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The polylogarithm is a generalization of the Riemann Zeta Function and is a special case of the Lerch Transcendent, defined for $\set {z \in \C: 0 \le z < 1}$ as the series:

$\ds \Li_s \paren z = \sum_{n \mathop = 1}^\infty \frac {z^n} {n^s}$

Analytic Continuation

By Analytic Continuations of Polylogarithm, $\Li_s \paren z$ has a unique analytic continuation to $\C$.

This analytic continuation is still called the polylogarithm and still denoted $\Li_s \paren z$.