Geometry and String Theory
The Dilogarithm Function (Rev #2)

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}

Revision from September 30, 2010 16:32:48 by Jacques Distler