Editing Polylogarithm (section)

==Series representations==
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As noted under [[#Integral representations|integral representations]] above, the Bose–Einstein integral representation of the polylogarithm may be extended to negative orders ''s'' by means of [[Hankel contour]] integration:
<math display="block">\operatorname{Li}_s(e^\mu) = -{\Gamma(1 - s) \over 2\pi i} \oint_H {(-t)^{s-1} \over e^{t-\mu}-1} dt,</math>

where ''H'' is the Hankel contour, ''s'' ≠ 1, 2, 3, …, and the ''t'' = ''μ'' pole of the integrand does not lie on the non-negative real axis. The [[methods of contour integration|contour]] can be modified so that it encloses the [[pole (complex analysis)|poles]] of the integrand at ''t'' − ''μ'' = 2''kπi'', and the integral can be evaluated as the sum of the [[residue (complex analysis)|residues]] ({{harvnb|Wood|1992|loc=§ 12, 13}}; {{harvnb|Gradshteyn|Ryzhik|2015}}):
<math display="block">\operatorname{Li}_s(e^\mu) = \Gamma(1 - s) \sum_{k=-\infty}^\infty (2k \pi i - \mu)^{s-1}.</math>

This will hold for Re(''s'') < 0 and all ''μ'' except where ''e''<sup>''μ''</sup> = 1. For 0 < Im(''μ'') ≤ 2''π'' the sum can be split as:
<math display="block">\operatorname{Li}_s(e^\mu) = \Gamma(1-s) \left[ (-2\pi i)^{s-1} \sum_{k=0}^\infty \left(k + {\mu \over {2\pi i}} \right)^{s-1} + (2\pi i)^{s-1} \sum_{k=0}^\infty \left(k+1- {\mu \over {2\pi i}} \right)^{s-1} \right],</math>

where the two series can now be identified with the [[Hurwitz zeta function]]:
<math display="block">\operatorname{Li}_s(e^\mu) = {\Gamma(1 - s) \over (2\pi)^{1-s}} \left[i^{1-s} ~\zeta \left(1 - s, ~{\mu \over {2\pi i}} \right) + i^{s-1} ~\zeta \left(1 - s, ~1 - {\mu \over {2\pi i}} \right) \right] \qquad (0 < \operatorname{Im}(\mu) \leq 2\pi) .</math>

This relation, which has already been given under [[#Relationship to other functions|relationship to other functions]] above, holds for all complex ''s'' ≠ 0, 1, 2, 3, … and was first derived in {{harv|Jonquière|1889|loc=eq. 6}}.
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In order to represent the polylogarithm as a power series about ''μ'' = 0, we write the series derived from the Hankel contour integral as:
<math display="block">\operatorname{Li}_s(e^\mu) = \Gamma(1 - s) (-\mu)^{s-1} + \Gamma(1 - s) \sum_{h=1}^\infty \left[(-2 h \pi i - \mu)^{s-1} + (2 h \pi i - \mu)^{s-1} \right] .</math>

When the binomial powers in the sum are expanded about ''μ'' = 0 and the order of summation is reversed, the sum over ''h'' can be expressed in closed form:
<math display="block">\operatorname{Li}_s(e^\mu) = \Gamma(1 - s) (-\mu)^{s-1} + \sum_{k=0}^\infty {\zeta(s-k) \over k!} \mu^k .</math>

This result holds for |''μ''| < 2''π'' and, thanks to the analytic continuation provided by the [[Riemann zeta function|zeta functions]], for all ''s'' ≠ 1, 2, 3, … . If the order is a positive integer, ''s'' = ''n'', both the term with ''k'' = ''n'' − 1 and the [[gamma function]] become infinite, although their sum does not. One obtains ({{harvnb|Wood|1992|loc=§ 9}}; {{harvnb|Gradshteyn|Ryzhik|2015}}):
<math display="block">\lim_{s \to k+1} \left[ {\zeta(s-k) \over k!} \mu^k + \Gamma(1 - s) (-\mu)^{s-1} \right] = {\mu^k \over k!} \left[\sum_{h=1}^k {1 \over h} - \ln(-\mu) \right],</math>

where the sum over ''h'' vanishes if ''k'' = 0. So, for positive integer orders and for |''μ''| < 2''π'' we have the series:
<math display="block">\operatorname{Li}_{n}(e^\mu) = {\mu^{n-1} \over (n-1)!} \left[ H_{n-1} - \ln(-\mu) \right] + \sum_{k=0,k\ne n-1}^\infty {\zeta(n-k) \over k!} \mu^k ,</math>

where ''H''<sub>''n''</sub> denotes the ''n''th [[harmonic number]]:
<math display="block">H_n = \sum_{h=1}^n {1 \over h}, \qquad H_0 = 0.</math>

The problem terms now contain −ln(−''μ'') which, when multiplied by ''μ''<sup>''n''−1</sup>, will tend to zero as ''μ'' → 0, except for ''n'' = 1. This reflects the fact that Li<sub>''s''</sub>(''z'') exhibits a true [[mathematical singularity|logarithmic singularity]] at ''s'' = 1 and ''z'' = 1 since:
<math display="block">\lim_{\mu \to 0} \Gamma(1-s)(-\mu)^{s-1} = 0 \qquad (\operatorname{Re}(s) > 1).</math>

For ''s'' close, but not equal, to a positive integer, the divergent terms in the expansion about ''μ'' = 0 can be expected to cause computational difficulties {{harv|Wood|1992|loc=§ 9}}. Erdélyi's corresponding expansion {{harv|Erdélyi et al.|1981|loc=§ 1.11-15}} in powers of ln(''z'') is not correct if one assumes that the principal branches of the polylogarithm and the logarithm are used simultaneously, since ln(<sup>1</sup>⁄<sub>''z''</sub>) is not uniformly equal to −ln(''z'').

For nonpositive integer values of ''s'', the zeta function ζ(''s'' − ''k'') in the expansion about ''μ'' = 0 reduces to [[Bernoulli numbers]]: ζ(−''n'' − ''k'') = −B<sub>1+''n''+''k''</sub> / (1 + ''n'' + ''k''). Numerical evaluation of Li<sub>−''n''</sub>(''z'') by this series does not suffer from the cancellation effects that the finite rational expressions given under [[#Particular values|particular values]] above exhibit for large ''n''.
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By use of the identity
<math display="block">1 = {1 \over \Gamma(s)} \int_0^\infty e^{-t} t^{s-1} dt \qquad (\operatorname{Re}(s) > 0) ,</math>

the Bose–Einstein integral representation of the polylogarithm ([[#Integral representations|see above]]) may be cast in the form:
<math display="block">\operatorname{Li}_s(z) = \tfrac{1}{2}z + {z \over 2 \Gamma(s)} \int_0^\infty e^{-t} t^{s-1} \coth{t - \ln z \over 2} dt \qquad (\operatorname{Re}(s) > 0).</math>

Replacing the hyperbolic cotangent with a bilateral series,
<math display="block">\coth{t-\ln z \over 2} = 2 \sum_{k = -\infty}^\infty {1 \over 2 k \pi i + t - \ln z} ,</math>

then reversing the order of integral and sum, and finally identifying the summands with an integral representation of the [[incomplete gamma function|upper incomplete gamma function]], one obtains:
<math display="block">\operatorname{Li}_s(z) = \tfrac{1}{2}z + \sum_{k = -\infty}^\infty {\Gamma(1-s, 2k \pi i - \ln z) \over (2k \pi i - \ln z)^{1-s}}.</math>

For both the bilateral series of this result and that for the hyperbolic cotangent, symmetric partial sums from −''k''<sub>max</sub> to ''k''<sub>max</sub> converge unconditionally as ''k''<sub>max</sub> → ∞. Provided the summation is performed symmetrically, this series for Li<sub>''s''</sub>(''z'') thus holds for all complex ''s'' as well as all complex ''z''.
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Introducing an explicit expression for the [[Stirling numbers of the second kind]] into the finite sum for the polylogarithm of nonpositive integer order ([[#Particular values|see above]]) one may write:
<math display="block">\operatorname{Li}_{-n}(z) = \sum_{k=0}^n \left( {-z \over 1-z} \right)^{k+1} \sum_{j=0}^k (-1)^{j+1} {k \choose j} (j+1)^n \qquad (n=0,1,2,\ldots).</math>

The infinite series obtained by simply extending the outer summation to ∞ {{harv|Guillera|Sondow|2008|loc=Theorem 2.1}}:
<math display="block">\operatorname{Li}_s(z) = \sum_{k=0}^\infty \left( {-z \over 1-z} \right)^{k+1} ~\sum_{j=0}^k (-1)^{j+1} {k \choose j} (j+1)^{-s} ,</math>

turns out to converge to the polylogarithm for all complex ''s'' and for complex ''z'' with Re(''z'') < <sup>1</sup>⁄<sub>2</sub>, as can be verified for |<sup>−''z''</sup>⁄<sub>(1−''z'')</sub>| < <sup>1</sup>⁄<sub>2</sub> by reversing the order of summation and using:
<math display="block">\sum_{k=j}^\infty {k \choose j} \left( {-z \over 1-z} \right)^{k+1} = \left[ \left( {-z \over 1-z} \right)^{-1} -1 \right]^{-j-1} = (-z)^{j+1}.</math>

The inner coefficients of these series can be expressed by [[Stirling numbers of the first kind|Stirling-number-related]] formulas involving the generalized [[harmonic numbers]]. For example, see [[Generating function transformation#Derivative transformations|generating function transformations]] to find proofs (references to proofs) of the following identities:
<math display="block">\begin{align}
\operatorname{Li}_2(z) &= \sum_{j \geq 1} \frac{(-1)^{j-1}}{2} \left(H_j^2+H_j^{(2)}\right) \frac{z^j}{(1-z)^{j+1}} \\ 
\operatorname{Li}_3(z) &= \sum_{j \geq 1} \frac{(-1)^{j-1}}{6} \left(H_j^3+3H_j H_j^{(2)} + 2 H_j^{(3)}\right) \frac{z^j}{(1-z)^{j+1}}. 
\end{align}</math>

For the other arguments with Re(''z'') < <sup>1</sup>⁄<sub>2</sub> the result follows by [[analytic continuation]]. This procedure is equivalent to applying [[binomial transform|Euler's transformation]] to the series in ''z'' that defines the polylogarithm.
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