Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2011, Article ID 421601, 10 pages
doi:10.1155/2011/421601
Research Article
Further Results on the Dilogarithm Integral
Biljana Jolevska-Tuneska1 and Brian Fisher2
1
Faculty of Electrical Engineering and Informational Technologies,
Ss. Cyril and Methodius University in Skopje, Karpos II bb, 1000 Skopje, Macedonia
2
Department of Mathematics, University of Leicester, Leicester LE1 7RH, England, UK
Correspondence should be addressed to Biljana Jolevska-Tuneska, biljanaj@feit.ukim.edu.mk
Received 5 May 2011; Accepted 4 July 2011
Academic Editor: Ch Tsitouras
Copyright q 2011 B. Jolevska-Tuneska and B. Fisher. 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.
The dilogarithm integral Lixs and its associated functions Li xs and Li− xs are defined as
locally summable functions on the real line. Some convolutions and neutrix convolutions of these
functions and other functions are then found.
1. Introduction
The dilogarithm integral Lix is defined by
Lix −
x
0
ln|1 − t|
dt
t
1.1
see 1. More generally, we have
Lix −
s
xs
0
ln|1 − t|
dt −s
t
x
0
ln|1 − ts |
dt
t
1.2
for s 1, 2, . . . .
The associated functions Li xs and Li− xs are defined by
Li xs HxLixs ,
Li− xs H−xLixs Lixs − Li xs ,
where Hx denotes Heaviside’s function.
1.3
2
Journal of Applied Mathematics
Next, we define the distribution ln |1 − xs |x−1 by
ln |1 − xs |x−1 −s−1 Lixs ,
1.4
and its associated distributions ln |1 − xs |x−1 and ln |1 − xs |x−−1 are defined by
ln|1 − xs |x−1 Hx ln|1 − xs |x−1 −s−1 Li xs ,
ln|1 − xs |x−−1 H−x ln|1 − xs |x−1 −s−1 Li− x .
1.5
The classical definition of the convolution product of two functions f and g is as
follows.
Definition 1.1. Let f and g be functions. Then the convolution f ∗ g is defined by
f ∗ g x ∞
−∞
ftgx − tdt
1.6
for all points x for which the integral exist.
It follows easily from the definition that if f ∗ g exists then g ∗ f exists and
f ∗ g g ∗ f,
1.7
and if f ∗ g and f ∗ g or f ∗ g exists, then
f ∗g
f ∗ g
or f ∗ g .
1.8
Definition 1.1 can be extended to define the convolution f ∗ g of two distributions f and g in
D with the following definition; see Gel’fand and Shilov 2.
Definition 1.2. Let f and g be distributions in D . Then the convolution f ∗ g is defined by the
equation
f ∗ g x, ϕ f y , gx, ϕ x y
1.9
for arbitrary ϕ in D, provided f and g satisfy either of the following conditions:
a either f or g has bounded support,
b the supports of f and g are bounded on the same side.
It follows that if the convolution f ∗ g exists by this definition then 1.7 and 1.8 are
satisfied.
Journal of Applied Mathematics
3
In order to extend Definition 1.2 to distributions which do not satisfy conditions a or
b, let τ be a function in D, see 3, satisfying the conditions:
i τx τ−x,
ii 0 ≤ τx ≤ 1,
iii τx 1 for |x| ≤ 1/2,
iv τx 0 for |x| ≥ 1.
The function τn is then defined by
⎧
⎪
1,
⎪
⎪
⎨
τn x τ nn x − nn1 ,
⎪
⎪
⎪
⎩ n
τ n x nn1 ,
|x| ≤ n,
x > n,
1.10
x < −n,
for n 1, 2, . . . .
The following definition of the noncommutative neutrix convolution was given in 4.
Definition 1.3. Let f and g be distributions in D , and let fn fτn for n 1, 2, . . . . Then the
noncommutative neutrix convolution f g is defined as the neutrix limit of the sequence {fn ∗ g},
provided the limit h exists in the sense that
N − lim fn ∗ g, ϕ h, ϕ
n→∞
1.11
for all ϕ in D, where N is the neutrix, see van der Corput [5], having domain N the positive reals and
range N the real numbers, with negligible functions finite linear sums of the functions
nλ lnr−1 n,
lnr n :
λ > 0, r 1, 2, . . .
1.12
and all functions which converge to zero in the normal sense as n tends to infinity.
In particular, if
lim fn ∗ g, ϕ h, ϕ
n→∞
1.13
exists, we say that the non-commutative convolution f g exists.
It is easily seen that any results proved with the original definition of the convolution
hold with the new definition of the neutrix convolution. Note also that because of the lack of
symmetry in the definition of f g the neutrix convolution is in general non-commutative.
The following results proved in 4 hold, first showing that the neutrix convolution is
a generalization of the convolution.
Theorem 1.4. Let f and g be distributions in D , satisfying either condition (a) or condition (b) of
Gel’fand and Shilov’s definition. Then the neutrix convolution f g exists and
f g f ∗ g.
1.14
4
Journal of Applied Mathematics
Theorem 1.5. Let f and g be distributions in D and suppose that the neutrix convolution f g
exists. Then the neutrix convolution f g exists and
f g
f g.
1.15
Note however that f g is not necessarily equal to f g but we do have the
following theorem.
Theorem 1.6. Let f and g be distributions in D and suppose that the neutrix convolution f g
exists. If N − limn → ∞ fτn ∗ g, ϕ exists and equals h, ϕ for all ϕ in D, then f g exists and
f g
f g h.
1.16
2. Main Result
We define the function Is,r x by
Is,r x x
ur ln|1 − us | du
2.1
0
for r 0, 1, 2, . . . and s 1, 2, . . . . In particular, we define the function Ir x by
Ir x I1,r x
2.2
for r 0, 1, 2, . . . .
The following theorem was proved in 6.
Theorem 2.1. The convolutions Li x ∗ xr and ln |1 − x|x−1 ∗ xr exist and
Li x ∗
xr
r
r1
1 1 r1
x Li x
−1r−i Ir−i xxi r 1 i0
r
1 i
2.3
for r 0, 1, 2, . . . and
ln|1 −
x|x−1
∗
xr
r−1 r
i0
i
−1r−i Ir−i−1 xxi − Li xxr
2.4
for r 1, 2, . . . .
We now prove the following generalization of Theorem 2.1.
Theorem 2.2. The convolutions Li xs ∗ xr and ln|1 − xs |x−1 ∗ xr exist and
Li x ∗
s
xr
r
r1
s 1
Li xs xr1
−1r−i Is,r−i xxi r 1 i0
r1
i
2.5
Journal of Applied Mathematics
5
for r 0, 1, 2, . . . , s 1, 2, . . . and
ln|1 − x
s
|x−1
∗
xr
r−1 r
1
−1r−i Is,r−i−1 xxi − Li xs xr
s
i
i0
2.6
for r, s 1, 2, . . . .
Proof. It is obvious that Li xs ∗ xr 0 if x < 0.
When x > 0, we have
Li x ∗
s
xr
−s
x
x − t
r
0
−s
x
t
u−1 ln|1 − us |du dt
0
u−1 ln|1 − us |
0
x
x − tr dt du
u
x
r1
r1
s r−i i
ur−i ln|1 − us |du
−1 x
r 1 i0
i
0
r
r1
1 r1
s x Lixs −1r−i xi Is,r−i x r 1 i0
r
1
i
2.7
proving 2.5.
Next, using 1.8 and 2.5, we have
−s ln|1 − xs |x−1 ∗ xr rLi xs ∗ xr−1
r−1 r
s
−1r−i−1 Is,r−i−1 xxi Li xs xr ,
i
i0
2.8
and 2.6 follows.
Corollary 2.3. The convolutions Li− xs ∗ x−r and ln|1 − xs |x−−1 ∗ x−r exist and
Li− x ∗
s
x−r
r
r1
s 1
Li− xs x−r1
−1r−i1 Is,r−i xx−i r 1 i0
r
1
i
2.9
for r 0, 1, 2, . . . , s 1, 2, . . . and
ln|1 − x
s
|x−−1
∗
x−r
r−1 r
i0
for r, s 1, 2, . . . .
1
−1r−i1 Is,r−i−1 xx−i − Li− xs x−r
s
i
2.10
6
Journal of Applied Mathematics
Proof. Equations 2.9 and 2.10 are obtained applying a similar procedure as used in
obtaining 2.5 and 2.6.
The next two theorems were proved in 6 and to prove it, our set of negligible
functions was extended to include finite linear sums of the functions ns Linr for s 0, 1, 2, . . .
and r 1, 2, . . . .
Theorem 2.4. The convolution Li x xr exists and
r
r1
1 −1r−i
xi
Li x x r 1 i0
i
r − i 12
r
2.11
for r 0, 1, 2, . . . .
Theorem 2.5. The convolution ln|1 − x|x−1 xr exists and
ln|1 −
x|x−1
x r
r−1 r
−1r−i1
i0
i
r − i2
xi
2.12
for r 1, 2, . . . .
Before proving some further results, we need the following lemma.
Lemma 2.6. If r 1/s ∈ N for r, s 1, 2, . . . , then
N − limIs,r n −
n→∞
s
r 12
.
2.13
Proof. Because
Is,r n ns
1 − tr1/s
1 r1
1
n − 1 ln|1 − ns | −
dt
r1
r1 0
1−t
2.14
when r 1/s ∈ N, we have
Is,r n r1/s−1
nsis
1 r1
1
n − 1 ln|1 − ns | −
r1
r 1 i0 i 1
r1/s−1
nsis
1 r1
1
n − 1 s ln n ln1 − n−s −
,
r1
r 1 i0 i 1
and 2.13 follows.
We now prove the following generalization of Theorems 2.4 and 2.5.
2.15
Journal of Applied Mathematics
7
Theorem 2.7. The neutrix convolution Li xs xr exists when r 1/s ∈ N and
r
r1
s2 −1r−i
Li x x xi
r 1 i0
i
r − i 12
s
r
2.16
for r, s 1, 2, . . . .
Proof. We put Li xs n Li xs τn x. Then the convolution Li xs n ∗ xr exists by
Definition 1.1 and
Li x n ∗ x s
r
n
r
Lit x − t dt s
nn−n
τn tLits x − tr dt
n
2.17
ln|1 − us |
du dt
u
0
0
n
ln|1 − us | n
−s
x − tr dt du
u
0
u
n
r1
r1
s r−i i
u−1 ln|1 − us | ur−i1 − nr−i1 du
−1 x
r 1 i0
i
0
r
r1
s 1 r1
x Lins −1r−i xi Is,r−i n r 1 i0
r
1
i
2.18
0
I1 I2 ,
where
I1 n
0
−s
Lits x − tr dt
n
x − t
r
t
r
r1
1 −1r−i xi Lins nr−i1 .
r 1 i0
i
Thus, using Lemma 2.6, we have
r
r 1 −1r−i1 i
s2 N − limI1 x.
n→∞
r 1 i0
i
r − i 12
2.19
Further, it is easily seen that
nn−n
lim
n→∞
τn tLits x − tr dt 0,
n
and 2.16 follows from 2.17, 2.19, and 2.20, proving the theorem.
2.20
8
Journal of Applied Mathematics
Theorem 2.8. The neutrix convolution ln|1 − xs |x−1 xr exists when r 1/s ∈ N and
ln|1 − x
s
|x−1
r−1 r
−1r−i i
x s
x
i r − i2
i0
r
2.21
for r, s 1, 2, . . . .
Proof. Using Theorems 1.5 and 1.6, we have
−s ln|1 − xs |x−1 xr rLi xs xr−1 N − lim Li xs τn x ∗ xr ,
n→∞
2.22
where, on integration by parts, we have
Li xs τn x ∗ xr nn−n
n
τn tLits x − tr dt
r
−Lin x − n − s
s
nn−n
ln|1 − ts |t−1 x − tr τn tdt
2.23
n
r
nn−n
Lits x − tr−1 τn tdt.
n
It is clear that
nn−n
lim
n→∞
lim
ln|1 − ts |t−1 x − tr τn tdt 0
n
nn−n
n→∞ n
Lit x − t
s
r−1
2.24
τn tdt 0 .
It now follows from 2.23 and 2.24 that
N − lim Li xs τn x ∗ xr 0.
n→∞
2.25
Equation 2.21 now follows directly from 2.16 and 2.22, proving the theorem.
Corollary 2.9. The neutrix convolutions Li− xs xr and ln |1 − xs |x−−1 xr exist when r 1/s ∈ N
and
r
r 1 −1r−i1 i
s2 x− ,
Li− x x r 1 i0
i
r − i 12
r−1
r −1r−i1 i
x−
ln|1 − xs |x−−1 xr s
i0
i r − i2
s
for r, s 1, 2, . . .
r
2.26
Journal of Applied Mathematics
9
Proof. Equations 2.26 are obtained applying a similar procedure as used in obtaining 2.16
and 2.21.
Corollary 2.10. The neutrix convolutions Li xs x−r and Li− xs xr exist when r 1/s ∈ N
and
Li x s
x−r
r
r1
s −1i s − r − i 12 Is,r−i x xi
2
r 1 i0
i
r − i 1
−1r1
Li xs xr1 ,
r1
r
r1
s −1i1 2
s
r
x−i
Li− x x s
−
−
i
1
I
r
x
s,r−i
r 1 i0
i
r − i 12
2.27
−1r1
Li− xs xr1
r1
for r, s 1, 2, . . . .
Proof. Since the neutrix convolution product is distributive with respect to addition, we have
Li xs xr Li xs ∗ xr −1r Li xs x−r ,
2.28
and 2.27 follows from 2.16 and 2.5. Equation 27 is obtained applying similar procedure
as in the case of 2.27.
Corollary 2.11. The neutrix convolutions ln|1 − xs |x−1 x−r and ln|1 − xs |x−−1 xr exist when
r 1/s ∈ N and
ln|1 − x
s
|x−1
x−r
r−1 r
−1i i0
i
r − i2
s − r − i2 Is,r−i−1 x xi
−1r
Li xs xr ,
s
r−1 r
−1i1 2
s −1
r
x−i
ln|1 − x |x− x s
−
−
i
I
r
x
s,r−i−1
2
i
−
i
r
i0
2.29
−1r
Li− xs xr
s
for r, s 1, 2, . . .
Proof. Equation 2.29 follows from 2.21 and 2.6. Equation 29 is obtained applying
similar procedure as in the case of 2.29.
10
Journal of Applied Mathematics
Acknowledgment
This research was supported by FEIT, University of Ss. Cyril and Methodius in Skopje,
Republic of Macedonia, Project no. 08-3619/8.
References
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Mathematical Tables, Dover, New York, NY, USA, 9th edition, 1972.
2 I. M. Gel’fand and G. E. Shilov, Generalized Functions, vol. 1, chapter 1, Academic Press, New York, NY,
USA, 1964.
3 D. S. Jones, “The convolution of generalized functions,” The Quarterly Journal of Mathematics, vol. 24,
pp. 145–163, 1973.
4 B. Fisher, “Neutrices and the convolution of distributions,” Univerzitet u Novom Sadu. Zbornik Radova
Prirodno-Matematičkog Fakulteta. Serija za Matemati, vol. 17, no. 1, pp. 119–135, 1987.
5 J. G. van der Corput, “Introduction to the neutrix calculus,” Journal d’Analyse Mathématique, vol. 7, pp.
291–398, 1959.
6 B. Jolevska-Tuneska, B. Fisher, and E. Özçağ, “On the dilogarithm integral,” International Journal of
Applied Mathematics, vol. 24, no. 3, pp. 361–369, 2011.
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