Editing Polylogarithm (section)

==Integral representations==
Any of the following integral representations furnishes the [[analytic continuation]] of the polylogarithm beyond the circle of convergence |''z''| = 1 of the defining power series.
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The polylogarithm can be expressed in terms of the integral of the [[Bose–Einstein distribution]]:
<math display="block">\operatorname{Li}_{s}(z) = {1 \over \Gamma(s)} \int_0^\infty {t^{s-1} \over e^t/z-1} dt .</math>

This converges for Re(''s'') >&nbsp;0 and all ''z'' except for ''z'' real and ≥&nbsp;1. The polylogarithm in this context is sometimes referred to as a Bose integral but more commonly as a '''Bose–Einstein integral''' ({{harvnb|Dingle|1957a}}, {{harvnb|Dingle|Arndt|Roy|1957}}).<ref group="note">Bose integral is result of multiplication between Gamma function and Zeta function.

One can begin with equation for Bose integral, then use series equation.

<math display="block">\int_{0}^{\infty}\frac{x^s}{e^x-1}dx = \int_{0}^{\infty}x^s\frac{1}{e^x-1}dx = \int_{0}^{\infty}\frac{x^s}{e^x}\frac{1}{1-\frac{1}{e^x}} dx\quad \wedge \quad \frac{1}{1-r} = \sum_{n=0}^{\infty}r^n</math>

<math display="block">\int_{0}^{\infty}\frac{x^s}{e^x}\sum_{n=0}^{\infty} \left(\frac{1}{e^x}\right)^n dx = \int_{0}^{\infty}\frac{x^s}{e^x}\sum_{n=0}^{\infty}e^{-nx}dx = \sum_{n=0}^{\infty}\int_{0}^{\infty}x^s e^{-nx} e^{-x}dx </math>

Secondly, regroup expressions.

<math display="block">\sum_{n=0}^{\infty}\int_{0}^{\infty}x^s e^{-(n+1)x}dx\quad\wedge\quad u=(n+1)x,du=(n+1)dx \Rightarrow dx=\frac{du}{n+1}</math>

<math display="block">\sum_{n=0}^{\infty}\int_{0}^{\infty}\left(\frac{u}{n+1}\right)^s e^{-u}\frac{du}{n+1} = \sum_{n=0}^{\infty}\int_{0}^{\infty}\frac{1}{(n+1)^{s+1}}u^s e^{-u}du</math>

<math display="block">\sum_{n=0}^{\infty}\frac{1}{(n+1)^{s+1}} \left(\int_{0}^{\infty} u^s e^{-u} du \right) = \left(\int_{0}^{\infty}u^s e^{-u}du \right) \left(\sum_{n=0}^{\infty}\frac{1}{(n+1)^{s+1}}\right)=</math>

<math display="block"> \left(\int_{0}^{\infty}u^{(s+1)-1}e^{-u}du\right) \left(\sum_{k=1}^{\infty}\frac{1}{k^{s+1}}\right) = \Gamma(s+1)\zeta (s+1).</math></ref> Similarly, the polylogarithm can be expressed in terms of the integral of the [[Fermi–Dirac distribution]]:
<math display="block">-\operatorname{Li}_{s}(-z) = \frac{1}{\Gamma(s)} \int_0^\infty {t^{s-1} \over e^t/z+1} dt .</math>

This converges for {{math|Re(''s'') > 0}} and all {{mvar|z}} except for ''z'' real and ≤&nbsp;−1. The polylogarithm in this context is sometimes referred to as a Fermi integral or a [[complete Fermi–Dirac integral|Fermi–Dirac integral]] ({{harvnb|GSL|2010}}, {{harvnb|Dingle|1957b}}). These representations are readily verified by [[Taylor series|Taylor expansion]] of the integrand with respect to ''z'' and termwise integration. The papers of Dingle contain detailed investigations of both types of integrals.

The polylogarithm is also related to the integral of the [[Maxwell–Boltzmann distribution]]:
<math display="block">
\lim_{z \to 0} \frac{\operatorname{Li}_{s}(z)}{z} = {1 \over \Gamma(s)}
\int_0^\infty {t^{s-1} e^{-t}} dt = 1 .
</math>

This also gives the [[asymptotic behavior]] of polylogarithm at the vicinity of origin.
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A complementary integral representation applies to Re(''s'') <&nbsp;0 and to all ''z'' except to ''z'' real and ≥&nbsp;0:
<math display="block">\operatorname{Li}_{s}(z) =\int_0^\infty {t^{-s} \sin[s \pi /2 - t \ln(-z)] \over \sinh(\pi t)} dt .</math>

This integral follows from the general relation of the polylogarithm with the [[Hurwitz zeta function]] ([[#Relationship to other functions|see above]]) and a familiar integral representation of the latter.
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The polylogarithm may be quite generally represented by a [[Hankel contour]] integral {{harv|Whittaker|Watson|1927|loc=§ 12.22, § 13.13}}, which extends the Bose–Einstein representation to negative orders ''s''. As long as the ''t'' = ''μ'' [[pole (complex analysis)|pole]] of the integrand does not lie on the non-negative real axis, and ''s'' ≠ 1, 2, 3, …, we have:
<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'' represents the Hankel contour. The integrand has a cut along the real axis from zero to infinity, with the axis belonging to the lower half plane of ''t''. The integration starts at +∞ on the upper half plane (Im(''t'') >&nbsp;0), circles the origin without enclosing any of the poles ''t'' = ''μ'' + 2''kπi'', and terminates at +∞ on the lower half plane (Im(''t'') <&nbsp;0). For the case where ''μ'' is real and non-negative, we can simply subtract the contribution of the enclosed ''t'' = ''μ'' pole:
<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 - 2\pi i R</math>

where ''R'' is the [[residue (complex analysis)|residue]] of the pole:
<math display="block">R = {i \over 2\pi} \Gamma(1 - s) (-\mu)^{s-1} .</math>
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When the [[Abel–Plana formula]] is applied to the defining series of the polylogarithm, a [[Charles Hermite|Hermite]]-type integral representation results that is valid for all complex ''z'' and for all complex ''s'':
<math display="block">\operatorname{Li}_s(z) = \tfrac{1}{2}z + {\Gamma(1 - s, -\ln z) \over (-\ln z)^{1-s}} + 2z \int_0^\infty \frac{\sin(s\arctan t - t\ln z)} {(1+t^2)^{s/2} (e^{2\pi t}-1)} dt</math>

where Γ is the [[incomplete gamma function|upper incomplete gamma-function]]. All (but not part) of the ln(''z'') in this expression can be replaced by −ln(<sup>1</sup>⁄<sub>''z''</sub>). A related representation which also holds for all complex ''s'',
<math display="block">\operatorname{Li}_s(z) = \tfrac{1}{2}z + z \int_0^\infty \frac{\sin[s \arctan t - t \ln(-z)]} {(1+t^2)^{s/2} \sinh(\pi t)} dt ,</math>

avoids the use of the incomplete gamma function, but this integral fails for ''z'' on the positive real axis if Re(''s'') ≤&nbsp;0. This expression is found by writing 2<sup>''s''</sup> Li<sub>''s''</sub>(−''z'') / (−''z'') = Φ(''z''<sup>2</sup>, ''s'', <sup>1</sup>⁄<sub>2</sub>) − ''z'' Φ(''z''<sup>2</sup>, ''s'', 1), where Φ is the [[Lerch transcendent]], and applying the Abel–Plana formula to the first Φ series and a complementary formula that involves 1 / (''e''<sup>2''πt''</sup> + 1) in place of 1 / (''e''<sup>2''πt''</sup> − 1) to the second Φ series.
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We can express an integral for the polylogarithm by integrating the ordinary [[geometric series]] termwise for <math>s \in \mathbb{N}</math> as {{harv|Borwein|Borwein|Girgensohn|1995|loc=§2, eqn. 4}}
<math display="block">\operatorname{Li}_{s+1}(z) = \frac{z \cdot (-1)^s}{s!} \int_0^1 \frac{\log^s(t)}{1-tz} dt. </math>
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