Polylogaritmit

Polylogaritmit, also known as polylogs or Landen's function, are a family of special functions denoted by Li_n(z), where n is a positive integer and z is a complex number. The polylogarithm is defined by the power series:

Li_n(z) = sum_{k=1 to infinity} z^k / k^n

This series converges for |z| < 1. For n = 1, the polylogarithm is the negative natural logarithm, Li_1(z) = -ln(1-z). For n = 2, it is the dilogarithm, Li_2(z), which has several interesting properties and connections to geometry and physics. Higher-order polylogarithms (n > 2) are generalizations of these functions.

The polylogarithm can be expressed in terms of an integral representation:

Li_n(z) = z * integral from 0 to 1 of (ln(1-zx)^(n-1) / (1-x)) dx

This integral representation allows the definition of the polylogarithm to be extended to values of z outside the unit circle, |z| > 1, by analytic continuation. Polylogarithms appear in various fields of mathematics and physics, including quantum field theory, statistical mechanics, and number theory. They are related to other special functions and play a role in the evaluation of certain integrals and series. The study of polylogarithms often involves exploring their functional equations, differential equations, and their relationships to other mathematical objects.