Editing Polylogarithm (section)

==Asymptotic expansions==
For |''z''| ≫ 1, the polylogarithm can be expanded into [[asymptotic expansion|asymptotic series]] in terms of ln(−''z''):

<math display="block">\operatorname{Li}_s(z) = {\pm i\pi \over \Gamma(s)} [\ln(-z) \pm i\pi]^{s-1} - \sum_{k = 0}^\infty (-1)^k (2\pi)^{2k} {B_{2k} \over (2k)!} {[\ln(-z) \pm i\pi]^{s-2 k} \over \Gamma(s+1-2k)},</math>
<math display="block">\operatorname{Li}_s(z) = \sum_{k=0}^\infty (-1)^k (1-2^{1-2k}) (2\pi)^{2k} {B_{2k} \over (2k)!} {[\ln(-z)]^{s-2 k} \over \Gamma(s+1-2k)},</math>

where ''B''<sub>2''k''</sub> are the [[Bernoulli numbers]]. Both versions hold for all ''s'' and for any arg(''z''). As usual, the summation should be terminated when the terms start growing in magnitude. For negative integer ''s'', the expansions vanish entirely; for non-negative integer ''s'', they break off after a finite number of terms. {{harvtxt|Wood|1992|loc=§ 11}} describes a method for obtaining these series from the Bose–Einstein integral representation (his equation 11.2 for Li<sub>''s''</sub>(''e''<sup>''μ''</sup>) requires −2''π'' < Im(''μ'') ≤ 0).