Issues with an equation involving Logs and PolyLogs

I want to check a claimed identity that involves Log and PolyLog functions. The relation can be found on this paper and more specifically eq.(5.7) on page 35.

The description of the task at hand:

The object

$$$$\Phi(z,\bar{z}) = \frac{1}{z-\bar{z}} \left(\log(z \bar{z}) \log\left(\frac{1-z}{1-\bar{z}}\right) + 2 \text{Li}(z) - 2 \text{Li}(\bar{z}) \right)$$$$

satisfies the differential equation

$$$$\partial_z \Phi = - \frac{1}{z-\bar{z}}\Phi - \frac{1}{z(z-\bar{z})}\log(-1+z)(-1+\bar{z}) + \frac{1}{(-1+z)(z-\bar{z})}\log(z\bar{z})$$$$

The way I went about it is the following:

Define the $$\Phi(z,\bar{z})$$

Phi[z_][zbar_] :=
1/(z - zbar) (Log[z zbar] Log[(1 - z)/(1 - zbar)] +
2 PolyLog[2, z] - 2 PolyLog[2, zbar])


Define the LHS and RHS of the differential equation that we want to show

eqn1 = D[Phi[z][zbar], z];
eqn2 = -(1/(z - zbar)) Phi[z][zbar] -
1/(z (z - zbar)) Log[(-1 + z) (-1 + zbar)] +
1/((-1 + z) (z - zbar)) Log[z zbar];


Subtract eqn2 from eqn1 and get a zero

eqn1 - eqn2 // FullSimplify // PowerExpand


This final command, however, does not return a zero.

Any ideas? Is this possible that this is not implemented into Mathematica? I am using 12.0.0 for Linux x86 (64-bit)

• The authors are quite sloppy unfortunately. They never exactly define what is Li (is it the polylog with n=2?) and in the first term with the log, it's unclear to me if the (-1 + zbar) is inside the log or not. Neither case works btw. Aug 3 at 10:48
• Seeing their Ref[72], I think Li is indeed Polylog with n=2, and it seems that paper is more clear, albeit with a slightly different notation. Aug 3 at 10:50
• @HansOlo Li is indeed the Polylog with n=2. I have checked other sources as well. They call it the dilogarithm, so I guess a bit of sloppy notation does not really matter. I checked that pulling the (-1 + zbar) ourside of the Log only complicates things. Aug 3 at 10:57
• I would try substituting numerical values Aug 3 at 11:56

Clear["Global*"]

Phi[z_, zbar_] :=
1/(z - zbar) (Log[z*zbar] Log[(1 - z)/(1 - zbar)] + 2 PolyLog[2, z] -
2 PolyLog[2, zbar])

eqn1 = D[Phi[z, zbar], z];
eqn2 = -(1/(z - zbar)) Phi[z, zbar] -
1/(z (z - zbar)) Log[(-1 + z) (-1 + zbar)] +
1/((-1 + z) (z - zbar)) Log[z*zbar];


Presumably, there are some assumptions given in the paper. Let's assume that Phi is real. This requires

cons = FunctionDomain[Phi[z, zbar], {z, zbar}]

(* (0 < z < 1 && 0 < zbar < 1 && z - zbar != 0 && z zbar > 0) || (0 < z < 1 &&
zbar < 0 && z - zbar != 0 && z zbar > 0) || (z < 0 && 0 < zbar < 1 &&
z - zbar != 0 && z zbar > 0) || (z < 0 && zbar < 0 && z - zbar != 0 &&
z zbar > 0) *)


Then

eqn1 - eqn2 // FullSimplify[#, cons] &

(* 0 *)
`