Further Results on the Dilogarithm Integral
The dilogarithm integral Li() and its associated functions and are defined as locally summable functions on the real line. Some convolutions and neutrix convolutions of these functions and other functions are then found.
The dilogarithm integral is defined by (see ). More generally, we have for .
The associated functions and are defined by where denotes Heaviside's function.
Next, we define the distribution by and its associated distributions and are defined by
The classical definition of the convolution product of two functions and is as follows.
Definition 1.1. Let and be functions. Then the convolution is defined by for all points for which the integral exist.
It follows easily from the definition that if exists then exists and and if and exists, then Definition 1.1 can be extended to define the convolution of two distributions and in with the following definition; see Gel'fand and Shilov .
Definition 1.2. Let and be distributions in . Then the convolution is defined by the equation for arbitrary in , provided and satisfy either of the following conditions:(a)either or has bounded support,(b)the supports of and are bounded on the same side.
The function is then defined by for .
The following definition of the noncommutative neutrix convolution was given in .
Definition 1.3. Let and be distributions in , and let for . Then the noncommutative neutrix convolution is defined as the neutrix limit of the sequence , provided the limit exists in the sense that
for all in , where is the neutrix, see van der Corput , having domain the positive reals and range the real numbers, with negligible functions finite linear sums of the functions
and all functions which converge to zero in the normal sense as tends to infinity.
In particular, if exists, we say that the non-commutative convolution 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 the neutrix convolution is in general non-commutative.
The following results proved in  hold, first showing that the neutrix convolution is a generalization of the convolution.
Theorem 1.4. Let and be distributions in , satisfying either condition (a) or condition (b) of Gel'fand and Shilov's definition. Then the neutrix convolution exists and
Theorem 1.5. Let and be distributions in and suppose that the neutrix convolution exists. Then the neutrix convolution exists and
Note however that is not necessarily equal to but we do have the following theorem.
Theorem 1.6. Let and be distributions in and suppose that the neutrix convolution exists. If exists and equals for all in , then exists and
2. Main Result
We define the function by for and . In particular, we define the function by for .
The following theorem was proved in .
Theorem 2.1. The convolutions for and for .
We now prove the following generalization of Theorem 2.1.
Theorem 2.2. The convolutions and exist and for , and for .
Corollary 2.3. The convolutions and exist and for , and for .
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  and to prove it, our set of negligible functions was extended to include finite linear sums of the functions for and .
Theorem 2.4. The convolution exists and for .
Theorem 2.5. The convolution exists and for .
Before proving some further results, we need the following lemma.
Lemma 2.6. If for , then
Proof. Because when , we have and (2.13) follows.
Theorem 2.7. The neutrix convolution exists when and for .
Proof. We put . Then the convolution exists by Definition 1.1 and
Thus, using Lemma 2.6, we have
Further, it is easily seen that and (2.16) follows from (2.17), (2.19), and (2.20), proving the theorem.
Theorem 2.8. The neutrix convolution exists when and for .
Proof. Using Theorems 1.5 and 1.6, we have
where, on integration by parts, we have
It is clear that It now follows from (2.23) and (2.24) that Equation (2.21) now follows directly from (2.16) and (2.22), proving the theorem.
Corollary 2.9. The neutrix convolutions and exist when and for
Corollary 2.10. The neutrix convolutions and exist when and for .
Proof. Since the neutrix convolution product is distributive with respect to addition, we have 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 and exist when and for
This research was supported by FEIT, University of Ss. Cyril and Methodius in Skopje, Republic of Macedonia, Project no. 08-3619/8.
M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, Dover, New York, NY, USA, 9th edition, 1972.
I. M. Gel'fand and G. E. Shilov, Generalized Functions, vol. 1, chapter 1, Academic Press, New York, NY, USA, 1964.
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.View at: Google Scholar
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.View at: Google Scholar