Define $$ \operatorname{Li}_2(z) \coloneqq \sum_{n\geq 1} \frac{z^n}{n^2},\qquad |z|\lt 1 $$ More generally, the polylogarithm $m=1,2,\dots$ $$ \operatorname{Li}_m(z) \coloneqq \sum_{n\geq 1} \frac{z^n}{n^m},\qquad |z|\lt 1 $$ Note that $$ \operatorname{Li}_1(z) = -\log(1-z) $$ and $$ \frac{d}{d z} \operatorname{Li}_m(z) = \operatorname{Li}_{m-1}(z) $$ So we get an analytic continuation $$ \operatorname{Li}_2(Z) = -\int_0^z \log(1-u) \frac{d u}{u} $$ where the path from $0$ to $z$ is in $\mathbb{C}\setminus [1,\infty)$ Functional equations: $$ \begin{gathered} \operatorname{Li}_1(1-x y) = \operatorname{Li}_1(1-x) + \operatorname{Li}_1(1-y)\\ \operatorname{Li}_2 = \text{5 terms (Spence 1809, Abel 1828, ...)} \end{gathered} $$ Monodromy (on $\operatorname{Li}_2(x),\log(x),1$) $$ \gamma_0=\begin{pmatrix}1&0&0\\0&1&2\pi i\\0&0&1\end{pmatrix}, \gamma_1=\begin{pmatrix}1&-2\pi i&0\\0&1&0\\0&0&1\end{pmatrix} $$ generate a Heisenberg group $$ \begin{pmatrix}1&\mathbb{Z}(1)&\mathbb{Z}(2)\\ 0&1&\mathbb{Z}(1)\\0&0&1\end{pmatrix} $$