# Further Results on the Dilogarithm Integral

Authors: Biljana Jolevska-Tuneska; Brian Fisher
Journal: Journal of Applied Mathematics
 (2011)

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## Abstract

The dilogarithm integral Li(xs) 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.


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## Body

## 1. Introduction

The dilogarithm integralLi(x) is defined by(1.1)Li(x)=-∫0xln⁡|1-t|tdt
(see [1]). More generally, we have  (1.2)Li(xs)=-∫0xsln⁡|1-t|tdt=-s∫0xln⁡|1-ts|tdt
for s=1,2,….The associated functionsLi+(xs) and Li-(xs) are defined by (1.3)Li+(xs)=H(x)Li(xs),Li-(xs)=H(-x)Li(xs)=Li(xs)-Li+(xs),
where H(x) denotes Heaviside's function.Next, we define the distributionln|1-xs|x-1 by (1.4)ln⁡|1-xs|x-1=-s-1[Li(xs)]′,
and its associated distributions ln|1-xs|x+-1 and ln|1-xs|x--1 are defined by (1.5)ln⁡|1-xs|x+-1=H(x)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)]′.The classical definition of the convolution product of two functionsf and g is as follows.Definition 1.1.
Letfandgbe functions. Then the convolutionf*gis defined by(1.6)(f*g)(x)=∫-∞∞f(t)g(x-t)dt
for all points x for which the integral exist.It follows easily from the definition that iff*g exists then g*f exists and  (1.7)f*g=g*f,
and if (f*g)′ and f*g′(orf′*g) exists, then  (1.8)(f*g)′=f*g′(orf′*g).
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.
Letfandgbe distributions in𝒟′. Then the  convolutionf*gis defined by the equation(1.9)〈(f*g)(x),φ〉=〈f(y),〈g(x),φ(x+y)〉〉
for arbitrary φ in 𝒟, provided f and g satisfy either of the following conditions:(a)
eitherf or g has bounded support,(b)
the supports off and g are bounded on the same side.It follows that if the convolutionf*g exists by this definition then (1.7) and (1.8) are satisfied.In order to extend Definition1.2 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 functionτn is then defined by (1.10)τn(x)={1,|x|≤n,τ(nnx-nn+1),x>n,τ(nnx+nn+1),x<-n,
for n=1,2,….The following definition of the noncommutative neutrix convolution was given in [4].Definition 1.3.
Letfandgbe distributions in𝒟′, and letfn=fτnforn=1,2,…. Then the noncommutative neutrix convolutionf⊛gis defined as the neutrix limit of the sequence{fn*g}, provided the limithexists in the sense that(1.11)N-lim⁡n→∞〈fn*g,φ〉=〈h,φ〉
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
(1.12)nλln⁡r-1n,ln⁡rn:λ>0,r=1,2,…
and all functions which converge to zero in the normal sense as n tends to infinity.
In particular, if(1.13)lim⁡n→∞〈fn*g,φ〉=〈h,φ〉
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 off⊛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.
Letf 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
(1.14)f⊛g=f*g.Theorem 1.5.
Letf and  g be distributions in  𝒟′ and suppose that the neutrix convolution  f⊛g exists. Then the neutrix convolution  f⊛g′ exists and
(1.15)(f⊛g)′=f⊛g′.Note however that(f⊛g)′ is not necessarily equal to f′⊛g but we do have the following theorem.Theorem 1.6.
Letf 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
(1.16)(f⊛g)′=f′⊛g+h.


## 2. Main Result

We define the functionIs,r(x) by  (2.1)Is,r(x)=∫0xurln⁡|1-us|du
for r=0,1,2,… and s=1,2,…. In particular, we define the function Ir(x) by (2.2)Ir(x)=I1,r(x)
for r=0,1,2,….The following theorem was proved in [6].Theorem 2.1.
The convolutionsLi+(x)*x+randln|1-x|x+-1*x+rexistand(2.3)Li+(x)*x+r=1r+1∑i=0r(r+1i)(-1)r-iIr-i(x)x+i+1r+1x+r+1Li+(x)
for r=0,1,2,… and
(2.4)ln⁡|1-x|x+-1*x+r=∑i=0r-1(ri)(-1)r-iIr-i-1(x)x+i-Li+(x)x+r
for r=1,2,….We now prove the following generalization of Theorem2.1.Theorem 2.2.
The convolutionsLi+(xs)*x+r and  ln|1-xs|x+-1*x+r exist and
(2.5)Li+(xs)*x+r=sr+1∑i=0r(r+1i)(-1)r-iIs,r-i(x)x+i+1r+1Li+(xs)x+r+1
for r=0,1,2,…,s=1,2,… and
(2.6)ln⁡|1-xs|x+-1*x+r=∑i=0r-1(ri)(-1)r-iIs,r-i-1(x)x+i-1sLi+(xs)x+r
for r,s=1,2,….Proof.
It is obvious thatLi+(xs)*x+r=0 if x<0.
Whenx>0, we have
(2.7)Li+(xs)*x+r=-s∫0x(x-t)r∫0tu-1ln⁡|1-us|dudt=-s∫0xu-1ln⁡|1-us|∫ux(x-t)rdtdu=sr+1∑i=0r+1(-1)r-ixi(r+1i)∫0xur-iln⁡|1-us|du=sr+1∑i=0r(r+1i)(-1)r-ixiIs,r-i(x)+1r+1xr+1Li(xs)
proving (2.5).
Next, using (1.8) and (2.5), we have
(2.8)-sln⁡|1-xs|x+-1*x+r=rLi+(xs)*x+r-1=s∑i=0r-1(ri)(-1)r-i-1Is,r-i-1(x)x+i+Li+(xs)xr,
and (2.6) follows.Corollary 2.3.
The convolutionsLi-(xs)*x-r and  ln|1-xs|x--1*x-r exist and
(2.9)Li-(xs)*x-r=sr+1∑i=0r(r+1i)(-1)r-i+1Is,r-i(x)x-i+1r+1Li-(xs)x-r+1
for r=0,1,2,…,s=1,2,… and
(2.10)ln⁡|1-xs|x--1*x-r=∑i=0r-1(ri)(-1)r-i+1Is,r-i-1(x)x-i-1sLi-(xs)x-r
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 nsLi(nr) for s=0,1,2,… and r=1,2,….Theorem 2.4.
The convolutionLi+(x)⊛xr exists and
(2.11)Li+(x)⊛xr=1r+1∑i=0r(r+1i)(-1)r-i(r-i+1)2xi
for r=0,1,2,….Theorem 2.5.
The convolutionln|1-x|x+-1⊛xr exists and
(2.12)ln⁡|1-x|x+-1⊛xr=∑i=0r-1(ri)(-1)r-i+1(r-i)2xi
for r=1,2,….Before proving some further results, we need the following lemma.Lemma 2.6.
Ifr+1/s∈N for r,s=1,2,…, then
(2.13)N-limn→∞Is,r(n)=-s(r+1)2.Proof.
Because(2.14)Is,r(n)=1r+1(nr+1-1)ln⁡|1-ns|-1r+1∫0ns1-t(r+1)/s1-tdt
when r+1/s∈N, we have
(2.15)Is,r(n)=1r+1(nr+1-1)ln⁡|1-ns|-1r+1∑i=0(r+1)/s-1nsi+si+1=1r+1(nr+1-1)(sln⁡n+ln⁡|1-n-s|)-1r+1∑i=0(r+1)/s-1nsi+si+1,
and (2.13) follows.We now prove the following generalization of Theorems2.4 and 2.5.Theorem 2.7.
The neutrix convolutionLi+(xs)⊛xr exists when  r+1/s∈N and
(2.16)Li+(xs)⊛xr=s2r+1∑i=0r(r+1i)(-1)r-i(r-i+1)2x+i
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
(2.17)[Li+(xs)]n*xr=∫0nLi(ts)(x-t)rdt+∫nn+n-nτn(t)Li(ts)(x-t)rdt=I1+I2,
where
(2.18)I1=∫0nLi(ts)(x-t)rdt=-s∫0n(x-t)r∫0tln⁡|1-us|ududt=-s∫0nln⁡|1-us|u∫un(x-t)rdtdu=sr+1∑i=0r+1(-1)r-ixi(r+1i)∫0nu-1ln⁡|1-us|{ur-i+1-nr-i+1}du=sr+1∑i=0r(r+1i)(-1)r-ixiIs,r-i(n)+1r+1xr+1Li(ns)+1r+1∑i=0r(r+1i)(-1)r-ixiLi(ns)nr-i+1.
Thus, using Lemma 2.6, we have
(2.19)N-lim⁡n→∞I1=s2r+1∑i=0r(r+1i)(-1)r-i+1(r-i+1)2xi.
Further, it is easily seen that(2.20)lim⁡n→∞∫nn+n-nτn(t)Li(ts)(x-t)rdt=0,
and (2.16) follows from (2.17), (2.19), and (2.20), proving the theorem.Theorem 2.8.
The neutrix convolutionln|1-xs|x+-1⊛xr exists when  r+1/s∈N and
(2.21)ln⁡|1-xs|x+-1⊛xr=s∑i=0r-1(ri)(-1)r-i(r-i)2x+i
for r,s=1,2,….Proof.
Using Theorems1.5 and 1.6, we have
(2.22)-sln⁡|1-xs|x+-1⊛xr=rLi+(xs)⊛xr-1+N-lim⁡n→∞[Li+(xs)τn′(x)]*xr,
where, on integration by parts, we have
(2.23)[Li+(xs)τn′(x)]*xr=∫nn+n-nτn′(t)Li(ts)(x-t)rdt=-Li(ns)(x-n)r-s∫nn+n-nln⁡|1-ts|t-1(x-t)rτn(t)dt+r∫nn+n-nLi(ts)(x-t)r-1τn(t)dt.
It is clear that(2.24)lim⁡n→∞∫nn+n-nln⁡|1-ts|t-1(x-t)rτn(t)dt=0lim⁡n→∞∫nn+n-nLi(ts)(x-t)r-1τn(t)dt=0.
It now follows from (2.23) and (2.24) that
(2.25)N-lim⁡n→∞[Li+(xs)τn′(x)]*xr=0.
Equation (2.21) now follows directly from (2.16) and (2.22), proving the theorem.Corollary 2.9.
The neutrix convolutionsLi-(xs)⊛xr and  ln|1-xs|x--1⊛xr exist when  r+1/s∈N and
(2.26)Li-(xs)⊛xr=s2r+1∑i=0r(r+1i)(-1)r-i+1(r-i+1)2x-i,ln|1-xs|x--1⊛xr=s∑i=0r-1(ri)(-1)r-i+1(r-i)2x-i
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 convolutionsLi+(xs)⊛x-r and  Li-(xs)⊛x+r exist when  r+1/s∈N and
(2.27)Li+(xs)⊛x-r=sr+1∑i=0r(r+1i)(-1)i(r-i+1)2[s-(r-i+1)2Is,r-i(x)]x+i+(-1)r+1r+1Li+(xs)xr+1,Li-(xs)⊛x+r=sr+1∑i=0r(r+1i)(-1)i+1(r-i+1)2[s-(r-i+1)2Is,r-i(x)]x-i+(-1)r+1r+1Li-(xs)xr+1
for r,s=1,2,….Proof.
Since the neutrix convolution product is distributive with respect to addition, we have(2.28)Li+(xs)⊛xr=Li+(xs)*x+r+(-1)rLi+(xs)⊛x-r,
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 convolutionsln|1-xs|x+-1⊛x-r and  ln|1-xs|x--1⊛x+r exist when  r+1/s∈N and
(2.29)ln⁡|1-xs|x+-1⊛x-r=∑i=0r-1(ri)(-1)i(r-i)2[s-(r-i)2Is,r-i-1(x)]x+i+(-1)rsLi+(xs)xr,ln⁡|1-xs|x--1⊛x+r=∑i=0r-1(ri)(-1)i+1(r-i)2[s-(r-i)2Is,r-i-1(x)]x-i+(-1)rsLi-(xs)xr
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).



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