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TagSet and Unprotect

• To: mathgroup at smc.vnet.net
• Subject: [mg77594] TagSet and Unprotect
• From: dimitris <dimmechan at yahoo.com>
• Date: Wed, 13 Jun 2007 07:41:30 -0400 (EDT)

Following a idea by Raymond Manzoni
let me prove that

- PolyLog[2, -(1/8)] + 3*PolyLog[2, 1/4]=Pi^2/6 - (3*Log[2]^2)/2

Indeed

Off[N::meprec] (*turn off a message*)
PolyLog[2, z/(z + 1)] + PolyLog[2, z/(z - 1)] - (1/2)*PolyLog[2, z^2/
(z^2 - 1)] + (1/4)*Log[(1 + z)/(1 - z)]^2 == 0
(*an identity of dilogarithms*)
(FunctionExpand[Expand[2*#1]] & ) /@ (% /. z -> -3^(-1)) (*Mathematica does
the rest*)
Reverse[(Plus @@ Cases[%, (a_)*PolyLog[x_, y_], Infinity] - #1 & ) /@
%]

(1/4)*Log[(1 + z)/(1 - z)]^2 + PolyLog[2, z/(-1 + z)] + PolyLog[2, z/
(1 + z)] - (1/2)*PolyLog[2, z^2/(-1 + z^2)] == 0

-(Pi^2/6) + (3*Log[2]^2)/2 - PolyLog[2, -(1/8)] + 3*PolyLog[2, 1/4] ==
0

-PolyLog[2, -(1/8)] + 3*PolyLog[2, 1/4] == Pi^2/6 - (3*Log[2]^2)/2

I try to add this result to Plus, so that
in the following to have additional substitution.

rul1 = {PolyLog[2, z_] /; z < -3 -> -PolyLog[2, 1/z] - Pi^2/6 -
(1/2)*Log[-z]^2,
    PolyLog[2, z_] -> PolyLog[2, 1/(1 - z)] - Pi^2/6 + (1/2)*Log[1 -
z]*Log[(1 - z)/z^2]};
o = Expand[3*PolyLog[2, -3] + PolyLog[2, -8] /. rul1]

-((2*Pi^2)/3) - (3/2)*Log[9/4]*Log[4] - Log[8]^2/2 - PolyLog[2, -
(1/8)] + 3*PolyLog[2, 1/4]

That is, I want Mathematica to substitute - PolyLog[2, -(1/8)] +
3*PolyLog[2, 1/4]
by Pi^2/6 - (3*Log[2]^2)/2 in the expression o.

In the beggining I tried:

Unprotect[PolyLog];
PolyLog /: -PolyLog[2, -(1/8)] + 3*PolyLog[2, 1/4] = Pi^2/6 -
(3*Log[2]^2)/2

TagSet::tagpos : Tag PolyLog in -PolyLog[2, -(1/8)] + 3*PolyLog[2,
1/4] is \
too deep for an assigned rule to be found.
Pi^2/6 - (3*Log[2]^2)/2

I understood the presence of this message.
That's why I try to assign this equality to Plus.

Unprotect[Plus];
Plus /: -PolyLog[2, -(1/8)] + 3*PolyLog[2, 1/4] = Pi^2/6 -
(3*Log[2]^2)/2
TagSet::write: Tag Plus in-PolyLog[2, -(1/8)] + 3*PolyLog[2, 1/4] is
Protected

Now I don't understand the presence of this message.
I can't assign an UpValue to a built in function?

Can somebody explain me what is going on?
And how I must work in order to overcome the problem?

Thanks
Dimitris

• Follow-Ups:
  • Re: TagSet and Unprotect
    • From: Andrzej Kozlowski <akoz@mimuw.edu.pl>