Inverse of the polylogarithm

Question

The polylogarithm can be defined using the power series $$ \operatorname{Li}_s(z) = \sum_{k=1}^\infty {z^k \over k^s}. $$ Contiguous polylogs have the ladder operators $$ \operatorname{Li}_{s+1}(z) = \int_0^z \frac {\operatorname{Li}_s(t)}{t}\,\mathrm{d}t\,, \qquad \operatorname{Li}_{s-1}(z) = z \,{\partial \operatorname{Li}_s(z) \over \partial z}\ , $$ and the sequence can be started with either $$ \operatorname{Li}_{1}(z) = -\ln(1-z)\,,\qquad \operatorname{Li}_{0}(z) = {z \over 1-z} \ . $$

Both $\operatorname{Li}_0$ and $\operatorname{Li}_1$ have inverse functions (up to a choice of branchcut) $$ \operatorname{Li}_0^{-1}(z)=\frac{z}{z+1}\,,\quad \operatorname{Li}_1^{-1}(z)=1-e^{-z}\,, $$ $$ \operatorname{Li}_0\left(\frac{z}{z+1}\right) =z= \operatorname{Li}_1\left(1-e^{-z}\right) + 2 n \pi i\,,\quad n\in\mathbb{Z} $$

Is there a nice/useful inverse function for the dilog ($\operatorname{Li}_2(z)$) and higher polylogs?

Answer 1

In astrophysics, specifically in partially degenerated matter, are used what is called Fermi-Dirac Integrals, which are written in terms of polylogaritms, and the z-value is a degeneracy parameter. In some papers I found that in fact they need the inverse of the Fermi-Dirac Integrals, that is, the inverse of the Polylogarithm.

Answer 2

The answer is yes. Polylogarithms are positive for all parameter values s and all z > 0. Thus the integrals used in the ladder operators should all have well defined inverses when the argument is positive, as the integrands are all monotonically increasing functions of z. It strikes me as similar to how the Elliptic Functions are also defined as inverses of integrals of certain functions. Analytic continuation should then extend this to the whole complex plane (assuming a branch cut somewhere along the negative real axis), though as @Jeffrey-Hersh points out using symbolic series inversion techniques may be complicated. I wonder if there are other functional equations we could exploit?

How to numerically compute this inverse generically even for positive z seems like another question entirely. Its likely its own class of special function. That said, numerically it shouldn't be that hard to implement a low order inverse like $Li_2^{-1}(x)$. If you have numerical versions of polylog that can handle fractional orders, the method of numerical integral inversion should work for those too, assuming fractional polylogs are also monotonically increasing on the relevant domains.

I fear you'll probably have to write your own functions in whatever symbolic computing environment you're using, as I'm unaware of any libraries specialized to this problem. If you find an implementation, please share!

Answer 3

Answering this for complex $z$ on the open unit disk, for $s>0,$ the polylogarithm is a convex function which is a special subclass of univalent (one to one) functions. It has an inverse but such an inverse (as shown above) is not analytically tractable.

Source: https://academic.oup.com/jlms/article-abstract/s2-27/3/435/828376