The Polylogarithm Function

This entry provides a definition of the Polylogarithm function, commonly denoted as $\text{Li}_s(z)$. Here, $z$ is a complex number and $s$ an integer parameter. This function can be defined by the power series expression $\text{Li}_s(z) = \sum_{k=1}^\infty \frac{z^k}{k^s}$ for $|z| < 1$ and analytically extended to the entire complex plane, except for a branch cut on $\mathbb{R}_{\geq 1}$.

Several basic properties are also proven, such as the relationship to the Eulerian polynomials via $\text{Li}_{-k}(z) = z (1 - z)^{k-1} A_k(z)$ for $k\geq 0$, the derivative formula $\frac{d}{dz} \text{Li}_s(z) = \frac{1}{z} \text{Li}_{s-1}(z)$, the relation to the “normal” logarithm via $\text{Li}_1(z) = -\ln (1 - z)$, and the duplication formula $\text{Li}_s(z) + \text{Li}_s(-z) = 2^{1-s} \text{Li}_s(z^2)$.