For questions about or related to polylogarithm functions.

The polylogarithm function $\operatorname{Li}_s$ is defined by the infinite sum

\begin{align*} \sum_{k = 1}^{\infty} \frac{z^k}{k^s} \end{align*}

for all $|z| < 1$ and complex order $s$, and obtained by analytic continuation of the sum. Depending on the order $s$, a branch cut must be taken for the logarithm.

In particular cases, the polylogarithm may have simpler representations; for example,

\begin{align*} \operatorname{Li}_1(z) &= -\ln{(1 - z)} \\ \operatorname{Li}_0(z) &= \frac{z}{1 - z} \\ \end{align*}

In the cases $s = 2$ and $s = 3$, the function is called the dilogarithm and trilogarithm, respectively.

The polylogarithm functions arise in quantum statistics and electrodynamics, and are related to the Fermi-Dirac integral.