I Independent arguments of dilogarithms

[CAF123]

I have a series of dilogarithms with various arguments and I was just wondering if it is possible to tell if they are all independent or if there is a way to reduce them to a smaller minimal set?

The dilogs in question are of the form $\text{Li}_2(X_i)$, where $X_i$ is an argument $i=1,\dots,10$ belonging to the set $$X_i \in \left\{\frac{v - u v}{1 - u v}, \frac{-1 + u v}{-u + u v}, \frac{-1 + u v}{u + u v}, \frac{-v + u v}{-u + u v},\frac{v + u v}{-1 + u v}, \frac{v + u v}{u + u v}, \frac{1 + u}{1-w}, \frac{-1 + u}{u - w}, \frac{1 + u}{u - w}, \frac{-1 + u}{u + w}\right\}$$

I have tried various transformation laws of the dilogs as described in the wolfram pages but none seem to relate the above arguments. Thanks for any comments.

 

[mfb]

What do you mean by independent? They are clearly not all independent in terms of statistics - you only have two variables but 10 distributions.
They might be linearly dependent in the vector space of distributions - that should be easy to check with the standard methods.

 

[CAF123]

What do you mean by independent? They are clearly not all independent in terms of statistics - you only have two variables but 10 distributions.
They might be linearly dependent in the vector space of distributions - that should be easy to check with the standard methods.

By ‘independent’, I meant if I could relate any of the dilogs with the above arguments to each other using transformation laws amongst dilogs, as given e.g in wolfram documentation. As far as I can see, they are not related but I have a hunch that they must be so was wondering if you (or anyone) could see a way in which my number of dilogs can be reduced through dilog identities.

 

[mfb]

With suitable ranges the dilogarithm is bijective, so clearly it is possible, but not necessarily in a useful way.

 

[CAF123]

With suitable ranges the dilogarithm is bijective, so clearly it is possible, but not necessarily in a useful way.

Could you give me an example? I've done some numerical tests and I fail to find a dilog relation between any two arguments presented.
I should also say I'm doing such an exercise from a physics background, the number of dilog arguments I have here exceeds the amount I naively expected in my analysis. However there is a symmetry u->-u and v->-v between pairs and I am wondering if this symmetry protects the reduction of dilog arguments further.

 

[mfb]

You can always take the inverse dilogarithm to get rid of the dilogarithm. That is probably not what you want, but it is possible. Afterwards it is a "solve for u,v,w" situation with multiple options to do so.

2/(X₉-X₈) = u-w

1/(1/(2X₁₀) + 1/(2X₉) - 1) + 1 = u

Add the dilogarithm and its inverse everywhere.

With that you can express w and u with these things, and find an independent way to express X₇.

 

[CAF123]

Thanks, if I understand properly in the example above you found two equations relating some X_i in terms of parameters u,w. Then you can solve for u and w in terms of these X_i which may e.g be subbed into X₇ to get this in terms of other X_i. This is one manner of simplification but indeed,as you surmised, not the one I was looking for since it probably is not the case that there exists dilog relations between the arguments obtained.

Just to be crystal clear about what I want to do: Given an argument X_i I want to find a relation e.g. $$\text{Li}_2(X_i) = \text{Li}_2 \left(\frac{1}{1-X_i}\right) + \dots,$$ where 1/(1-X_i) will coincide with some X_j for suitable i and j. I have hacked up lots of dilog identities and I fail to find a case where e.g 1/(1-X_i) will coincide with an X_j.