Editing Polylogarithm (section)

==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>