Polylogarithm

The polylogarithm is a special function denoted by Li_s(z), where s is a complex parameter and z is a complex argument. It generalizes the logarithm and the classical polylogarithm functions and is defined by the infinite series:

Li_s(z) = ∑_{k=1}^∞ (z^k) / (k^s),

which converges for complex |z| < 1 and can be analytically continued beyond this disk, except for certain branch points and singularities.

The polylogarithm arises frequently in various areas of mathematics and physics, including number theory, combinatorics, and quantum field theory. For real values of s and z within the unit disk, it connects to the Riemann zeta function, with the special case Li_s(1) equal to ζ(s) when the real part of s exceeds 1.

Several notable special cases include Li_1(z) = -ln(1 - z), the ordinary logarithm function, and Li_2(z), known as the dilogarithm, which appears in algebraic K-theory, polyhedral geometry, and hyperbolic geometry.

Analytic properties of the polylogarithm include functional equations, monodromy, and series expansions that facilitate computation. Various identities relate polylogarithms at different points and with different parameters, aiding in their evaluation and application.

Overall, the polylogarithm serves as a fundamental special function with extensive applications across mathematics and theoretical physics, providing a bridge between classical functions and advanced theoretical concepts.