0 and so 2.1 holds for x > −1.
Now, suppose that ϕ is an arbitrary function in Da, b, where a < b < 0. Then,
b
s
δn
−1
sinh x
r
ϕx dx n
b
s1
a
ρs 0ϕxdx
2.17
a
and so
N − lim
b
n→∞
s
δn
sinh−1 x
r
2.18
ϕxdx 0.
a
r
It follows that δs sinh−1 x 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.
r
Corollary 2.2. The neutrix composition δs sinh−1 |x| exists and
δs
sinh−1 |x|
r
srr−1 k
k0 i0
for s 0, 1, 2, . . . and r 1, 2, . . ..
k
k −1 1 cr,s,k,i
i
2k1 k!
δk x,
2.19
8
Journal of Applied Mathematics
In particular, the composition δsinh−1 |x| exists and
δ sinh−1 |x| 1
δx.
2
2.20
Proof. To prove 2.19, we note that
1
−1
s
δn
sinh−1 |x|
r
xk dx ns1
1
−1
ρs
n sinh−1 |x|
r
xk dx
1
r
ρs n sinh−1 x xk dx,
ns1 1 −1k
2.21
0
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 δ2s−1 sinh−1 sgn x · x2 exists and
δ
2s−1
2s−1ik1 k
−1k bs,k,i k
−1
2
sinh sgn x · x
δ x,
i 2k1 2k 1!
k0 i0
2.22
for s 1, 2, . . ., where
2.23
bs,k,i k − 2i 12s−1 k − 2i − 12s−1 .
Proof. To prove 2.22, we now have to evaluate
1
−1
2s−1
δn
sinh−1 sgn x · x2 xk dx.
2.24
We have
1
−1
2s−1
δn
1
ρ2s−1 n sinh−1 sgn x · x2 xk dx
sinh−1 sgn x · x2 xk dx n2s
−1
⎧
1
⎪
⎪
⎨2n2s ρ2s−1 n sinh−1 x2 xk dx,
⎪
⎪
⎩0,
k odd,
0
k even.
2.25
Journal of Applied Mathematics
9
Making the substitution t nsinh−1 x2 , we have for large enough n
1
−1
2s−1
δn
2n
sinh−1 sgn x · x2 xk dx
1
2s
ρ2s−1 n sinh−1 x2 x2k1 dx
2.26
0
n2s−1
2k1
k
k −1i
i
i0
1#
exp
0
k − 2i − 1t
k − 2i 1t
exp
n
n
$
ρ2s−1 tdt,
where
1#
n2s−1
exp
0
∞
k − 2i − 1t
k − 2i 1t
exp
n
n
1 k − 2i 1p k − 2i − 1p tp
p0
p!np−2s1
0
$
ρs tdt
2.27
ρ2s−1 tdt.
It follows that
N − lim n2s−1
n→∞
1#
$
k − 2i − 1t
k − 2i 1t
exp
ρs t dt
exp
n
n
0
N − lim
n→∞
∞
1 p0
0
k − 2i 1p k − 2i − 1p tp 2s−1
ρ
t dt
p!np−2s1
−k − 2i 12s−1 k − 2i − 12s−1
2
bs,k,i
,
2
2.28
and so by using the neutrix limit, we have
N − lim
n→∞
for k 0, 1, 2, . . ..
1
−1
2s−1
δn
sinh−1 sgn x · x2 x2k1 dx k
i0
k −1i1 bs,k,i
i
2k1
,
2.29
10
Journal of Applied Mathematics
When k 2s, we have
1
1 2s−1
sinh−1 sgn x · x2 x4s1 dx n2s
ρ2s−1 n sinh−1 x2 x4s1 dx
δn
−1
−1
1 2t 2s 2s−1 ρ
t dt
1 − exp −
n
−1 1 2s
2t
O n−2
exps 1
ρ2s−1 tdt
n
−1
n2s−1
≤ s−1 exps 1
2
n2s−1
2s−1
≤2
1 n exps 1
1 O n−2/r ρ2s−1 t dt
2s1 −1
−1
O n−1 .
2.30
Thus, if ψ is an arbitrary continuous function, then
1
lim
n→∞
−1
2s−1
δn
sinh−1 sgn x · x2 x4s1 ψxdx 0.
2.31
If now ϕ is an arbitrary function in D−1, 1, then by Taylor’s Theorem, we have
ϕx ϕk 0 k
x4s1
x ϕ4s1 ξx,
k!
1!
4s
k0
4s
2.32
where 0 < ξ < 1, and so
2s−1
N − lim δn
sinh−1 sgn x · x2 , ϕx
n→∞
ϕ2k1 0
N − lim
n→∞
2k 1!
k0
2s−1
1
lim
n → ∞ 4s 1!
2s−1 k
k0 i0
1
−1
1
−1
4s1
δn
2s−1
δn
sinh−1 sgn x · x2 x4s1 ϕ4s1 ξxdx
k −1i1 bs,k,i ϕk 0
i
sinh−1 sgn x · x2 x2k1 dx
2k1 2k 1!
2.33
0
k −1ik1 bs,k,i k
,
δ
ϕx
x,
i 2k1 2k 1!
k0 i0
2s−1 k
on using 2.25 to 2.31, proving 2.22 on the interval −1, 1. However, it is clear that
2s−1
δn
sinh−1 sgnx · x2 0 for |x| > 0 and so 2.22 holds on the real line, completing
the proof of the theorem.
Journal of Applied Mathematics
11
Corollary 2.4. The composition δ sinh−1 sgn x · x2 exists and
δ x
− 2δx.
δ sinh−1 sgn x · x2 4.3!
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 referees 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.
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μ
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http://www.hindawi.com
Volume 2014
Optimization
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014