---
tag: 2XB240D6MP
kind: definition
name: Polylogarithm
msc: ["33B30", "11G55"]
defines: 96CHDZKGQ0
uses: [0D6KXBHKC0, 0M3YQJ81NB]
sources:
  - url: https://en.wikipedia.org/wiki/Polylogarithm
---

## Definition

Let $s$ be a complex number.
For $z \in \C$ with $\abs{z} < 1$, the *polylogarithm of order $s$* is the [[power series|0D6KXBHKC0]]
$$\operatorname{Li}_s(z) = \sum_{k=1}^\infty \frac{z^k}{k^s},$$
together with its [[analytic continuation|0M3YQJ81NB]] in $z$ to $\C \setminus [1, \infty)$.

## Remarks

The series has radius of convergence $1$ for every $s$, because $\abs{k^{-s}}^{1/k} = k^{-\operatorname{Re} s/k} \to 1$.

Example: for $s = 1$ the series is $-\ln(1 - z)$, and for $s = 0$ and $s = -1$ it gives $\operatorname{Li}_0(z) = \frac{z}{1 - z}$ and $\operatorname{Li}_{-1}(z) = \frac{z}{(1 - z)^2}$.
For $\operatorname{Re} s > 1$ the series also converges at $z = 1$, and $\operatorname{Li}_s(1) = \zeta(s)$, the [[Riemann zeta function|1XWBY4N0VT]].

Non-example of a valid argument: for $s = 1$ the series diverges at $z = 1$ (it is the harmonic series), and $-\ln(1 - z)$ has a singularity there.

Termwise differentiation gives $z \frac{d}{dz} \operatorname{Li}_s(z) = \operatorname{Li}_{s-1}(z)$, so $\operatorname{Li}_{s+1}(z) = \int_0^z \frac{\operatorname{Li}_s(t)}{t}\, dt$.
For $s = 2$ this is the [[dilogarithm|G0X0NWSDB7]].