Editing Polylogarithm (section)

==Properties==
In the case where the order <math>s</math> is an integer, it will be represented by <math>s=n</math> (or <math>s=-n</math> when negative). It is often convenient to define <math>\mu=\ln(z)</math> where <math>\ln(z)</math> is the [[principal branch]] of the [[complex logarithm]] <math>\operatorname{Ln}(z)</math> so that <math>-\pi< \operatorname{Im}(\mu) \le \pi.</math> Also, all exponentiation will be assumed to be single-valued: <math>z^s = \exp(s\ln(z)).</math>

Depending on the order <math>s</math>, the polylogarithm may be multi-valued. The ''principal branch'' of <math>\operatorname{Li}_s(z)</math> is taken to be given for <math>| z | < 1</math> by the above series definition and taken to be continuous except on the positive real axis, where a cut is made from <math>z = 1</math> to <math>\infty</math> such that the axis is placed on the lower half plane of {{nowrap|<math>z</math>.}} In terms of {{nowrap|<math>\mu</math>,}} this amounts to <math>-\pi < \arg(-\mu)\le \pi </math>. The discontinuity of the polylogarithm in dependence on <math>\mu</math> can sometimes be confusing.

For real argument <math>z</math>, the polylogarithm of real order <math>s</math> is real if {{nowrap|<math>z < 1</math>,}} and its imaginary part for <math>z \ge 1</math> is {{harv|Wood|1992|loc=§3}}:

<math display="block">\operatorname{Im}\left( \operatorname{Li}_s(z) \right) = -{{\pi \mu^{s-1}}\over{\Gamma(s)}}.</math>

Going across the cut, if ''ε'' is an infinitesimally small positive real number, then:

<math display="block">\operatorname{Im}\left( \operatorname{Li}_s(z+i\epsilon) \right) = {{\pi \mu^{s-1}}\over{\Gamma(s)}}.</math>

Both can be concluded from the series expansion ([[#Series representations|see below]]) of {{nowrap|Li<sub>''s''</sub>(''e''{{i sup|''μ''}})}} about {{nowrap|1=''μ'' = 0.}}

The derivatives of the polylogarithm follow from the defining power series:

<math display="block">z \frac{\partial \operatorname{Li}_s(z) }{ \partial z} = \operatorname{Li}_{s-1}(z)</math>
<math display="block">\frac{\partial \operatorname{Li}_s(e^\mu) }{ \partial \mu} = \operatorname{Li}_{s-1}(e^\mu).</math>

The square relationship is seen from the series definition, and is related to the [[duplication formula]] (see also {{harvtxt|Clunie|1954}}, {{harvtxt|Schrödinger|1952}}):

<math display="block">\operatorname{Li}_s(-z) + \operatorname{Li}_s(z) = 2^{1-s} \operatorname{Li}_s(z^2).</math>

[[Kummer's function]] obeys a very similar duplication formula. This is a special case of the [[multiplication formula]], for any positive integer ''p'':

<math display="block">\sum_{m=0}^{p-1} \operatorname{Li}_s(z e^{2\pi i m/p}) = p^{1-s} \operatorname{Li}_s(z^p),</math>

which can be proved using the series definition of the polylogarithm and the orthogonality of the exponential terms (see e.g. [[discrete Fourier transform]]).

Another important property, the inversion formula, involves the [[Hurwitz zeta function]] or the [[Bernoulli polynomials]] and is found under [[#Relationship to other functions|relationship to other functions]] below.