Further Results on the Dilogarithm Integral

Abstract

The dilogarithm integral Li(𝑥^𝑠) and its associated functions Li₊(𝑥^𝑠) and Li₋(𝑥^𝑠) 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 Li(𝑥) is defined byLi(𝑥)=−^𝑥₀ln|1−𝑡|𝑡𝑑𝑡 (1.1) (see [1]). More generally, we have Li𝑥^𝑠=−^𝑥𝑠₀ln|1−𝑡|𝑡𝑑𝑡=−𝑠^𝑥₀ln1−𝑡^𝑠𝑡𝑑𝑡 (1.2) for 𝑠=1,2,….

The associated functions Li₊(𝑥^𝑠) and Li₋(𝑥^𝑠) are defined by Li₊𝑥^𝑠=𝐻(𝑥)Li𝑥^𝑠,Li₋𝑥^𝑠=𝐻(−𝑥)Li𝑥^𝑠=Li𝑥^𝑠−Li₊𝑥^𝑠, (1.3) where 𝐻(𝑥) denotes Heaviside's function.

Next, we define the distribution ln|1−𝑥^𝑠|𝑥⁻¹ by ln1−𝑥^𝑠𝑥⁻¹=−𝑠⁻¹Li𝑥^𝑠′, (1.4) and its associated distributions ln|1−𝑥^𝑠|𝑥₊⁻¹ and ln|1−𝑥^𝑠|𝑥₋⁻¹ are defined by ln1−𝑥^𝑠𝑥₊⁻¹=𝐻(𝑥)ln1−𝑥^𝑠𝑥⁻¹=−𝑠⁻¹Li₊𝑥^𝑠′,ln1−𝑥^𝑠𝑥₋⁻¹=𝐻(−𝑥)ln1−𝑥^𝑠𝑥⁻¹=−𝑠⁻¹Li₋(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)=^∞_−∞𝑓(𝑡)𝑔(𝑥−𝑡)𝑑𝑡 (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′(or𝑓^′∗𝑔) exists, then 𝑓∗𝑔′=𝑓∗𝑔^′or𝑓^′∗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=𝑓𝑦,𝑔(𝑥),𝜑𝑥+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.

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) 𝜏(𝑥)=𝜏(−𝑥),

(ii) 0≤𝜏(𝑥)≤1,

(iii) 𝜏(𝑥)=1for|𝑥|≤1/2,

(iv) 𝜏(𝑥)=0for|𝑥|≥1.

The function 𝜏_𝑛 is then defined by 𝜏_𝑛(𝑥)=|𝑥|𝜏𝑛1,≤𝑛,_𝑛𝑥−𝑛^𝑛+1𝜏𝑛,𝑥>𝑛,_𝑛𝑥+𝑛^𝑛+1,𝑥<−𝑛, (1.10) 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 𝑓_𝑛=𝑓𝜏_𝑛 for 𝑛=1,2,…. Then the noncommutative neutrix convolution 𝑓⊛𝑔 is defined as the neutrix limit of the sequence {𝑓_𝑛∗𝑔}, provided the limit ℎ exists in the sense that 𝑁−lim_𝑛→∞𝑓_𝑛∗𝑔,𝜑=ℎ,𝜑 (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 𝑛^𝜆ln^𝑟−1𝑛,ln^𝑟𝑛∶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_𝑛→∞𝑓_𝑛∗𝑔,𝜑=ℎ,𝜑 (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)

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−lim_𝑛→∞⟨(𝑓𝜏^′_𝑛)∗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 𝐼_𝑠,𝑟(𝑥) by 𝐼_𝑠,𝑟(𝑥)=^𝑥₀𝑢^𝑟ln1−𝑢^𝑠𝑑𝑢 (2.1) for 𝑟=0,1,2,… and 𝑠=1,2,…. In particular, we define the function 𝐼_𝑟(𝑥) by 𝐼_𝑟(𝑥)=𝐼_1,𝑟(𝑥) (2.2) for 𝑟=0,1,2,….

The following theorem was proved in [6].

Theorem 2.1. The convolutions Li₊(𝑥)∗𝑥^𝑟₊𝑎𝑛𝑑ln|1−𝑥|𝑥₊⁻¹∗𝑥^𝑟₊𝑒𝑥𝑖𝑠𝑡𝑎𝑛𝑑Li₊(𝑥)∗𝑥^𝑟₊=1𝑟+1^𝑟_𝑖=0𝑖()𝑟+1−1^𝑟−𝑖𝐼_𝑟−𝑖(𝑥)𝑥^𝑖₊+1𝑥𝑟+1^+𝑟+1Li₊(x) (2.3) for 𝑟=0,1,2,… and ln|1−𝑥|𝑥₊⁻¹∗𝑥^𝑟₊=^𝑟−1_𝑖=0ri(−1)^𝑟−𝑖𝐼_{𝑟−𝑖−1}(𝑥)𝑥^𝑖₊−Li₊(𝑥)𝑥^𝑟₊ (2.4) for 𝑟=1,2,….

We now prove the following generalization of Theorem 2.1.

Theorem 2.2. The convolutions Li₊(𝑥^𝑠)∗𝑥^𝑟₊ and ln|1−𝑥^𝑠|𝑥₊⁻¹∗𝑥^𝑟₊ exist and Li₊𝑥^𝑠∗𝑥^𝑟₊=s𝑟+1^𝑟_𝑖=0𝑖()r+1−1^r−𝑖𝐼_{𝑠,𝑟−𝑖}(𝑥)𝑥^𝑖₊+1𝑟+1Li₊𝑥^𝑠𝑥₊^𝑟+1 (2.5) for 𝑟=0,1,2,…,𝑠=1,2,… and ln1−𝑥^s𝑥₊⁻¹∗𝑥^𝑟₊=^𝑟−1_𝑖=0𝑟𝑖(−1)^𝑟−𝑖𝐼_{𝑠,𝑟−𝑖−1}(𝑥)𝑥^𝑖₊−1𝑠Li₊𝑥^𝑠𝑥^𝑟₊ (2.6) for 𝑟,𝑠=1,2,….

Proof. It is obvious that Li₊(𝑥^𝑠)∗𝑥^𝑟₊=0 if 𝑥<0.
When 𝑥>0, we have Li₊𝑥^𝑠∗𝑥^𝑟₊=−𝑠^𝑥₀(𝑥−𝑡)^𝑟𝑡₀𝑢⁻¹ln1−𝑢^𝑠𝑑𝑢𝑑𝑡=−𝑠^𝑥₀𝑢⁻¹ln1−𝑢^𝑠𝑥_𝑢(𝑥−𝑡)^𝑟=𝑠𝑑𝑡𝑑𝑢𝑟+1^𝑟+1_𝑖=0(−1)^𝑟−𝑖𝑥^𝑖𝑖𝑟+1^𝑥₀𝑢^𝑟−𝑖ln1−𝑢^𝑠=𝑠𝑑𝑢𝑟+1^𝑟_𝑖=0𝑖()𝑟+1−1^𝑟−𝑖𝑥^𝑖𝐼_{𝑠,𝑟−𝑖}(𝑥)+1𝑥𝑟+1^𝑟+1Li𝑥^𝑠 (2.7) proving (2.5).
Next, using (1.8) and (2.5), we have −𝑠ln1−𝑥^𝑠𝑥₊⁻¹∗𝑥^𝑟₊=𝑟Li₊𝑥^𝑠∗𝑥₊^𝑟−1=𝑠^𝑟−1_𝑖=0𝑟𝑖(−1)^{𝑟−𝑖−1}𝐼_{𝑠,𝑟−𝑖−1}(𝑥)𝑥^𝑖₊+Li₊𝑥^𝑠𝑥^𝑟, (2.8) and (2.6) follows.

Corollary 2.3. The convolutions Li₋(𝑥^𝑠)∗𝑥^𝑟₋ and ln|1−𝑥^𝑠|𝑥₋⁻¹∗𝑥^𝑟₋ exist and Li₋𝑥^𝑠∗𝑥^𝑟₋=𝑠𝑟+1^𝑟_𝑖=0𝑖()𝑟+1−1^{𝑟−𝑖+1}𝐼_{𝑠,𝑟−𝑖}(𝑥)𝑥^𝑖₋+1𝑟+1Li₋𝑥^𝑠𝑥₋^𝑟+1 (2.9) for 𝑟=0,1,2,…,𝑠=1,2,… and ln1−𝑥^𝑠𝑥₋⁻¹∗𝑥^𝑟₋=^𝑟−1_𝑖=0𝑟𝑖(−1)^{𝑟−𝑖+1}𝐼_{𝑠,𝑟−𝑖−1}(𝑥)𝑥^𝑖₋−1𝑠L𝑖₋𝑥^𝑠𝑥^𝑟₋ (2.10) for 𝑟,𝑠=1,2,….

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 𝑛^𝑠Li(𝑛^𝑟) for 𝑠=0,1,2,… and 𝑟=1,2,….

Theorem 2.4. The convolution Li₊(𝑥)⊛𝑥^𝑟 exists and Li₊(𝑥)⊛𝑥^𝑟=1𝑟+1^𝑟_𝑖=0𝑖𝑟+1(−1)^𝑟−𝑖(𝑟−𝑖+1)²𝑥^𝑖 (2.11) for 𝑟=0,1,2,….

Theorem 2.5. The convolution ln|1−𝑥|𝑥₊⁻¹⊛𝑥^𝑟 exists and ln|1−𝑥|𝑥₊⁻¹⊛𝑥^𝑟=^𝑟−1_𝑖=0𝑟𝑖(−1)^{𝑟−𝑖+1}(𝑟−𝑖)²𝑥^𝑖 (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−lim_𝑛→∞𝐼_𝑠,𝑟(𝑛)𝑠=−(𝑟+1)². (2.13)

Proof. Because 𝐼_𝑠,𝑟(𝑛)=1𝑛𝑟+1^𝑟+1−1ln1−𝑛^𝑠−1𝑟+1^𝑛𝑠₀1−𝑡^{(𝑟+1)/𝑠}1−𝑡𝑑𝑡 (2.14) when 𝑟+1/𝑠∈𝐍, we have 𝐼_𝑠,𝑟(𝑛)=1𝑛𝑟+1^𝑟+1−1ln1−𝑛^𝑠−1𝑟+1^{(𝑟+1)/𝑠−1}_𝑖=0𝑛^{𝑠𝑖+𝑠}=1𝑖+1𝑛𝑟+1^𝑟+1−1𝑠ln𝑛+ln1−𝑛^−𝑠−1𝑟+1^{(𝑟+1)/𝑠−1}_𝑖=0𝑛^{𝑠𝑖+𝑠},𝑖+1 (2.15) and (2.13) follows.

We now prove the following generalization of Theorems 2.4 and 2.5.

Theorem 2.7. The neutrix convolution Li₊(𝑥^𝑠)⊛𝑥^𝑟 exists when 𝑟+1/𝑠∈𝐍 and Li₊𝑥^𝑠⊛𝑥^𝑟=𝑠²𝑟+1^𝑟_𝑖=0𝑖𝑟+1(−1)^𝑟−𝑖(𝑟−𝑖+1)²𝑥^𝑖₊ (2.16) for 𝑟,𝑠=1,2,….

Proof. We put [Li₊(𝑥^𝑠)]_𝑛=Li₊(𝑥^𝑠)𝜏_𝑛(𝑥). Then the convolution [Li₊(𝑥^𝑠)]_𝑛∗𝑥^𝑟 exists by Definition 1.1 and Li₊𝑥^𝑠_𝑛∗𝑥^𝑟=^𝑛₀Li𝑡^𝑠(𝑥−𝑡)^𝑟𝑑𝑡+^{𝑛+𝑛−𝑛}_𝑛𝜏_𝑛(𝑡)Li𝑡^𝑠(𝑥−𝑡)^𝑟𝑑𝑡=𝐼₁+𝐼₂, (2.17) where 𝐼₁=^𝑛₀Li𝑡^𝑠(𝑥−𝑡)^𝑟𝑑𝑡=−𝑠^𝑛₀(𝑥−𝑡)^𝑟𝑡₀ln1−𝑢^𝑠𝑢𝑑𝑢𝑑𝑡=−𝑠^𝑛₀ln1−𝑢^𝑠𝑢^𝑛_𝑢(𝑥−𝑡)^𝑟=𝑠𝑑𝑡𝑑𝑢𝑟+1^𝑟+1_𝑖=0(−1)^𝑟−𝑖𝑥^𝑖𝑖𝑟+1^𝑛₀𝑢⁻¹ln1−𝑢^𝑠𝑢^{𝑟−𝑖+1}−𝑛^{𝑟−𝑖+1}=𝑠𝑑𝑢𝑟+1^𝑟_𝑖=0𝑖()𝑟+1−1^𝑟−𝑖𝑥^𝑖𝐼_{𝑠,𝑟−𝑖}(𝑛)+1𝑥𝑟+1^𝑟+1Li𝑛^𝑠+1𝑟+1^𝑟_𝑖=0𝑖()𝑟+1−1^𝑟−𝑖𝑥^𝑖Li𝑛^𝑠𝑛^{𝑟−𝑖+1}. (2.18) Thus, using Lemma 2.6, we have N−lim_𝑛→∞𝐼₁=𝑠²𝑟+1^𝑟_𝑖=0𝑖𝑟+1(−1)^{𝑟−𝑖+1}(𝑟−𝑖+1)²𝑥^𝑖. (2.19)
Further, it is easily seen that lim_𝑛→∞^{𝑛+𝑛−𝑛}_𝑛𝜏_𝑛(𝑡)Li𝑡^𝑠(𝑥−𝑡)^𝑟𝑑𝑡=0, (2.20) and (2.16) follows from (2.17), (2.19), and (2.20), proving the theorem.

Theorem 2.8. The neutrix convolution ln|1−𝑥^𝑠|𝑥₊⁻¹⊛𝑥^𝑟 exists when 𝑟+1/𝑠∈𝐍 and ln1−𝑥^𝑠𝑥₊⁻¹⊛𝑥^𝑟=𝑠^𝑟−1_𝑖=0𝑟𝑖(−1)^𝑟−𝑖(𝑟−𝑖)²𝑥^𝑖₊ (2.21) for 𝑟,𝑠=1,2,….

Proof. Using Theorems 1.5 and 1.6, we have −𝑠ln1−𝑥^𝑠𝑥₊⁻¹⊛𝑥^𝑟=𝑟Li₊𝑥^𝑠⊛𝑥^𝑟−1+𝑁−lim_𝑛→∞Li₊𝑥^𝑠𝜏^′_𝑛(𝑥)∗𝑥^𝑟, (2.22) where, on integration by parts, we have Li₊𝑥^𝑠𝜏^′_𝑛(𝑥)∗𝑥^𝑟=^{𝑛+𝑛−𝑛}_𝑛𝜏^′_𝑛(𝑡)Li𝑡^𝑠(𝑥−𝑡)^𝑟𝑛𝑑𝑡=−Li^𝑠(𝑥−𝑛)^𝑟−𝑠^{𝑛+𝑛−𝑛}_𝑛ln1−𝑡^𝑠𝑡⁻¹(𝑥−𝑡)^𝑟𝜏_𝑛(𝑡)𝑑𝑡+𝑟^{𝑛+𝑛−𝑛}_𝑛Li𝑡^𝑠(𝑥−𝑡)^𝑟−1𝜏_𝑛(𝑡)𝑑𝑡. (2.23)
It is clear that lim_𝑛→∞^{𝑛+𝑛−𝑛}_𝑛ln1−𝑡^𝑠𝑡⁻¹(𝑥−𝑡)^𝑟𝜏_𝑛(𝑡)𝑑𝑡=0lim_𝑛→∞∫^{𝑛+𝑛−𝑛}_𝑛Li𝑡^𝑠(𝑥−𝑡)^𝑟−1𝜏_𝑛(𝑡)𝑑𝑡=0. (2.24) It now follows from (2.23) and (2.24) that 𝑁−lim_𝑛→∞Li₊𝑥^𝑠𝜏^′_𝑛(𝑥)∗𝑥^𝑟=0. (2.25) Equation (2.21) now follows directly from (2.16) and (2.22), proving the theorem.

Corollary 2.9. The neutrix convolutions 𝐿𝑖₋(𝑥^𝑠)⊛𝑥^𝑟 and ln|1−𝑥^𝑠|𝑥₋⁻¹⊛𝑥^𝑟 exist when 𝑟+1/𝑠∈𝐍 and 𝐿𝑖₋𝑥^𝑠⊛𝑥^𝑟=𝑠²𝑟+1^𝑟_𝑖=0𝑖𝑟+1(−1)^{𝑟−𝑖+1}(𝑟−𝑖+1)²𝑥^𝑖₋,ln1−𝑥^𝑠𝑥₋⁻¹⊛𝑥^𝑟=𝑠∑^𝑟−1_𝑖=0𝑟𝑖(−1)^{𝑟−𝑖+1}(𝑟−𝑖)²𝑥^𝑖₋ (2.26) for 𝑟,𝑠=1,2,…

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₊(𝑥^𝑠)⊛𝑥^𝑟₋ and 𝐿𝑖₋(𝑥^𝑠)⊛𝑥^𝑟₊ exist when 𝑟+1/𝑠∈𝐍 and 𝐿𝑖₊𝑥^𝑠⊛𝑥^𝑟₋=𝑠𝑟+1^𝑟_𝑖=0𝑖𝑟+1(−1)^𝑖(𝑟−𝑖+1)²𝑠−(𝑟−𝑖+1)²𝐼_{𝑠,𝑟−𝑖}(𝑥)𝑥^𝑖₊+(−1)^𝑟+1𝑟+1𝐿𝑖₊𝑥^𝑠𝑥^𝑟+1,𝐿𝑖₋𝑥^𝑠⊛𝑥^𝑟₊=𝑠𝑟+1^𝑟_𝑖=0𝑖𝑟+1(−1)^𝑖+1(𝑟−𝑖+1)²𝑠−(𝑟−𝑖+1)²𝐼_{𝑠,𝑟−𝑖}(𝑥)𝑥^𝑖₋+(−1)^r+1𝑟+1𝐿𝑖₋𝑥^𝑠𝑥^𝑟+1 (2.27) for 𝑟,𝑠=1,2,….

Proof. Since the neutrix convolution product is distributive with respect to addition, we have Li₊𝑥^𝑠⊛𝑥^𝑟=Li₊𝑥^𝑠∗𝑥^𝑟₊+(−1)^𝑟Li₊𝑥^𝑠⊛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−𝑥^𝑠|𝑥₊⁻¹⊛𝑥^𝑟₋ and ln|1−𝑥^𝑠|𝑥₋⁻¹⊛𝑥^𝑟₊ exist when 𝑟+1/𝑠∈𝐍 and ln1−𝑥^𝑠𝑥₊⁻¹⊛𝑥^𝑟₋=^𝑟−1_𝑖=0𝑟𝑖(−1)^𝑖(𝑟−𝑖)²𝑠−(𝑟−𝑖)²𝐼_{𝑠,𝑟−𝑖−1}(𝑥)𝑥^𝑖₊+(−1)^𝑟𝑠Li₊𝑥^𝑠𝑥^𝑟,ln1−𝑥^𝑠𝑥₋⁻¹⊛𝑥^𝑟₊=^𝑟−1_𝑖=0𝑟𝑖(−1)^𝑖+1(𝑟−𝑖)²𝑠−(𝑟−𝑖)²𝐼_{𝑠,𝑟−𝑖−1}(𝑥)𝑥^𝑖₋+(−1)^𝑟𝑠Li₋𝑥^𝑠𝑥^𝑟 (2.29) 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).

Acknowledgment

This research was supported by FEIT, University of Ss. Cyril and Methodius in Skopje, Republic of Macedonia, Project no. 08-3619/8.