Editing Polylogarithm (section)

==Particular values==
[[File:Polylogarithm plot negative.svg|right]]

For particular cases, the polylogarithm may be expressed in terms of other functions ([[#Relationship to other functions|see below]]). Particular values for the polylogarithm may thus also be found as particular values of these other functions.

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For integer values of the polylogarithm order, the following explicit expressions are obtained by repeated application of ''z''·∂/∂''z'' to Li<sub>1</sub>(''z''):
<math display="block">\operatorname{Li}_{1}(z) = -\ln(1-z)</math>
<math display="block">\operatorname{Li}_{0}(z) = {z \over 1-z}</math>
<math display="block">\operatorname{Li}_{-1}(z) = {z \over (1-z)^2}</math>
<math display="block">\operatorname{Li}_{-2}(z) = {z (1+z) \over (1-z)^3}</math>
<math display="block">\operatorname{Li}_{-3}(z) = {z (1+4z+z^2) \over (1-z)^4}</math>
<math display="block">\operatorname{Li}_{-4}(z) = {z (1+z) (1+10z+z^2) \over (1-z)^5} .</math>

Accordingly the polylogarithm reduces to a ratio of polynomials in ''z'', and is therefore a [[rational function]] of ''z'', for all nonpositive integer orders. The general case may be expressed as a finite sum:
<math display="block">\operatorname{Li}_{-n}(z) = \left(z {\partial \over \partial z} \right)^n {z \over {1-z}} = \sum_{k=0}^n k! S(n+1, k+1) \left({z \over {1-z}} \right)^{k+1} \qquad (n=0,1,2,\ldots),</math>
where ''S''(''n'',''k'') are the [[Stirling numbers of the second kind]]. Equivalent formulae applicable to negative integer orders are {{harv|Wood|1992|loc=§ 6}}:
<math display="block">\operatorname{Li}_{-n}(z) = (-1)^{n+1} \sum_{k=0}^n k! S(n+1, k+1) \left({{-1} \over {1-z}} \right)^{k+1} \qquad (n=1,2,3,\ldots),</math>

and:
<math display="block">\operatorname{Li}_{-n}(z) = {1 \over (1-z)^{n+1}} \sum_{k=0}^{n-1} \left\langle {n \atop k} \right\rangle z^{n-k} \qquad (n=1,2,3,\ldots),</math>

where <math>\scriptstyle \left\langle {n \atop k} \right\rangle</math> are the [[Eulerian numbers]]. All roots of Li<sub>−''n''</sub>(''z'') are distinct and real; they include ''z'' = 0, while the remainder is negative and centered about ''z'' = −1 on a logarithmic scale. As ''n'' becomes large, the numerical evaluation of these rational expressions increasingly suffers from cancellation {{harv|Wood|1992|loc=§ 6}}; full accuracy can be obtained, however, by computing Li<sub>−''n''</sub>(''z'') via the general relation with the Hurwitz zeta function ([[#Relationship to other functions|see below]]).
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Some particular expressions for half-integer values of the argument ''z'' are:
<math display="block">\operatorname{Li}_1(\tfrac12) = \ln 2</math>
<math display="block">\operatorname{Li}_2(\tfrac12) = \tfrac1{12} \pi^2 - \tfrac12 (\ln 2)^2</math>
<math display="block">\operatorname{Li}_3(\tfrac12) = \tfrac16 (\ln 2)^3 - \tfrac1{12} \pi^2 \ln 2 + \tfrac78 \zeta(3) ,</math>

where ''ζ'' is the [[Riemann zeta function]]. No formulae of this type are known for higher integer orders {{harv|Lewin|1991|p=2}}, but one has for instance {{harv|Borwein|Borwein|Girgensohn|1995}}:
<math display="block">\operatorname{Li}_4(\tfrac12) = \tfrac 1{360} \pi^4 - \tfrac 1{24}(\ln 2)^4 + \tfrac1{24} \pi^2 (\ln 2)^2 - \tfrac 1 2 \zeta(\bar3, \bar1),</math>

which involves the alternating double sum 
<math display="block">\zeta(\bar3, \bar1)=\sum_{m>n>0} (-1)^{m+n} m^{-3} n^{-1}.</math>

In general one has for integer orders ''n'' ≥ 2 {{harv|Broadhurst|1996|p=9}}:
<math display="block">\operatorname{Li}_n(\tfrac12) = -\zeta(\bar1, \bar1, \left\{ 1 \right\}^{n-2}),</math>

where ''ζ''(''s''<sub>1</sub>, …, ''s''<sub>''k''</sub>) is the [[multiple zeta function]]; for example:
<math display="block">\operatorname{Li}_5(\tfrac12) = -\zeta(\bar1, \bar1, 1,1,1).</math>
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As a straightforward consequence of the series definition, values of the polylogarithm at the ''p''th complex [[roots of unity]] are given by the [[discrete Fourier transform|Fourier sum]]:
<math display="block"> \operatorname{Li}_s(e^{2 \pi i m/p}) = p^{-s} \sum_{k=1}^p e^{2 \pi i m k/p} \zeta(s, \tfrac {k}{p}) \qquad (m = 1, 2, \dots, p-1),</math>

where ''ζ'' is the [[Hurwitz zeta function]]. For Re(''s'') > 1, where Li<sub>''s''</sub>(1) is finite, the relation also holds with ''m'' = 0 or ''m'' = ''p''. While this formula is not as simple as that implied by the more general relation with the Hurwitz zeta function listed under [[#Relationship to other functions|relationship to other functions]] below, it has the advantage of applying to non-negative integer values of ''s'' as well. As usual, the relation may be inverted to express ζ(''s'', <sup>''m''</sup>⁄<sub>''p''</sub>) for any ''m'' = 1, …, ''p'' as a Fourier sum of Li<sub>''s''</sub>(exp(2''πi'' <sup>''k''</sup>⁄<sub>''p''</sub>)) over ''k'' = 1, …, ''p''.
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