Polylogarithms

The classical polylogarithms are special functions of a complex variable $z$, defined by the series \begin{equation} \label{polylog} \operatorname{Li}_n(z) = \sum_{k=1}^{\infty} \frac{z^k}{k^n}, \end{equation} for $z$ a complex number inside the unit disk and $n$ an integer at least equal to $1$. These functions satisfy the differential equations \begin{equation} \frac{d}{dz} \operatorname{Li}_n(z) = \frac{\operatorname{Li}_{n-1}(z)}{z}, \end{equation} for $n\geq 2$. It follows that they can be extended by analytic continuation into multivalued functions on $\CC \setminus \{0,1\}$.

The classical polylogarithms are useful in many areas of mathematics, including hyperbolic geometry. They are in particular closely related to algebraic K-theory.

For $k=1$, the function $\operatorname{Li}_1(z)=-\log(1-z)$ is essentially the usual logarithm function.

For $k=2$, the function $\operatorname{Li}_2(z)$ is known as the dilogarithm. It has been studied by Spence, Abel and many others, see [z1]. It satisfies a five-term functional equation. The dilogarithm has several variants, including the Rogers dilogarithm and the Bloch-Wigner dilogarithm.

The classical polylogarithms can be seen as iterated integrals. As such, they are part of the larger class of multiple polylogarithms, which are functions of several complex variables, introduced by A.B. Goncharov.

References

• [l1] L. Lewin, "Polylogarithms and associated functions", Elsevier (1981) Zbl 0465.33001
• [z1] D. Zagier, "The dilogarithm function", Frontiers in number theory, physics, and geometry, vol II. Zbl 1176.11026