Geometry and String Theory
The Dilogarithm Function (Rev #1, changes)

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Define

Li 2(z)≔∑ n≥1z nn 2,|z|<1 \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,…m=1,2,\dots

Li m(z)≔∑ n≥1z nn m,|z|<1 \operatorname{Li}_m(z) \coloneqq \sum_{n\geq 1} \frac{z^n}{n^m},\qquad |z|\lt 1

Note that

Li 1(z)=−log(1−z) \operatorname{Li}_1(z) = -\log(1-z)

and

ddzLi m(z)=Li m−1(z) \frac{d}{d z} \operatorname{Li}_m(z) = \operatorname{Li}_{m-1}(z)

So we get an analytic continuation

Li 2(Z)=−∫ 0 zlog(1−u)duu \operatorname{Li}_2(Z) = -\int_0^z \log(1-u) \frac{d u}{u}

where the path from 00 to zz is in ℂ∖[1,∞)\mathbb{C}\setminus [1,\infty)

Functional equations:

Li 1(1−xy)=Li 1(1−x)+Li 1(1−y) Li−2=5 terms (Spence 1809, Abel 1828, ...) \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}
Revision from September 30, 2010 15:53:17 by Jacques Distler
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