Polylog

Polylog, in mathematics and physics, refers to the polylogarithm function of a complex order s and argument z, commonly written Li_s(z). It is defined by the power series Li_s(z) = sum_{k=1}^∞ z^k / k^s, which converges for |z| < 1 and can be extended by analytic continuation to a multi-valued function on the complex plane with a branch cut typically taken along the real axis from 1 to ∞.

For the unit disk, the series provides the principal branch of Li_s. Notable special cases include Li_1(z) = -ln(1−z). At z = 1, Li_s(1) = ζ(s) for Re(s) > 1, linking the polylogarithm to the Riemann zeta function. Li_s(−1) relates to the Dirichlet eta function. When s is a positive integer, Li_s is often referred to as the dilogarithm (s = 2), trilogarithm (s = 3), and so on. The polylogarithm also satisfies various functional equations and analytic continuation formulas, and it can be written in terms of the Lerch transcendent: Li_s(z) = Φ(z, s, 1).

In addition to its intrinsic mathematical interest, the polylogarithm appears in diverse applications. It arises in quantum field theory and statistical mechanics, for example in Fermi-Dirac and Bose-Einstein integrals, and in knot theory and algebraic geometry through special values and relations to volumes and regulators.

Overall, Li_s(z) provides a unifying framework for a family of special functions that generalize logarithms and zeta values, with rich analytic structure and broad interdisciplinary uses.