Editing Polylogarithm

{{Short description|Special mathematical function}}
{{Distinguish|polylogarithmic function|logarithmic integral function}}
{{Use American English|date = March 2019}}

In [[mathematics]], the '''polylogarithm''' (also known as '''Jonquière's function''', for Alfred Jonquière) is a [[special function]] {{math|Li{{sub|''s''}}(''z'')}} of order {{mvar|s}} and argument {{mvar|z}}. Only for special values of {{mvar|s}} does the polylogarithm reduce to an [[elementary function]] such as the [[natural logarithm]] or a [[rational function]]. In [[quantum statistics]], the polylogarithm function appears as the closed form of [[integral]]s of the [[Fermi–Dirac distribution]] and the [[Bose–Einstein distribution]], and is also known as the '''Fermi–Dirac integral''' or the '''Bose–Einstein integral'''. In [[quantum electrodynamics]], polylogarithms of positive [[integer]] order arise in the calculation of processes represented by higher-order [[Feynman diagram]]s.

The polylogarithm function is equivalent to the [[Hurwitz zeta function]] — either [[Function (mathematics)|function]] can be expressed in terms of the other — and both functions are special cases of the [[Lerch transcendent]]. Polylogarithms should not be confused with [[polylogarithmic function]]s, nor with the [[offset logarithmic integral]] {{math|Li(''z'')}}, which has the same notation without the subscript.

<gallery mode="packed" heights="140px" caption="Different polylogarithm functions in the complex plane">
File:Complex polylogminus3.jpg|{{math|Li{{hairsp}}{{sub|–3}}(''z'')}}
File:Complex polylogminus2.jpg|{{math|Li{{hairsp}}{{sub|–2}}(''z'')}}
File:Complex polylogminus1.jpg|{{math|Li{{hairsp}}{{sub|–1}}(''z'')}}
File:Complex polylog0.jpg|{{math|Li{{sub|0}}(''z'')}}
File:Complex polylog1.jpg|{{math|Li{{sub|1}}(''z'')}}
File:Complex polylog2.jpg|{{math|Li{{sub|2}}(''z'')}}
File:Complex polylog3.jpg|{{math|Li{{sub|3}}(''z'')}}
</gallery>
The polylogarithm function is defined by a [[power series]] in {{mvar|z}}, which is also a [[Dirichlet series]] in {{mvar|s}}:
<math display="block">\operatorname{Li}_s(z) = \sum_{k=1}^\infty {z^k \over k^s} = z + {z^2 \over 2^s} + {z^3 \over 3^s} + \cdots</math>

This definition is valid for arbitrary [[Complex numbers|complex]] order {{mvar|s}} and for all complex arguments {{mvar|z}} with {{math|{{abs|''z''}} < 1}}; it can be extended to {{math|{{abs|''z''}} ≥ 1}} by the process of [[analytic continuation]]. (Here the denominator {{mvar|k{{sup|s}}}} is understood as {{math|exp(''s'' ln ''k'')}}). The special case {{math|1=''s'' = 1}} involves the ordinary [[natural logarithm]], {{math|1=Li{{sub|1}}(''z'') = −ln(1−''z'')}}, while the special cases {{math|1=''s'' = 2}} and {{math|1=''s'' = 3}} are called the [[dilogarithm]] (also referred to as Spence's function) and trilogarithm respectively. The name of the function comes from the fact that it may also be defined as the repeated [[indefinite integral|integral]] of itself:
<math display="block">\operatorname{Li}_{s+1}(z) = \int_0^z \frac{\operatorname{Li}_s(t)}{t} dt</math>
thus the dilogarithm is an integral of a function involving the logarithm, and so on. For nonpositive integer orders {{mvar|s}}, the polylogarithm is a [[rational function]].

==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.

==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.

<ol>
<li>
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]]).
</li>
<li>
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>
</li>
<li>
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''.
</li>
</ol>

==Relationship to other functions==

* For {{math|1=''z'' = 1}}, the polylogarithm reduces to the [[Riemann zeta function]] <math display="block">\operatorname{Li}_s(1) = \zeta(s) \qquad (\operatorname{Re}(s)>1).</math>
* The polylogarithm is related to [[Dirichlet eta function]] and the [[Dirichlet beta function]]: <math display="block"> \operatorname{Li}_s(-1) = -\eta(s),</math> where {{math|''η''(''s'')}} is the Dirichlet eta function. For pure imaginary arguments, we have: <math display="block">\operatorname{Li}_s(\pm i) = -2^{-s} \eta(s) \pm i\beta(s),</math> where {{math|''β''(''s'')}} is the Dirichlet beta function.
* The polylogarithm is related to the [[complete Fermi–Dirac integral]] as: <math display="block">F_s(\mu) = -\operatorname{Li}_{s+1}(-e^\mu).</math>
* The polylogarithm is related to the complete Bose–Einstein integral as: <math display="block">G_s(\mu) = \operatorname{Li}_{s+1}(e^\mu).</math>
* The polylogarithm is a special case of the [[incomplete polylogarithm]] function <math display="block"> \operatorname{Li}_s(z) = \operatorname{Li}_s(0,z) .</math>
* The polylogarithm is a special case of the [[Lerch transcendent]] {{harv|Erdélyi et al.|1981|loc=§ 1.11-14}} <math display="block">\operatorname{Li}_s(z) = z\Phi(z,s,1).</math>
* The polylogarithm is related to the [[Hurwitz zeta function]] by: <math display="block">\operatorname{Li}_s(z) = {\Gamma(1 - s) \over (2\pi)^{1-s}} \left[i^{1-s} \zeta \left(1 - s, \frac{1}{2} + {\ln(-z) \over {2\pi i}} \right) + i^{s-1} ~\zeta \left(1 - s, \frac{1}{2} - {\ln(-z) \over {2\pi i}} \right) \right],</math> which relation, however, is invalidated at positive integer ''s'' by [[pole (complex analysis)|poles]] of the [[gamma function]] {{math|Γ(1 − ''s'')}}, and at {{math|1=''s'' = 0}} by a pole of both zeta functions; a derivation of this formula is given under [[#Series representations|series representations]] below. With a little help from a functional equation for the Hurwitz zeta function, the polylogarithm is consequently also related to that function via {{harv|Jonquière|1889}}: <math display="block">
i^{-s} \operatorname{Li}_s(e^{2\pi i x}) + i^s \operatorname{Li}_s(e^{-2\pi i x}) = \frac{(2\pi)^s}{\Gamma(s)} \zeta(1 - s, x),
</math> which relation holds for {{math|0 ≤ Re(''x'') < 1}} if {{math|Im(''x'') ≥ 0}}, and for {{math|0 < Re(''x'') ≤ 1}} if {{math|Im(''x'') < 0}}. Equivalently, for all complex ''s'' and for complex {{math|''z'' ∉ {{open-closed|0, 1}}}}, the inversion formula reads <math display="block">
\operatorname{Li}_s(z) + (-1)^s \operatorname{Li}_s(1/z) = \frac{(2\pi i)^s}{\Gamma(s)} ~\zeta \left(1 - s, ~\frac{1}{2} + {\ln(-z) \over {2\pi i}} \right),
</math> and for all complex ''s'' and for complex {{math|''z'' ∉ {{open-open|1, ∞}}}} <math display="block"> \operatorname{Li}_s(z) + (-1)^s \operatorname{Li}_s(1/z) = {(2\pi i)^s \over \Gamma(s)} ~\zeta \left(1 - s, ~\frac{1}{2} - {\ln(-1/z) \over {2\pi i}} \right) .
</math> For {{math|''z'' ∉ {{open-open|0, ∞}}}}, one has {{math|1=ln(−''z'') = −ln(−<sup>1</sup>⁄<sub>''z''</sub>)}}, and both expressions agree. These relations furnish the analytic continuation of the polylogarithm beyond the circle of convergence |''z''| = 1 of the defining power series. (The corresponding equation of {{harvtxt|Jonquière|1889|loc=eq. 5}} and {{harvtxt|Erdélyi et al.|1981|loc=§ 1.11-16}} is not correct if one assumes that the principal branches of the polylogarithm and the logarithm are used simultaneously.) See the next item for a simplified formula when ''s'' is an integer.
* For positive integer polylogarithm orders ''s'', the Hurwitz zeta function ζ(1−''s'', ''x'') reduces to [[Bernoulli polynomials]], {{math|1=ζ(1−''n'', ''x'') = −B<sub>''n''</sub>(''x'') / ''n''}}, and Jonquière's inversion formula for ''n'' = 1, 2, 3, … becomes: <math display="block">\operatorname{Li}_{n}(e^{2\pi i x}) + (-1)^n \operatorname{Li}_{n}(e^{-2\pi i x}) = -{(2\pi i)^n \over n!} B_n(x),</math> where again 0 ≤ Re(''x'') < 1 if Im(''x'') ≥ 0, and 0 < Re(''x'') ≤ 1 if Im(''x'') < 0. Upon restriction of the polylogarithm argument to the unit circle, Im(''x'') = 0, the left hand side of this formula simplifies to 2 Re(Li<sub>''n''</sub>(''e''<sup>2''πix''</sup>)) if ''n'' is even, and to 2''i'' Im(Li<sub>''n''</sub>(''e''<sup>2''πix''</sup>)) if ''n'' is odd. For negative integer orders, on the other hand, the divergence of Γ(''s'') implies for all ''z'' that {{harv|Erdélyi et al.|1981|loc=§ 1.11-17}}: <math display="block">\operatorname{Li}_{-n}(z) + (-1)^n \operatorname{Li}_{-n}(1/z) = 0 \qquad (n = 1,2,3,\ldots).
</math> More generally, one has for {{math|1=''n'' = 0, ±1, ±2, ±3, …}}: <math display="block">\begin{align}
\operatorname{Li}_{n}(z) + (-1)^n \operatorname{Li}_{n}(1/z) &= -\frac{(2\pi i)^n}{n!} B_n \left( \frac{1}{2} + {\ln(-z) \over {2\pi i}} \right) & (z \not\in ]0;1]), \\
\operatorname{Li}_{n}(z) + (-1)^n \operatorname{Li}_{n}(1/z) &= -\frac{(2\pi i)^n}{n!} B_n \left( \frac{1}{2} - {\ln(-1/z) \over {2\pi i}} \right) & (z \not\in ~]1;\infty[),
\end{align}</math> where both expressions agree for {{math|''z'' ∉ {{open-open|0, ∞}}}}. (The corresponding equation of {{harvtxt|Jonquière|1889|loc=eq. 1}} and {{harvtxt|Erdélyi et al.|1981|loc=§ 1.11-18}} is again not correct.)
* The polylogarithm with pure imaginary ''μ'' may be expressed in terms of the [[Clausen function]]s ''Ci''<sub>''s''</sub>(θ) and ''Si''<sub>''s''</sub>(θ), and vice versa ({{harvnb|Lewin|1958|loc=Ch. VII § 1.4}}; {{harvnb|Abramowitz|Stegun|1972|loc=§ 27.8}}):<math display="block">\operatorname{Li}_s(e^{\pm i \theta}) = Ci_s(\theta) \pm i Si_s(\theta).</math>
* The [[inverse tangent integral]] {{math|''Ti''<sub>''s''</sub>(''z'')}} {{harv|Lewin|1958|loc=Ch. VII § 1.2}} can be expressed in terms of polylogarithms: <math display="block">\operatorname{Ti}_s(z) = {1 \over 2i} \left[ \operatorname{Li}_s(i z) - \operatorname{Li}_s(-i z) \right].</math> The relation in particular implies: <math display="block">\operatorname{Ti}_0(z) = {z \over 1+z^2}, \quad \operatorname{Ti}_1(z) = \arctan z, \quad \operatorname{Ti}_2(z) = \int_0^z {\arctan t \over t} dt, \quad \ldots~ \quad \operatorname{Ti}_{n+1}(z) = \int_0^z \frac{\operatorname{Ti}_n(t)}{t} dt,</math> which explains the function name.
* The [[Legendre chi function]] ''χ''<sub>''s''</sub>(''z'') ({{harvnb|Lewin|1958|loc=Ch. VII § 1.1}}; {{harvnb|Boersma|Dempsey|1992}}) can be expressed in terms of polylogarithms: <math display="block">
\chi_s(z) = \tfrac {1}{2} \left[ \operatorname{Li}_s(z) - \operatorname{Li}_s(-z) \right].</math>
* The polylogarithm of integer order can be expressed as a [[generalized hypergeometric function]]: <math display="block">\begin{align}
\operatorname{Li}_n(z) &= z _{n+1}F_{n} (1,1,\dots,1; 2,2,\dots,2; z) & (n = 0,1,2,\ldots), \\
\operatorname{Li}_{-n}(z) &= z _{n}F_{n-1} (2,2,\dots,2; 1,1,\dots,1; z) & (n = 1,2,3,\ldots) ~.
\end{align}</math>
* In terms of the [[Riemann zeta function#Generalizations|incomplete zeta functions]] or "[[Debye function]]s" {{harv|Abramowitz|Stegun|1972|loc=§ 27.1}}: <math display="block">
Z_n(z) = {1 \over (n - 1)!} \int_z^\infty {t^{n-1} \over e^t-1} dt \qquad (n = 1,2,3,\ldots) ,
</math> the polylogarithm Li<sub>''n''</sub>(''z'') for positive integer n may be expressed as the finite sum {{harv|Wood|1992|loc=§16}}: <math display="block">
\operatorname{Li}_{n}(e^\mu) = \sum_{k=0}^{n-1} Z_{n-k}(-\mu) {\mu^k \over k!} \qquad (n = 1,2,3,\ldots) .
</math> A remarkably similar expression relates the "Debye functions" ''Z''<sub>''n''</sub>(''z'') to the polylogarithm: <math display="block">Z_n(z) = \sum_{k=0}^{n-1} \operatorname{Li}_{n-k}(e^{-z}) {z^k \over k!} \qquad (n = 1,2,3,\ldots) .</math>
* Using [[Lambert series]], if <math>J_s(n)</math> is [[Jordan's totient function]], then <math display="block">
\sum_{n=1}^\infty\frac{z^nJ_{-s}(n)}{1-z^n}=\operatorname{Li}_{s}(z).</math>

==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.
<ol>
<li>
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.
</li>
<li>
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.
</li>
<li>
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>
</li>
<li>
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.
</li>
<li>
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>
</li>
</ol>

==Series representations==
<ol>
<li>
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}}.
</li>
<li>
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''.
</li>
<li>
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''.
</li>
<li>
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.
</li>
</ol>

==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).

==Limiting behavior==
The following [[limit (mathematics)|limits]] result from the various representations of the polylogarithm {{harv|Wood|1992|loc=§ 22}}:

<math display="block">\operatorname{Li}_s(z) \sim_{|z|\to 0} z</math>
<math display="block">\operatorname{Li}_s(e^\mu) \sim_{|\mu|\to 0} \Gamma(1 - s) (-\mu)^{s-1} \qquad (\operatorname{Re}(s) < 1)</math>
<math display="block">\operatorname{Li}_s(\pm e^\mu) \sim_{\operatorname{Re}(\mu) \to \infty} -{\mu^s \over \Gamma(s+1)}\qquad (s \ne -1, -2, -3, \ldots)</math>
<math display="block">\operatorname{Li}_{-n}(e^\mu) \sim_{\operatorname{Re}(\mu) \to \infty} -(-1)^n e^{-\mu} \qquad (n = 1, 2, 3, \ldots)</math>
<math display="block">\operatorname{Li}_s(z) \sim_{\operatorname{Re}(s) \to \infty} z</math>
<math display="block">\operatorname{Li}_s(e^\mu) \sim_{\operatorname{Re}(s) \to -\infty}   \Gamma(1-s) (-\mu)^{s-1} \qquad (-\pi < \operatorname{Im}(\mu) < \pi)</math>
<math display="block">\operatorname{Li}_s(-e^\mu) \sim_{\operatorname{Re}(s) \to -\infty} \Gamma(1 - s) \left[ (-\mu - i\pi)^{s-1} + (-\mu + i\pi)^{s-1} \right] \qquad (\operatorname{Im}(\mu) = 0)</math>

Wood's first limit for {{math|Re(''μ'') → ∞}} has been corrected in accordance with his equation 11.3. The limit for {{math|Re(''s'') → −∞}} follows from the general relation of the polylogarithm with the [[Hurwitz zeta function]] ([[#Relationship to other functions|see above]]).

==Dilogarithm==
{{Main|Dilogarithm}}

The dilogarithm is the polylogarithm of order ''s'' = 2. An alternate integral expression of the dilogarithm for arbitrary complex argument ''z'' is {{harv|Abramowitz|Stegun|1972|loc=§ 27.7}}:
<math display="block">\operatorname{Li}_2 (z) = -\int_0^z{\ln (1-t) \over t} dt = -\int_0^1{\ln (1-zt) \over t} dt.</math>

A source of confusion is that some [[computer algebra system]]s define the dilogarithm as dilog(''z'') = Li<sub>2</sub>(1−''z'').

In the case of real ''z'' ≥ 1 the first integral expression for the dilogarithm can be written as
<math display="block">\operatorname{Li}_2(z) = \frac{\pi^2}{6} - \int_1^z{\ln(t-1) \over t} dt - i\pi \ln z</math>

from which expanding ln(''t''−1) and integrating term by term we obtain

<math display="block">\operatorname{Li}_2(z) = \frac{\pi^2}{3} - \frac{1}{2}(\ln z)^2 - \sum_{k=1}^\infty {1 \over k^2 z^k} - i\pi \ln z \qquad (z \ge 1).</math>

The ''[[Niels Henrik Abel|Abel]] identity'' for the dilogarithm is given by {{harv|Abel|1881}}

<math display="block">\operatorname{Li}_2 \left( \frac{x}{1-y} \right) + \operatorname{Li}_2 \left( \frac{y}{1-x} \right) - \operatorname{Li}_2 \left(\frac{xy}{(1-x)(1-y)} \right) = \operatorname{Li}_2(x) + \operatorname{Li}_2(y) + \ln(1-x) \ln(1-y)</math>

<math display="block">(\operatorname{Re}(x) \le \tfrac{1}{2} \wedge \operatorname{Re}(y) \le \tfrac{1}{2} \vee \operatorname{Im}(x) > 0 \wedge \operatorname{Im}(y) > 0 \vee \operatorname{Im}(x) < 0 \wedge \operatorname{Im}(y) < 0 \vee \ldots).</math>

This is immediately seen to hold for either ''x'' = 0 or ''y'' = 0, and for general arguments is then easily verified by differentiation ∂/∂''x'' ∂/∂''y''. For ''y'' = 1−''x'' the identity reduces to [[Leonhard Euler|Euler]]'s ''reflection formula''
<math display="block">\operatorname{Li}_2 \left(x \right) + \operatorname{Li}_2 \left(1-x\right) = \frac{1}{6} \pi^2 - \ln(x)\ln(1-x) ,</math>
where Li<sub>2</sub>(1) = ζ(2) = <sup>1</sup>⁄<sub>6</sub> ''π''<sup>2</sup> has been used and ''x'' may take any complex value.

In terms of the new variables ''u'' = ''x''/(1−''y''), ''v'' = ''y''/(1−''x'') the Abel identity reads
<math display="block">\operatorname{Li}_2(u) + \operatorname{Li}_2(v) - \operatorname{Li}_2(uv) = \operatorname{Li}_2 \left( \frac{u-uv}{1-uv} \right) + \operatorname{Li}_2 \left( \frac{v-uv}{1-uv} \right) + \ln \left( \frac{1-u}{1-uv} \right) \ln\left( \frac{1-v}{1-uv} \right),</math>
which corresponds to the ''pentagon identity'' given in {{harv|Rogers|1907}}.

From the Abel identity for ''x'' = ''y'' = 1−''z'' and the square relationship we have [[John Landen|Landen]]'s identity
<math display="block">\operatorname{Li}_2(1-z) + \operatorname{Li}_2 \left( 1-\frac{1}{z} \right) = - \frac{1}{2} (\ln z)^2 \qquad (z \not \in ~]-\infty; 0]) ,</math>
and applying the reflection formula to each dilogarithm we find the inversion formula
<math display="block">\operatorname{Li}_2(z) + \operatorname{Li}_2(1/z) = -\tfrac{1}{6} \pi^2 - \tfrac{1}{2} [\ln(-z)]^2 \qquad (z \not \in [0; 1[) ,</math>

and for real ''z'' ≥ 1 also
<math display="block">\operatorname{Li}_2(z) + \operatorname{Li}_2(1/z) = \tfrac{1}{3} \pi^2 - \tfrac{1}{2} (\ln z)^2 - i\pi \ln z .</math>

Known closed-form evaluations of the dilogarithm at special arguments are collected in the table below. Arguments in the first column are related by reflection ''x'' ↔ 1−''x'' or inversion ''x'' ↔ <sup>1</sup>⁄<sub>''x''</sub> to either ''x'' = 0 or ''x'' = −1; arguments in the third column are all interrelated by these operations.

{{harvtxt|Maximon|2003}} discusses the 17th to 19th century references. The reflection formula was already published by Landen in 1760, prior to its appearance in a 1768 book by Euler {{harv|Maximon|2003|loc=§ 10}}; an equivalent to Abel's identity was already published by [[William Spence (mathematician)|Spence]] in 1809, before Abel wrote his manuscript in 1826 {{harv|Zagier|1989|loc=§ 2}}. The designation ''bilogarithmische Function'' was introduced by [[:sv:Carl Johan Hill|Carl Johan Danielsson Hill]] (professor in Lund, Sweden) in 1828 {{harv|Maximon|2003|loc=§ 10}}. {{harvs | txt | author-link= Don Zagier | first= Don | last= Zagier | year= 1989}} has remarked that the dilogarithm is the only mathematical function possessing a sense of humor.

:{| class="wikitable" style="text-align: center;"
|+ '''Special values of the dilogarithm'''
|-
! <math>x </math>
! <math>\operatorname{Li}_2(x) </math>
! <math>x </math>
! <math>\operatorname{Li}_2(x) </math>
|-
| <math>-1 </math>
| <math>-\tfrac {1}{12} \pi^2 </math>
| <math>-\phi </math>
| <math>-\tfrac {1}{10} \pi^2 - \ln^2 \phi </math>
|-
| <math>0 </math>
| <math>0 </math>
| <math>-1 / \phi </math>
| <math>-\tfrac {1}{15} \pi^2 + \tfrac {1}{2} \ln^2 \phi </math>
|-
| <math>\tfrac {1}{2} </math>
| <math>\tfrac {1}{12} \pi^2 - \tfrac {1}{2} \ln^2 2 </math>
| <math>1 / \phi^2 </math>
| <math>\tfrac {1}{15} \pi^2 - \ln^2 \phi </math>
|-
| <math>1 </math>
| <math>\tfrac {1}{6} \pi^2 </math>
| <math>1 / \phi </math>
| <math>\tfrac {1}{10} \pi^2 - \ln^2 \phi </math>
|-
| <math>2 </math>
| <math>\tfrac {1}{4} \pi^2 - \pi i \ln 2 </math>
| <math>\phi </math>
| <math>\tfrac {11}{15} \pi^2 + \tfrac {1}{2} \ln^2(-1 / \phi) </math>
|-
| 
| 
| <math>\phi^2 </math>
| <math>-\tfrac {11}{15} \pi^2 - \ln^2(-\phi) </math>
|-
|}
:Here <math>\phi = \tfrac{1}{2} (\sqrt{5}+1)</math> denotes the [[golden ratio]].

==Polylogarithm ladders==

[[Leonard Lewin (telecommunications engineer)|Leonard Lewin]] discovered a remarkable and broad generalization of a number of classical relationships on the polylogarithm for special values. These are now called ''polylogarithm ladders''. Define <math>\rho = \tfrac{1}{2} (\sqrt{5}-1)</math> as the reciprocal of the [[golden ratio]]. Then two simple examples of dilogarithm ladders are

<math display="block">\operatorname{Li}_2(\rho^6) = 4 \operatorname{Li}_2(\rho^3) + 3 \operatorname{Li}_2(\rho^2) - 6 \operatorname{Li}_2(\rho) + \tfrac {7}{30} \pi^2</math>

given by {{harvs | txt | author-link= Harold Scott MacDonald Coxeter | last= Coxeter | year= 1935}} and

<math display="block">\operatorname{Li}_2(\rho) = \tfrac{1}{10} \pi^2 - \ln^2\rho</math>

given by [[John Landen|Landen]]. Polylogarithm ladders occur naturally and deeply in [[K-theory]] and [[algebraic geometry]]. Polylogarithm ladders provide the basis for the rapid computations of various mathematical constants by means of the [[BBP algorithm]] {{harv|Bailey|Borwein|Plouffe|1997}}.

==Monodromy==
The polylogarithm has two [[branch point]]s; one at ''z'' = 1 and another at ''z'' = 0. The second branch point, at ''z'' = 0, is not visible on the main sheet of the polylogarithm; it becomes visible only when the function is [[analytically continued]] to its other sheets. The [[monodromy]] group for the polylogarithm consists of the [[homotopy]] classes of loops that wind around the two branch points. Denoting these two by ''m''<sub>0</sub> and ''m''<sub>1</sub>, the monodromy group has the [[group presentation]]

<math display="block">\langle m_0, m_1 \vert w = m_0 m_1 m^{-1}_0 m^{-1}_1, w m_1 = m_1 w \rangle.</math>

For the special case of the dilogarithm, one also has that ''wm''<sub>0</sub> = ''m''<sub>0</sub>''w'', and the monodromy group becomes the [[Heisenberg group]] (identifying ''m''<sub>0</sub>, ''m''<sub>1</sub> and ''w'' with ''x'', ''y'', ''z'') {{harv|Vepstas|2008}}.

== Notes ==
{{reflist|group=note}}

==References==
{{Reflist}}
{{refbegin}}
* {{cite book | last= Abel | first= N.H. | author-link= Niels Henrik Abel | contribution= Note sur la fonction <math>\scriptstyle \psi x = x+ \frac{x^2}{2^2}+ \frac{x^3}{3^2}+ \cdots+ \frac{x^n}{n^2}+ \cdots</math> | language= fr | contribution-url= http://www.abelprisen.no/nedlastning/verker/oeuvres_1881_del2/oeuvres_completes_de_abel_nouv_ed_2_kap14_opt.pdf | contribution-format= PDF | editor1-last= Sylow | editor1-first= L. | editor2-last= Lie | editor2-first= S. | title= Œuvres complètes de Niels Henrik Abel − Nouvelle édition, Tome II | location= Christiania [Oslo] | publisher= Grøndahl & Søn | orig-year= 1826 | year= 1881 | pages= 189–193 }} (this 1826 manuscript was only published posthumously.)
* {{cite book | last1= Abramowitz | first1= M. | last2= Stegun | first2= I.A. | title= Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables | location= New York | publisher= Dover Publications | year= 1972 | isbn= 978-0-486-61272-0 | title-link= Abramowitz and Stegun }}
* {{dlmf | id= 25.12 | first= T.M. | last= Apostol }}
* {{cite journal | last1= Bailey | first1= D.H. | author1-link= David H. Bailey (mathematician) | last2= Borwein | first2= P.B. | author2-link= Peter Borwein | last3= Plouffe | first3= S. | author3-link= Simon Plouffe | title= On the Rapid Computation of Various Polylogarithmic Constants | url= http://crd.lbl.gov/~dhbailey/dhbpapers/digits.pdf | journal= Mathematics of Computation |date=April 1997 | volume= 66 | issue= 218 | pages= 903–913 | doi= 10.1090/S0025-5718-97-00856-9 | bibcode= 1997MaCom..66..903B| doi-access= free }}
* {{cite arXiv | last1= Bailey | first1= D.H. | last2= Broadhurst | first2= D.J. | title= A Seventeenth-Order Polylogarithm Ladder | eprint= math.CA/9906134 | date= June 20, 1999 }}
* {{cite book | last= Berndt | first= B.C. | title= Ramanujan's Notebooks, Part IV | location= New York | publisher= Springer-Verlag | year= 1994 | pages= 323–326 | isbn=978-0-387-94109-7 }}
* {{cite journal | author-link= Johannes Boersma | last1= Boersma | first1= J. | last2= Dempsey | first2= J.P. | title= On the evaluation of Legendre's chi-function | jstor= 2152987 | journal= Mathematics of Computation | year= 1992 | volume= 59 | issue= 199 | pages= 157–163 | doi= 10.2307/2152987 | url= https://research.tue.nl/nl/publications/on-the-numerical-evaluation-of-legendres-chifunction(cb45fc94-8446-4b7f-80f8-7bf3b220e62e).html }}
* {{cite journal | last1= Borwein | first1= D. | author1-link= David Borwein | last2= Borwein | first2= J.M. | author2-link= Jonathan Borwein | last3= Girgensohn | first3= R. | title= Explicit evaluation of Euler sums | url= http://docserver.carma.newcastle.edu.au/58/2/93_001-Borwein-Borwein-Girgensohn.pdf | journal= Proceedings of the Edinburgh Mathematical Society |series=Series 2 | year= 1995 | volume= 38 | issue= 2 | pages= 277–294 | doi= 10.1017/S0013091500019088 | doi-access= free }}
* {{cite journal | last1= Borwein | first1= J.M. | last2= Bradley | first2= D.M. | last3= Broadhurst | first3= D.J. | last4= Lisonek | first4= P. | title= Special Values of Multiple Polylogarithms | journal= Transactions of the American Mathematical Society | year= 2001 | volume= 353 | issue= 3 | pages= 907–941 | doi= 10.1090/S0002-9947-00-02616-7 | arxiv= math/9910045 | s2cid= 11373360 }}
* {{cite arXiv | last= Broadhurst | first= D.J. | title= On the enumeration of irreducible k-fold Euler sums and their roles in knot theory and field theory | eprint= hep-th/9604128 | date= April 21, 1996 }}
* {{cite journal | last= Clunie | first= J. | title= On Bose-Einstein functions | journal= Proceedings of the Physical Society | series = Series A | year= 1954 | volume= 67 | issue= 7 | pages= 632–636 | doi= 10.1088/0370-1298/67/7/308 | bibcode= 1954PPSA...67..632C }}
* {{cite journal | last1= Cohen | first1= H. | last2= Lewin | first2= L. | last3= Zagier | first3= D. | title= A Sixteenth-Order Polylogarithm Ladder | url= http://www.expmath.org/expmath/volumes/1/1.html | format= PS | journal= Experimental Mathematics | year= 1992 | volume= 1 | issue= 1 | pages= 25–34 }}
* {{cite journal | last= Coxeter | first= H.S.M. | author-link= Harold Scott MacDonald Coxeter | title= The functions of Schläfli and Lobatschefsky | journal= Quarterly Journal of Mathematics | year= 1935 | volume= 6 | issue= 1 | pages= 13–29 | doi= 10.1093/qmath/os-6.1.13 | jfm= 61.0395.02 | bibcode= 1935QJMat...6...13C }}
* {{cite journal | last1= Cvijovic | first1= D. | last2= Klinowski | first2= J. | title= Continued-fraction expansions for the Riemann zeta function and polylogarithms | url= https://www.ams.org/journals/proc/1997-125-09/S0002-9939-97-04102-6/S0002-9939-97-04102-6.pdf | journal= Proceedings of the American Mathematical Society | year= 1997 | volume= 125 | issue= 9 | pages= 2543–2550 | doi= 10.1090/S0002-9939-97-04102-6 | doi-access= free }}
* {{cite journal | last= Cvijovic | first= D. | title= New integral representations of the polylogarithm function | journal= [[Proceedings of the Royal Society A]] | year= 2007 | volume= 463 | issue= 2080 | pages= 897–905 | doi= 10.1098/rspa.2006.1794 | arxiv= 0911.4452 | bibcode= 2007RSPSA.463..897C | s2cid= 115156743 }}
* {{Cite journal |last=Dingle |first=R. B. |date=1955 |title=The evaluation of integrals containing a parameter |url=http://link.springer.com/10.1007/BF02920017 |journal=Applied Scientific Research, Section B |language=en |volume=4 |issue=1 |pages=401–410 |doi=10.1007/BF02920017 |issn=0365-7140}}
* {{Cite journal |last=Dingle |first=R. B. |date=1957a |title=The Bose-Einstein integrals <math>\mathcal{B}_p (\eta ) = (p!)^{ - 1} \int\limits_0^\infty {\varepsilon ^p (e^{\varepsilon - \eta } - 1} )^{ - 1} d\varepsilon </math> |url=http://link.springer.com/10.1007/BF02920380 |journal=Applied Scientific Research, Section B |language=en |volume=6 |issue=1 |pages=240–244 |doi=10.1007/BF02920380 |issn=0365-7140}}
* {{Cite journal |last=Dingle |first=R. B. |last2=Arndt |first2=Doreen |last3=Roy |first3=S. K. |date=1957 |title=The integrals <math>\mathfrak{A}_p (x) = (p!)^{ - 1} \int\limits_0^\infty {\varepsilon ^p (\varepsilon + x} )^{ - 1} e^{ - \varepsilon } d\varepsilon</math> and <math>\mathfrak{B}_p (x) = (p!)^{ - 1} \int\limits_0^\infty {\varepsilon ^p (\varepsilon + x} )^{ - 2} e^{ - \varepsilon } d\varepsilon</math> and their tabulation |url=http://link.springer.com/10.1007/BF02920371 |journal=Applied Scientific Research, Section B |language=en |volume=6 |issue=1 |pages=144–154 |doi=10.1007/BF02920371 |issn=0365-7140}}
* {{Cite journal |last=Dingle |first=R. B. |date=1957b |title=The Fermi-Dirac integrals <math>\mathcal{F}_p (\eta ) = (p!)^{ - 1} \int\limits_0^\infty {\varepsilon ^p (e^{\varepsilon - \eta } + 1} )^{ - 1} d\varepsilon </math> |url=http://link.springer.com/10.1007/BF02920379 |journal=Applied Scientific Research, Section B |language=en |volume=6 |issue=1 |pages=225–239 |doi=10.1007/BF02920379 |issn=0365-7140}}
* {{cite book | last1= Erdélyi | first1= A. | author1-link= Arthur Erdélyi | last2= Magnus | first2= W. | last3= Oberhettinger | first3= F. | last4= Tricomi | first4= F.G. | title= Higher Transcendental Functions, Vol. 1 | url= https://s3.amazonaws.com/apps.nrbook.com/bateman/Vol1.pdf | location= Malabar, FL | publisher= R.E. Krieger Publishing | year= 1981 | isbn= 978-0-89874-206-0 | ref= {{harvid|Erdélyi et al.|1981}}}} (this is a reprint of the McGraw–Hill original of 1953.)
* {{cite journal | last1= Fornberg | first1= B. | last2= Kölbig | first2= K.S. | title= Complex zeros of the Jonquière or polylogarithm function | jstor= 2005579 | journal= Mathematics of Computation | year= 1975 | volume= 29 | issue= 130 | pages= 582–599 | doi= 10.2307/2005579 | url= https://cds.cern.ch/record/880364 }}
* {{cite web | author= GNU Scientific Library | url= https://www.gnu.org/software/gsl/manual/gsl-ref.html#SEC117 | title= Reference Manual | year= 2010 | access-date= 2010-06-13 | ref= {{harvid|GSL|2010}} }}
*{{cite book
 | last1 = Gradshteyn | first1 = I. S.
 | last2 = Ryzhik | first2 = I. M.
 | contribution = 9.55 The function <math>\Phi(z,s,v)</math>: Series representation 9.553, 9.554
 | contribution-url = https://books.google.com/books?id=F7jiBQAAQBAJ&pg=PA1075
 | edition = 7th<!-- as numbered by MathSciNet -->
 | isbn = 978-0-12-384933-5
 | mr = 3307944
 | page = 1075
 | publisher = Elsevier/Academic Press | location = Amsterdam
 | title = Table of Integrals, Series, and Products
 | title-link = Gradshteyn and Ryzhik
 | year = 2015}}
* {{cite journal | last1= Guillera | first1= J. | last2= Sondow | first2= J. | title= Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent | arxiv= math.NT/0506319 | journal= The Ramanujan Journal | year= 2008 | volume= 16 | issue= 3 | pages= 247–270 | doi= 10.1007/s11139-007-9102-0 | s2cid= 119131640 }}
* {{cite arXiv | last= Hain | first= R.M. | title= Classical polylogarithms | eprint= alg-geom/9202022 | date= March 25, 1992 }}
* {{cite book | last1= Jahnke | first1= E. | last2= Emde | first2= F. | title= Tables of Functions with Formulae and Curves | edition= 4th | location= New York | publisher= Dover Publications | year= 1945 }}
* {{cite journal | last= Jonquière | first= A. | title= Note sur la série <math>\scriptstyle \sum_{n=1}^\infty \frac{x^n}{n^s}</math> | language= fr | url= http://archive.numdam.org/item?id=BSMF_1889__17__142_1 | format= PDF | journal= Bulletin de la Société Mathématique de France | year= 1889 | volume= 17 | pages= 142–152 | jfm= 21.0246.02 | doi= 10.24033/bsmf.392 | doi-access= free }}
* {{cite journal | last1= Kölbig | first1= K.S. | last2= Mignaco | first2= J.A. | last3= Remiddi | first3= E. | title= On Nielsen's generalized polylogarithms and their numerical calculation | journal= BIT | year= 1970 | volume= 10 | pages= 38–74 | doi= 10.1007/BF01940890 | s2cid= 119672619 | url= https://cds.cern.ch/record/350733 }}
* {{cite journal | last= Kirillov | first= A.N. | title= Dilogarithm identities | arxiv= hep-th/9408113 | journal= Progress of Theoretical Physics Supplement | year= 1995 | volume= 118 | pages= 61–142 | doi= 10.1143/PTPS.118.61 | bibcode= 1995PThPS.118...61K | s2cid= 119177149 }}
* {{cite book | last= Lewin | first= L. | title= Dilogarithms and Associated Functions | location= London | publisher= Macdonald | year= 1958 | mr= 0105524 }}
* {{cite book | last= Lewin | first= L. | title= Polylogarithms and Associated Functions | location= New York | publisher= North-Holland | year= 1981 | isbn= 978-0-444-00550-2 }}
* {{cite book | editor-last= Lewin | editor-first= L. | title= Structural Properties of Polylogarithms | series= Mathematical Surveys and Monographs | volume= 37 | location= Providence, RI | publisher= Amer. Math. Soc. | year= 1991 | isbn= 978-0-8218-1634-9 }}
* {{cite journal | last= Markman | first= B. | title= The Riemann Zeta Function | journal= BIT | year= 1965 | volume= 5 | pages= 138–141 }}
* {{cite journal | last= Maximon | first= L.C. | title= The Dilogarithm Function for Complex Argument | journal= [[Proceedings of the Royal Society A]] | year= 2003 | volume= 459 | issue= 2039 | pages= 2807–2819 | doi= 10.1098/rspa.2003.1156 | bibcode= 2003RSPSA.459.2807M | s2cid= 122271244 }}
* {{cite journal | last1= McDougall | first1= J. | last2= Stoner | first2= E.C. | title= The computation of Fermi-Dirac functions | journal= [[Philosophical Transactions of the Royal Society A]] | year= 1938 | volume= 237 | issue= 773 | pages= 67–104 | doi= 10.1098/rsta.1938.0004 | jfm= 64.1500.04 | bibcode= 1938RSPTA.237...67M | doi-access= free }}
* {{cite journal | author-link= Niels Nielsen (mathematician) | last= Nielsen | first= N. | title= Der Eulersche Dilogarithmus und seine Verallgemeinerungen. Eine Monographie | language= de | location= Halle – Leipzig, Germany | publisher= Kaiserlich-Leopoldinisch-Carolinische Deutsche Akademie der Naturforscher | journal= Nova Acta Leopoldina | year= 1909 | volume= XC | issue= 3 | pages= 121–212 | jfm= 40.0478.01 }}
* {{cite book | last1= Prudnikov | first1= A.P. | last2= Marichev | first2= O.I. | last3= Brychkov | first3= Yu.A. | title= Integrals and Series, Vol. 3: More Special Functions | location= Newark, NJ | publisher= Gordon and Breach | year= 1990 | isbn= 978-2-88124-682-1 }} (see § 1.2, "The generalized zeta function, Bernoulli polynomials, Euler polynomials, and polylogarithms", p.&nbsp;23.)
* {{cite journal | last= Robinson | first= J.E. | title= Note on the Bose-Einstein integral functions | journal= Physical Review | series = Series 2 | year= 1951 | volume= 83 | issue= 3 | pages= 678–679 | doi= 10.1103/PhysRev.83.678 | bibcode= 1951PhRv...83..678R }}
* {{cite journal | last= Rogers | first= L.J. | title= On function sum theorems connected with the series <math>\scriptstyle \sum_{n=1}^\infty \frac{x^n}{n^2}</math> | journal= Proceedings of the London Mathematical Society (2) | year= 1907 | volume= 4 | issue= 1 | pages= 169–189 | doi= 10.1112/plms/s2-4.1.169 | jfm= 37.0428.03 | url= https://zenodo.org/record/1447792 }}
* {{cite book | last= Schrödinger | first= E. | author-link= Erwin Schrödinger | title= Statistical Thermodynamics | edition= 2nd | location= Cambridge, UK | publisher= Cambridge University Press | year= 1952 }}
* {{cite journal | last= Truesdell | first= C. | title= On a function which occurs in the theory of the structure of polymers | jstor= 1969153 | journal= Annals of Mathematics |series=Second Series | year= 1945 | volume= 46 | issue= 1 | pages= 144–157 | doi= 10.2307/1969153 }}
* {{cite journal | last= Vepstas | first= L. | title= An efficient algorithm for accelerating the convergence of oscillatory series, useful for computing the polylogarithm and Hurwitz zeta functions | arxiv= math.CA/0702243 | journal= Numerical Algorithms | year= 2008 | volume= 47 | issue= 3 | pages= 211–252 | doi= 10.1007/s11075-007-9153-8 | bibcode= 2008NuAlg..47..211V | s2cid= 15131811 }}
* {{cite book | last1= Whittaker | first1= E.T. | author1-link= Edmund Taylor Whittaker | last2= Watson | first2= G.N. | author2-link= George Neville Watson | title= [[A Course of Modern Analysis]] | edition= 4th | location= Cambridge, UK | publisher= Cambridge University Press | year= 1927 }} (this edition has been reprinted many times, a 1996 paperback has {{isbn|0-521-09189-6}}.)
* {{cite journal | last= Wirtinger | first= W. | title= Über eine besondere Dirichletsche Reihe | language= de | journal= Journal für die Reine und Angewandte Mathematik | year= 1905 | volume= 1905 | issue= 129 | pages= 214–219 | jfm= 37.0434.01 | doi=10.1515/crll.1905.129.214| s2cid= 199545536 }}
* {{cite web | last= Wood | first= D.C. | title= The Computation of Polylogarithms. Technical Report 15-92* | url= http://www.cs.kent.ac.uk/pubs/1992/110 | format= PS | location= Canterbury, UK | publisher= University of Kent Computing Laboratory |date=June 1992 | access-date= 2005-11-01 }}
* {{cite conference | last= Zagier | first= D. | author-link= Don Zagier | title= The dilogarithm function in geometry and number theory | book-title= Number Theory and Related Topics: papers presented at the Ramanujan Colloquium, Bombay, 1988 | series= Studies in Mathematics | volume= 12 | location= Bombay | publisher= Tata Institute of Fundamental Research and Oxford University Press | year= 1989 | pages= 231–249 | isbn= 0-19-562367-3 }} (also appeared as "The remarkable dilogarithm" in ''Journal of Mathematical and Physical Sciences'' '''22''' (1988), pp.&nbsp;131–145, and as Chapter I of {{harv|Zagier|2007}}.)
* {{cite book | last= Zagier | first= D. | chapter= The Dilogarithm Function | chapter-url= http://mathlab.snu.ac.kr/~top/articles/zagier.pdf | editor1-last= Cartier | editor1-first= P.E. | editor2-last= Julia | editor2-first= B. | editor3-last= Moussa | editor3-first= P. | editor4-last= Vanhove | editor4-first= P. | display-editors= 1 | title= Frontiers in Number Theory, Physics, and Geometry II – On Conformal Field Theories, Discrete Groups and Renormalization | location= Berlin | publisher= Springer-Verlag | year= 2007 | pages= 3–65 | isbn= 978-3-540-30307-7 }}
{{refend}}

==External links==
* {{mathworld | urlname= Polylogarithm | title= Polylogarithm}}
* {{mathworld | urlname= Dilogarithm | title= Dilogarithm}}
* [https://launchpad.net/anant Algorithms in Analytic Number Theory] provides an arbitrary-precision, [[GNU Multiple Precision Arithmetic Library|GMP]]-based, [[GPL]]-licensed implementation.

[[Category:Special functions]]
[[Category:Zeta and L-functions]]
[[Category:Rational functions]]