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I'm struggling to grasp the polylogarithm's behavior around its branch points, particularly at $z=0$ and $z=1$. The branch point at $z=0$ seems "hidden" within the unit disk, and I'm having trouble understanding its implications.

I've consulted many books and articles, but unfortunately, it didn't clarify my doubts. Other related questions (Branch Points of the Polylog function, Extending the domain of the functional equation of Hurwitz zeta function., references for learning about branch cuts/ branch points in complex analysis) haven't provided the insight I need.

Specifically, I'd appreciate sources or explanations on:

  • The nature of the branch points at $z=0$.
  • The monodromy of the polylogarithm.
  • Understanding the multivaluedness, especially when looping around the origin within the unit disk.

Any book recommendations, research articles or detailed explanations would be greatly appreciated!

M. A.
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Dr Potato
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    If you're on the principal sheet and loop around the origin $n$ times within the unit disk, you'll return to where you started because you haven't crossed over the branch cut on $[1, \infty)$. The origin is not a branch point on the principal sheet. On the other hand, if you cross over the branch cut, the origin is a branch point on the next sheet. And if you then proceed to loop around the origin $n$ times within the unit disk, you'll be at $$\operatorname{Li}_{s}(z) - \frac{2 \pi i \left( \ln(z)+ 2 n \pi i \right)^{s-1}}{\Gamma(s)}$$ if you're moving in a counterclockwise direction. – Random Variable Oct 21 '24 at 17:34
  • So $z=0$ has a regular value in the main sheet because its value comes from a convergent series inside the unit disk, but what about its values in other sheets? nearby $z=1$ the points in the unit disk must vary since in the neighborhood of this point we have multivalues, and therefore we should have many values points inside the unit $z$-complex open disk. – Dr Potato Oct 23 '24 at 04:48
  • If you're on the principal sheet and cross over the branch cut on $[1, \infty)$ in the counterclockwise direction, the values on the next sheet are defined by $$\operatorname{Li}{s}(z) - \frac{2 \pi i \ln^{s-1}(z)}{\Gamma(s)}, $$ where $\operatorname{Li}{s}(z)$, $s>0$, is the principal branch of the polylogarithm, and $\ln(z)$ is the principal branch of the logarithm. On this sheet there is a branch cut on $[1, \infty)$ for the principal branch of $\operatorname{Li}_{s}(z)$, and a branch cut on the negative real axis for the principal branch of $\ln(z)$. – Random Variable Oct 23 '24 at 15:18
  • If you then cross over the negative real axis in the counterclockwise direction, the values on the next sheet are defined by $$\operatorname{Li}{s}(z) - \frac{2 \pi i \ln \left(\ln(z) + 2 \pi i \right)^{s-1}}{\Gamma(s)}, $$ where again $\operatorname{Li}{s}(z) $ is the principal branch of the polylogarithm, and $\ln(z)$ is the principal branch of the logarithm. And so on. – Random Variable Oct 23 '24 at 15:19
  • I thought it was easier: The standard branch cut for the re-parametrized logarithm $$Li_1(z)=\ln(1-z)$$ is $[1,\infty)$. The polylogarithms satisfy the stair-property for repeated integration $$(z \mathcal{D}z)^n Li{s+n}(z) = Li_s(z) $$ If I loop $m$ times around $z=1$, for $s=1$ finding $Li_{1+n}(z)$ is a matter of applying the "divide by $z$ and then integrate" $n$ times to $$\ln(1-z)+2\pi i m=Li_1(z)$$ ...or Is this operator not commutable with integration in closed loop $m$ times? if it commutes, then we can loop first and then jump to the corresponding $Li_{s+1}$ looping $(zD_z)$. – Dr Potato Oct 25 '24 at 01:03
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    That would show what happens when you loop around $z=1$ without also looping around $z=0$. The issue is that when you divide by $z$ and integrate, a term involving $\ln (z)$ gets introduced which has a branch point at the origin. – Random Variable Oct 25 '24 at 03:35
  • A clear description of the monodromy of Lerch transcendent function, for which a special case is the polylogarityhm, also closely related to the "periodic zeta function", is in: Lagarias, J. C., & Li, W.-C. W. (2015). The Lerch Zeta Function III: Polylogarithms and Special Values. arXiv:1506.06161v2 [math.NT]. Available at: https://arxiv.org/abs/1506.06161 – Dr Potato May 23 '25 at 01:24

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