Getting started

Either open the file logsine.m and click "Run Package" or use something like this:

<<"~/docs/math/mathematica/logsine.m"

General usage

Log-sine integrals can be expressed in terms of polylogarithms:

LsToLi[Ls[5,1,Pi/3]]

We can check this numerically:

Ls[5,1,Pi/3]//N
LsToLi[Ls[5,1,Pi/3]]//N
N[LsToLi[Ls[5,1,Pi/3]],100]

Combinations of Nielsen-type polylogarithms may be simplified:

LiReduce[Gl[{3,1},Pi/3]]

LiReduce[Cl[{5,1},t]]

Advanced options

LiReduce[Cl[{4,1,1},Pi/3]]
LiReduce[Cl[{4,1,1},Pi/3],UseReductionTable->False]

Comments on the reduction of Nielsen-type polylogarithms

NielsenArgsOfWeight[w_,t_]:=Table[{Join[{w-n+1},Table[1,{n-1}]],t},{n,1,w-1}];
NielsenArgsUpToWeight[w_,t_]:=Flatten[Table[NielsenArgsOfWeight[n,t],{n,2,w}],1];

The irreducible Clausen values of argument pi/3 and weight at most 10:

Select[Table[Cl@@a,{a,NielsenArgsUpToWeight[10,Pi/3]}],#===LiReduce[#]&]

The irreducible Glaisher values of argument pi/3 and weight at most 10:

Select[Table[Gl@@a,{a,NielsenArgsUpToWeight[10,Pi/3]}],#===LiReduce[#]&]

The Clausen values of argument pi/3 and weight at most 10 for which extra relations exist:

Select[Table[Cl@@a,{a,NielsenArgsUpToWeight[10,Pi/3]}],LiReduce[#,UseReductionTable->False]===#&&LiReduce[#]=!=#&]

The Glaisher values of argument pi/3 and weight at most 10 for which extra relations exist:

Select[Table[Gl@@a,{a,NielsenArgsUpToWeight[10,Pi/3]}],LiReduce[#,UseReductionTable->False]===#&&LiReduce[#]=!=#&]

Examples from "Special values of generalized log-sine integrals"

Table[LsToLi[Ls[n,0,Pi]],{n,2,8}]

-Ls[4,2,Pi]//LsToLi

-Ls[5,1,Pi]//N
-Ls[5,1,Pi]//LsToLi
%//N

-Ls[6,1,Pi]//LsToLi

Ls[5,2,2Pi]//LsToLi

$Assumptions=0<τ<Pi;
Ls[4,1,τ]//LsToLi

-Ls[4,1,Pi/3]//LsToLi

Table[LsToLi[Ls[n,0,Pi/3]],{n,2,7}]

Check numerically:

Table[Ls[n,0,Pi/3]-LsToLi[Ls[n,0,Pi/3]],{n,2,7}]//N

A result of Zucker

LsToLi[Ls[6,3,Pi/3]-2Ls[6,1,Pi/3]]

Reducing Nielsen polylogs at 1

Li[{9,1,1,1,1,1},1]//LiReduce

This is based on K. S. Koelbig's expression (1982) for Nielsen polylogarithms at 1.

Numerical usage

Ls[5,2,2Pi/3]//LsToLi

Ls[5,2,2Pi/3]//N

N[Ls[5,2,2Pi/3]//LsToLi,200]

For much higher precision, specialized code should be used for the evaluation of the multiple polylogarithms.

A random higher weight example

N[Ls[15,11,2Pi/3]//LsToLi,50]

Examples from "Log-sine evaluations of Mahler measures"

Table[LsToLi[Ls[n,0,Pi/3]-Ls[n,0,Pi]]/Pi,{n,2,7}]//Expand//Column

Evaluation of μ(1+x)

μk[k_]:=-1/Pi Ls[k+1,0,Pi]//LsToLi

-μk[5]
μk[6]

Reducibility

$Assumptions=0<τ<Pi;
Ls[3,0,τ]//LsToLi
Ls[3,1,τ]//LsToLi
Ls[4,0,τ]//LsToLi
Ls[4,1,τ]//LsToLi
Ls[4,2,τ]//LsToLi

Appendix A from "New results for the ϵ-expansion of certain one-, two- and three-loop Feynman diagrams" by A. Davydychev and M. Kalmykov

$Assumptions=0<τ<Pi;
Ls[3,1,τ]//LsToLi
Ls[4,2,τ]//LsToLi
Ls[5,3,τ]//LsToLi
Ls[6,4,τ]//LsToLi

Ls[2,0,Pi]//LsToLi
Ls[3,0,Pi]//LsToLi
Ls[4,0,Pi]//LsToLi
Ls[5,0,Pi]//LsToLi
Ls[6,0,Pi]//LsToLi

Ls[3,0,Pi/3]//LsToLi
Ls[4,1,Pi/3]//LsToLi

(A.9)

Ls[4,0,Pi/3]//LsToLi
Ls[6,0,Pi/3]//LsToLi

(A.10)

Ls[5,1,Pi/3]-Pi/3Ls[4,0,Pi/3]//LsToLi
Ls[6,1,Pi/3]-Pi/3Ls[5,0,Pi/3]//LsToLi
Ls[7,1,Pi/3]-Pi/3Ls[6,0,Pi/3]//LsToLi
Ls[8,1,Pi/3]-Pi/3Ls[7,0,Pi/3]//LsToLi

Note that the built-in reductions only reduce among polylogarithms of Nielsen type.  That's why Gl[{7,1},pi/3] is not reduced to zeta values including zeta(5,3).

Ls[9,1,Pi/3]-Pi/3Ls[8,0,Pi/3]//LsToLi

(A.11)

Ls[5,2,Pi/3]-2/3Ls[5,0,Pi/3]//LsToLi
Ls[6,2,Pi/3]+4/15Ls[6,0,Pi/3]-2/3Zeta[2]Ls[4,0,Pi/3]//LsToLi
Ls[6,3,Pi/3]-2/3Pi Ls[5,0,Pi/3]//LsToLi

(A.14)

Our package will provide evaluations in terms of Clausen/Glaisher values at pi/2.  These can be rewritten as polylogarithms at 1/2.

Ls[4,1,Pi/2]//LsToLi
Ls[3,0,Pi/2]//LsToLi

Ls[5,1,Pi/2]//LsToLi

Again, the alternating zeta values can be rewritten as polylogarithms at 1/2.

Ls[4,1,Pi]//LsToLi
Ls[5,1,Pi]//LsToLi
Ls[5,2,Pi]//LsToLi

Numerical values

N[Ls[3,0,Pi/2]//LsToLi,100]

N[Ls[4,0,Pi/2]//LsToLi,100]

N[Cl[{4},Pi/2],100]

N[Ls[5,0,Pi/2]//LsToLi,100]

N[Ls[5,2,Pi/2]//LsToLi,100]

Examples from “Explicit evaluations of some families of log-sine and log-cosine integrals”

Table[LsToLi[-Ls[n,1,2Pi]],{n,4,12}]//Column

Last three are given incorrectly in the reference.

LsToLi[-Ls[10,1,2Pi]]
N[%,100]
N[82691/40320Pi^10+3780Pi^2Zeta[3]Zeta[5]+315/4Pi^4Zeta[3]^2,100]
NIntegrate[t(Log[2Sin[t/2]])^8,{t,0,2Pi}]

Table[LsToLi[-Ls[n,2,2Pi]],{n,4,13}]//Column

Table[LsToLi[-Ls[n,3,2Pi]],{n,5,14}]//Column

Table[LsToLi[-Ls[n,4,2Pi]],{n,5,15}]//Column

Table[LsToLi[-Ls[n,5,2Pi]],{n,7,16}]//Column

Table[LsToLi[-Ls[n,6,2Pi]],{n,8,12}]//Column