Inmathematics, thepolylogarithm (also known asJonquière's function, for Alfred Jonquière) is aspecial functionLi_s(z) of orders and argumentz. Only for special values ofs does the polylogarithm reduce to anelementary function such as thenatural logarithm or arational function. Inquantum statistics, the polylogarithm function appears as the closed form ofintegrals of theFermi–Dirac distribution and theBose–Einstein distribution, and is also known as theFermi–Dirac integral or theBose–Einstein integral. Inquantum electrodynamics, polylogarithms of positiveinteger order arise in the calculation of processes represented by higher-orderFeynman diagrams.

The polylogarithm function is equivalent to theHurwitz zeta function — eitherfunction can be expressed in terms of the other — and both functions are special cases of theLerch transcendent. Polylogarithms should not be confused withpolylogarithmic functions, nor with theoffset logarithmic integralLi(z), which has the same notation without the subscript.

• Different polylogarithm functions in the complex plane
• 
  Li _–3(z)
• 
  Li _–2(z)
• 
  Li _–1(z)
• 
  Li₀(z)
• 
  Li₁(z)
• 
  Li₂(z)
• 
  Li₃(z)

The polylogarithm function is defined by apower series inz generalizing theMercator series, which is also aDirichlet series ins:$${\displaystyle {\mathrm{Li}}_s(z)=\sum _{k=1}^{\mathrm{\infty }}\frac{z^k}{k^s}=z+\frac{z^2}{2^s}+\frac{z^3}{3^s}+\cdots }$$

This definition is valid for arbitrarycomplex orders and for all complex argumentsz with|z| < 1; it can be extended to|z| ≥ 1 by the process ofanalytic continuation. (Here the denominatork^s is understood asexp(s lnk)). The special cases = 1 involves the ordinarynatural logarithm,Li₁(z) = −ln(1−z), while the special casess = 2 ands = 3 are called thedilogarithm (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 repeatedintegral of itself:$${\displaystyle {\mathrm{Li}}_{s+1}(z)={\int }_0^z\frac{{\mathrm{Li}}_s(t)}{t}dt}$$thus the dilogarithm is an integral of a function involving the logarithm, and so on. For nonpositive integer orderss, the polylogarithm is arational function.

In the case where the order${\displaystyle s}$ is an integer, it will be represented by${\displaystyle s=n}$ (or${\displaystyle s=-n}$ when negative). It is often convenient to define${\displaystyle \mu =\ln (z)}$ where${\displaystyle \ln (z)}$ is theprincipal branch of thecomplex logarithm${\displaystyle \mathrm{Ln}(z)}$ so that Also, all exponentiation will be assumed to be single-valued:${\displaystyle z^s=\exp (s\ln (z)).}$

Depending on the order${\displaystyle s}$, the polylogarithm may be multi-valued. Theprincipal branch of${\displaystyle {\mathrm{Li}}_s(z)}$ is taken to be given for by the above series definition and taken to be continuous except on the positive real axis, where a cut is made from${\displaystyle z=1}$ to${\displaystyle \mathrm{\infty }}$ such that the axis is placed on the lower half plane of${\displaystyle z}$. In terms of${\displaystyle \mu }$, this amounts to. The discontinuity of the polylogarithm in dependence on${\displaystyle \mu }$ can sometimes be confusing.

For real argument${\displaystyle z}$, the polylogarithm of real order${\displaystyle s}$ is real if, and its imaginary part for${\displaystyle z\ge 1}$ is (Wood 1992, §3):

$${\displaystyle \mathrm{Im}\left({\mathrm{Li}}_s(z)\right)=-\frac{\pi {\mu }^{s-1}}{\mathrm{\Gamma }(s)}.}$$

Going across the cut, ifε is an infinitesimally small positivereal number, then:

$${\displaystyle \mathrm{Im}\left({\mathrm{Li}}_s(z+iϵ)\right)=\frac{\pi {\mu }^{s-1}}{\mathrm{\Gamma }(s)}.}$$

Both can be concluded from the series expansion (see below) ofLi_s(e^μ) aboutμ = 0.

The derivatives of the polylogarithm follow from the defining power series:

$${\displaystyle z\frac{\mathrm{\partial }{\mathrm{Li}}_s(z)}{\mathrm{\partial }z}={\mathrm{Li}}_{s-1}(z)}$$$${\displaystyle \frac{\mathrm{\partial }{\mathrm{Li}}_s(e^{\mu })}{\mathrm{\partial }\mu }={\mathrm{Li}}_{s-1}(e^{\mu }).}$$

The square relationship is seen from the series definition, and is related to theduplication formula (see alsoClunie (1954),Schrödinger (1952)):

$${\displaystyle {\mathrm{Li}}_s(-z)+{\mathrm{Li}}_s(z)=2^{1-s}{\mathrm{Li}}_s(z^2).}$$

Kummer's function obeys a very similar duplication formula. This is a special case of themultiplication formula, for any positive integerp:

$${\displaystyle \sum _{m=0}^{p-1}{\mathrm{Li}}_s(z{e}^{2\pi im/p})=p^{1-s}{\mathrm{Li}}_s(z^p),}$$

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 theHurwitz zeta function or theBernoulli polynomials and is found underrelationship to other functions below.

For particular cases, the polylogarithm may be expressed in terms of other functions (see below). Particular values for the polylogarithm may thus also be found as particular values of these other functions.

1. For integer values of the polylogarithm order, the following explicit expressions are obtained by repeated application ofz·∂/∂z to Li₁(z):$${\displaystyle {\mathrm{Li}}_1(z)=-\ln (1-z)}$$$${\displaystyle {\mathrm{Li}}_0(z)=\frac{z}{1-z}}$$$${\displaystyle {\mathrm{Li}}_{-1}(z)=\frac{z}{(1-z{)}^2}}$$$${\displaystyle {\mathrm{Li}}_{-2}(z)=\frac{z(1+z)}{(1-z{)}^3}}$$$${\displaystyle {\mathrm{Li}}_{-3}(z)=\frac{z(1+4z+z^2)}{(1-z{)}^4}}$$$${\displaystyle {\mathrm{Li}}_{-4}(z)=\frac{z(1+z)(1+10z+z^2)}{(1-z{)}^5}.}$$Accordingly the polylogarithm reduces to a ratio of polynomials inz, and is therefore arational function ofz, for all nonpositive integer orders. The general case may be expressed as a finite sum:$${\displaystyle {\mathrm{Li}}_{-n}(z)={\left(z\frac{\mathrm{\partial }}{\mathrm{\partial }z}\right)}^n\frac{z}{1-z}=\sum _{k=0}^{n}k!S(n+1,k+1){\left(\frac{z}{1-z}\right)}^{k+1}(n=0,1,2,\dots ),}$$whereS(n,k) are theStirling numbers of the second kind. Equivalent formulae applicable to negative integer orders are (Wood 1992, § 6):$${\displaystyle {\mathrm{Li}}_{-n}(z)=(-1{)}^{n+1}\sum _{k=0}^{n}k!S(n+1,k+1){\left(\frac{-1}{1-z}\right)}^{k+1}(n=1,2,3,\dots ),}$$and:$${\displaystyle {\mathrm{Li}}_{-n}(z)=\frac{1}{(1-z{)}^{n+1}}\sum _{k=0}^{n-1}⟨\genfrac{}{}{0ex}{}{n}{k}⟩z^{n-k}(n=1,2,3,\dots ),}$$where${\displaystyle ⟨\genfrac{}{}{0ex}{}{n}{k}⟩}$ are theEulerian numbers. All roots of Li_−n(z) are distinct and real; they includez = 0, while the remainder is negative and centered aboutz = −1 on a logarithmic scale. Asn becomes large, the numerical evaluation of these rational expressions increasingly suffers from cancellation (Wood 1992, § 6); full accuracy can be obtained, however, by computing Li_−n(z) via the general relation with the Hurwitz zeta function (see below).
2. Some particular expressions for half-integer values of the argumentz are:$${\displaystyle {\mathrm{Li}}_1(\frac{1}{2})=\ln 2}$$$${\displaystyle {\mathrm{Li}}_2(\frac{1}{2})=\frac{1}{12}{\pi }^2-\frac{1}{2}(\ln 2{)}^2}$$$${\displaystyle {\mathrm{Li}}_3(\frac{1}{2})=\frac{1}{6}(\ln 2{)}^3-\frac{1}{12}{\pi }^2\ln 2+\frac{7}{8}\zeta (3),}$$whereζ is theRiemann zeta function. No formulae of this type are known for higher integer orders (Lewin 1991, p. 2), but one has for instance (Borwein, Borwein & Girgensohn 1995):$${\displaystyle {\mathrm{Li}}_4(\frac{1}{2})=\frac{1}{360}{\pi }^4-\frac{1}{24}(\ln 2{)}^4+\frac{1}{24}{\pi }^2(\ln 2{)}^2-\frac{1}{2}\zeta (\overline{3},\overline{1}),}$$which involves the alternating double sum$${\displaystyle \zeta (\overline{3},\overline{1})=\sum _{m>n>0}(-1{)}^{m+n}{m}^{-3}{n}^{-1}.}$$In general one has for integer ordersn ≥ 2 (Broadhurst 1996, p. 9):$${\displaystyle {\mathrm{Li}}_n(\frac{1}{2})=-\zeta (\overline{1},\overline{1},{\left\{1\right\}}^{n-2}),}$$whereζ(s₁, …,s_k) is themultiple zeta function; for example:$${\displaystyle {\mathrm{Li}}_5(\frac{1}{2})=-\zeta (\overline{1},\overline{1},1,1,1).}$$
3. As a straightforward consequence of the series definition, values of the polylogarithm at thepth complexroots of unity are given by theFourier sum:$${\displaystyle {\mathrm{Li}}_s(e^{2\pi im/p})=p^{-s}\sum _{k=1}^p{e}^{2\pi imk/p}\zeta (s,\frac{k}{p})(m=1,2,\dots ,p-1),}$$whereζ is theHurwitz zeta function. For Re(s) > 1, where Li_s(1) is finite, the relation also holds withm = 0 orm =p. While this formula is not as simple as that implied by the more general relation with the Hurwitz zeta function listed underrelationship to other functions below, it has the advantage of applying to non-negative integer values ofs as well. As usual, the relation may be inverted to express ζ(s,^m⁄_p) for anym = 1, …,p as a Fourier sum of Li_s(exp(2πi^k⁄_p)) overk = 1, …,p.

• Forz = 1, the polylogarithm reduces to theRiemann zeta function$${\displaystyle {\mathrm{Li}}_s(1)=\zeta (s)(\mathrm{Re}(s)>1).}$$
• The polylogarithm is related toDirichlet eta function and theDirichlet beta function:$${\displaystyle {\mathrm{Li}}_s(-1)=-\eta (s),}$$ whereη(s) is the Dirichlet eta function. For pure imaginary arguments, we have:$${\displaystyle {\mathrm{Li}}_s(\pm i)=-2^{-s}\eta (s)\pm i\beta (s),}$$ whereβ(s) is the Dirichlet beta function.
• The polylogarithm is related to thecomplete Fermi–Dirac integral as:$${\displaystyle F_s(\mu )=-{\mathrm{Li}}_{s+1}(-e^{\mu }).}$$
• The polylogarithm is related to the complete Bose–Einstein integral as:$${\displaystyle G_s(\mu )={\mathrm{Li}}_{s+1}(e^{\mu }).}$$
• The polylogarithm is a special case of theincomplete polylogarithm function$${\displaystyle {\mathrm{Li}}_s(z)={\mathrm{Li}}_s(0,z).}$$
• The polylogarithm is a special case of theLerch transcendent (Erdélyi et al. 1981, § 1.11-14)$${\displaystyle {\mathrm{Li}}_s(z)=z\mathrm{\Phi }(z,s,1).}$$
• The polylogarithm is related to theHurwitz zeta function by:$${\displaystyle {\mathrm{Li}}_s(z)=\frac{\mathrm{\Gamma }(1-s)}{(2\pi {)}^{1-s}}\left[i^{1-s}\zeta \left(1-s,\frac{1}{2}+\frac{\ln (-z)}{2\pi i}\right)+i^{s-1}\zeta \left(1-s,\frac{1}{2}-\frac{\ln (-z)}{2\pi i}\right)\right],}$$ which relation, however, is invalidated at positive integers bypoles of thegamma functionΓ(1 −s), and ats = 0 by a pole of both zeta functions; a derivation of this formula is given underseries 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 (Jonquière 1889):$${\displaystyle i^{-s}{\mathrm{Li}}_s(e^{2\pi ix})+i^s{\mathrm{Li}}_s(e^{-2\pi ix})=\frac{(2\pi {)}^s}{\mathrm{\Gamma }(s)}\zeta (1-s,x),}$$ which relation holds for0 ≤ Re(x) < 1 ifIm(x) ≥ 0, and for0 < Re(x) ≤ 1 ifIm(x) < 0. Equivalently, for all complexs and for complexz ∉(0, 1], the inversion formula reads$${\displaystyle {\mathrm{Li}}_s(z)+(-1{)}^s{\mathrm{Li}}_s(1/z)=\frac{(2\pi i{)}^s}{\mathrm{\Gamma }(s)}\zeta \left(1-s,\frac{1}{2}+\frac{\ln (-z)}{2\pi i}\right),}$$ and for all complexs and for complexz ∉(1, ∞)$${\displaystyle {\mathrm{Li}}_s(z)+(-1{)}^s{\mathrm{Li}}_s(1/z)=\frac{(2\pi i{)}^s}{\mathrm{\Gamma }(s)}\zeta \left(1-s,\frac{1}{2}-\frac{\ln (-1/z)}{2\pi i}\right).}$$ Forz ∉(0, ∞), one hasln(−z) = −ln(−¹⁄_z), 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 ofJonquière (1889, eq. 5) andErdélyi et al. (1981, § 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 whens is an integer.
• For positive integer polylogarithm orderss, the Hurwitz zeta function ζ(1−s,x) reduces toBernoulli polynomials,ζ(1−n,x) = −B_n(x) /n, and Jonquière's inversion formula forn = 1, 2, 3, … becomes:$${\displaystyle {\mathrm{Li}}_n(e^{2\pi ix})+(-1{)}^n{\mathrm{Li}}_n(e^{-2\pi ix})=-\frac{(2\pi i{)}^n}{n!}{B}_n(x),}$$ 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 theunit circle, Im(x) = 0, the left hand side of this formula simplifies to 2 Re(Li_n(e^2πix)) ifn is even, and to 2i Im(Li_n(e^2πix)) ifn is odd. For negative integer orders, on the other hand, the divergence of Γ(s) implies for allz that (Erdélyi et al. 1981, § 1.11-17):$${\displaystyle {\mathrm{Li}}_{-n}(z)+(-1{)}^n{\mathrm{Li}}_{-n}(1/z)=0(n=1,2,3,\dots ).}$$ More generally, one has forn = 0, ±1, ±2, ±3, …: where both expressions agree forz ∉(0, ∞). (The corresponding equation ofJonquière (1889, eq. 1) andErdélyi et al. (1981, § 1.11-18) is again not correct.)
• The polylogarithm with pure imaginaryμ may be expressed in terms of theClausen functionsCi_s(θ) andSi_s(θ), and vice versa (Lewin 1958, Ch. VII § 1.4;Abramowitz & Stegun 1972, § 27.8):$${\displaystyle {\mathrm{Li}}_s(e^{\pm i\theta })=C{i}_s(\theta )\pm iS{i}_s(\theta ).}$$
• Theinverse tangent integralTi_s(z) (Lewin 1958, Ch. VII § 1.2) can be expressed in terms of polylogarithms:$${\displaystyle {\mathrm{Ti}}_s(z)=\frac{1}{2i}\left[{\mathrm{Li}}_s(iz)-{\mathrm{Li}}_s(-iz)\right].}$$ The relation in particular implies:$${\displaystyle {\mathrm{Ti}}_0(z)=\frac{z}{1+z^2},{\mathrm{Ti}}_1(z)=\arctan z,{\mathrm{Ti}}_2(z)={\int }_0^z\frac{\arctan t}{t}dt,\dots {\mathrm{Ti}}_{n+1}(z)={\int }_0^z\frac{{\mathrm{Ti}}_n(t)}{t}dt,}$$ which explains the function name.
• TheLegendre chi functionχ_s(z) (Lewin 1958, Ch. VII § 1.1;Boersma & Dempsey 1992) can be expressed in terms of polylogarithms:$${\displaystyle {\chi }_s(z)=\frac{1}{2}\left[{\mathrm{Li}}_s(z)-{\mathrm{Li}}_s(-z)\right].}$$
• The polylogarithm of integer order can be expressed as ageneralized hypergeometric function:
• In terms of theincomplete zeta functions or "Debye functions" (Abramowitz & Stegun 1972, § 27.1):$${\displaystyle Z_n(z)=\frac{1}{(n-1)!}{\int }_z^{\mathrm{\infty }}\frac{t^{n-1}}{e^t-1}dt(n=1,2,3,\dots ),}$$ the polylogarithm Li_n(z) for positive integer n may be expressed as the finite sum (Wood 1992, §16):$${\displaystyle {\mathrm{Li}}_n(e^{\mu })=\sum _{k=0}^{n-1}{Z}_{n-k}(-\mu )\frac{{\mu }^k}{k!}(n=1,2,3,\dots ).}$$ A remarkably similar expression relates the "Debye functions"Z_n(z) to the polylogarithm:$${\displaystyle Z_n(z)=\sum _{k=0}^{n-1}{\mathrm{Li}}_{n-k}(e^{-z})\frac{z^k}{k!}(n=1,2,3,\dots ).}$$
• UsingLambert series, if${\displaystyle J_s(n)}$ isJordan's totient function, then$${\displaystyle \sum _{n=1}^{\mathrm{\infty }}\frac{z^n{J}_{-s}(n)}{1-z^n}={\mathrm{Li}}_s(z).}$$

Any of the following integral representations furnishes theanalytic continuation of the polylogarithm beyond the circle of convergence |z| = 1 of the defining power series.

1. The polylogarithm can be expressed in terms of the integral of theBose–Einstein distribution:$${\displaystyle {\mathrm{Li}}_s(z)=\frac{1}{\mathrm{\Gamma }(s)}{\int }_0^{\mathrm{\infty }}\frac{t^{s-1}}{e^t/z-1}dt.}$$This converges for Re(s) > 0 and allz except forz real and ≥ 1. The polylogarithm in this context is sometimes referred to as a Bose integral but more commonly as aBose–Einstein integral (Dingle 1957a,Dingle, Arndt & Roy 1957).^{[note 1]} Similarly, the polylogarithm can be expressed in terms of the integral of theFermi–Dirac distribution:$${\displaystyle -{\mathrm{Li}}_s(-z)=\frac{1}{\mathrm{\Gamma }(s)}{\int }_0^{\mathrm{\infty }}\frac{t^{s-1}}{e^t/z+1}dt.}$$This converges forRe(s) > 0 and allz except forz real and ≤ −1. The polylogarithm in this context is sometimes referred to as a Fermi integral or aFermi–Dirac integral (GSL 2010,Dingle 1957b). These representations are readily verified byTaylor expansion of the integrand with respect toz and termwise integration. The papers of Dingle contain detailed investigations of both types of integrals.The polylogarithm is also related to the integral of theMaxwell–Boltzmann distribution:$${\displaystyle \underset{z\to 0}{lim}\frac{{\mathrm{Li}}_s(z)}{z}=\frac{1}{\mathrm{\Gamma }(s)}{\int }_0^{\mathrm{\infty }}{t}^{s-1}{e}^{-t}dt=1.}$$This also gives theasymptotic behavior of polylogarithm at the vicinity of origin.
2. A complementary integral representation applies to Re(s) < 0 and to allz except toz real and ≥ 0:$${\displaystyle {\mathrm{Li}}_s(z)={\int }_0^{\mathrm{\infty }}\frac{t^{-s}\sin [s\pi /2-t\ln (-z)]}{\sinh (\pi t)}dt.}$$This integral follows from the general relation of the polylogarithm with theHurwitz zeta function (see above) and a familiar integral representation of the latter.
3. The polylogarithm may be quite generally represented by aHankel contour integral (Whittaker & Watson 1927, § 12.22, § 13.13), which extends the Bose–Einstein representation to negative orderss. As long as thet =μpole of the integrand does not lie on the non-negative real axis, ands ≠ 1, 2, 3, …, we have:$${\displaystyle {\mathrm{Li}}_s(e^{\mu })=-\frac{\mathrm{\Gamma }(1-s)}{2\pi i}{\oint }_H\frac{(-t{)}^{s-1}}{e^{t-\mu }-1}dt}$$whereH 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 oft. The integration starts at +∞ on the upper half plane (Im(t) > 0), circles the origin without enclosing any of the polest =μ + 2kπi, and terminates at +∞ on the lower half plane (Im(t) < 0). For the case whereμ is real and non-negative, we can simply subtract the contribution of the enclosedt =μ pole:$${\displaystyle {\mathrm{Li}}_s(e^{\mu })=-\frac{\mathrm{\Gamma }(1-s)}{2\pi i}{\oint }_H\frac{(-t{)}^{s-1}}{e^{t-\mu }-1}dt-2\pi iR}$$whereR is theresidue of the pole:$${\displaystyle R=\frac{i}{2\pi }\mathrm{\Gamma }(1-s)(-\mu {)}^{s-1}.}$$
4. When theAbel–Plana formula is applied to the defining series of the polylogarithm, aHermite-type integral representation results that is valid for all complexz and for all complexs:$${\displaystyle {\mathrm{Li}}_s(z)=\frac{1}{2}z+\frac{\mathrm{\Gamma }(1-s,-\ln z)}{(-\ln z{)}^{1-s}}+2z{\int }_0^{\mathrm{\infty }}\frac{\sin (s\arctan t-t\ln z)}{(1+t^2{)}^{s/2}(e^{2\pi t}-1)}dt}$$where Γ is theupper incomplete gamma-function. All (but not part) of the ln(z) in this expression can be replaced by −ln(¹⁄_z). A related representation which also holds for all complexs,$${\displaystyle {\mathrm{Li}}_s(z)=\frac{1}{2}z+z{\int }_0^{\mathrm{\infty }}\frac{\sin [s\arctan t-t\ln (-z)]}{(1+t^2{)}^{s/2}\sinh (\pi t)}dt,}$$avoids the use of the incomplete gamma function, but this integral fails forz on the positive real axis if Re(s) ≤ 0. This expression is found by writing 2^s Li_s(−z) / (−z) = Φ(z²,s,¹⁄₂) −z Φ(z²,s, 1), where Φ is theLerch transcendent, and applying the Abel–Plana formula to the first Φ series and a complementary formula that involves 1 / (e^2πt + 1) in place of 1 / (e^2πt − 1) to the second Φ series.
5. We can express an integral for the polylogarithm by integrating the ordinarygeometric series termwise for${\displaystyle s\in \mathbb{N}}$ as (Borwein, Borwein & Girgensohn 1995, §2, eqn. 4)$${\displaystyle {\mathrm{Li}}_{s+1}(z)=\frac{z\cdot (-1{)}^s}{s!}{\int }_0^1\frac{{\log }^s(t)}{1-tz}dt.}$$

1. As noted underintegral representations above, the Bose–Einstein integral representation of the polylogarithm may be extended to negative orderss by means ofHankel contour integration:$${\displaystyle {\mathrm{Li}}_s(e^{\mu })=-\frac{\mathrm{\Gamma }(1-s)}{2\pi i}{\oint }_H\frac{(-t{)}^{s-1}}{e^{t-\mu }-1}dt,}$$whereH is the Hankel contour,s ≠ 1, 2, 3, …, and thet =μ pole of the integrand does not lie on the non-negative real axis. Thecontour can be modified so that it encloses thepoles of the integrand att −μ = 2kπi, and the integral can be evaluated as the sum of theresidues (Wood 1992, § 12, 13;Gradshteyn & Ryzhik 2015):$${\displaystyle {\mathrm{Li}}_s(e^{\mu })=\mathrm{\Gamma }(1-s)\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}(2k\pi i-\mu {)}^{s-1}.}$$This will hold for Re(s) < 0 and allμ except wheree^μ = 1. For 0 < Im(μ) ≤ 2π the sum can be split as:$${\displaystyle {\mathrm{Li}}_s(e^{\mu })=\mathrm{\Gamma }(1-s)\left[(-2\pi i{)}^{s-1}\sum _{k=0}^{\mathrm{\infty }}{\left(k+\frac{\mu }{2\pi i}\right)}^{s-1}+(2\pi i{)}^{s-1}\sum _{k=0}^{\mathrm{\infty }}{\left(k+1-\frac{\mu }{2\pi i}\right)}^{s-1}\right],}$$where the two series can now be identified with theHurwitz zeta function:This relation, which has already been given underrelationship to other functions above, holds for all complexs ≠ 0, 1, 2, 3, … and was first derived in (Jonquière 1889, eq. 6).
2. In order to represent the polylogarithm as a power series aboutμ = 0, we write the series derived from the Hankel contour integral as:$${\displaystyle {\mathrm{Li}}_s(e^{\mu })=\mathrm{\Gamma }(1-s)(-\mu {)}^{s-1}+\mathrm{\Gamma }(1-s)\sum _{h=1}^{\mathrm{\infty }}\left[(-2h\pi i-\mu {)}^{s-1}+(2h\pi i-\mu {)}^{s-1}\right].}$$When the binomial powers in the sum are expanded aboutμ = 0 and the order of summation is reversed, the sum overh can be expressed in closed form:$${\displaystyle {\mathrm{Li}}_s(e^{\mu })=\mathrm{\Gamma }(1-s)(-\mu {)}^{s-1}+\sum _{k=0}^{\mathrm{\infty }}\frac{\zeta (s-k)}{k!}{\mu }^k.}$$This result holds for |μ| < 2π and, thanks to the analytic continuation provided by thezeta functions, for alls ≠ 1, 2, 3, … . If the order is a positive integer,s =n, both the term withk =n − 1 and thegamma function become infinite, although their sum does not. One obtains (Wood 1992, § 9;Gradshteyn & Ryzhik 2015):$${\displaystyle \underset{s\to k+1}{lim}\left[\frac{\zeta (s-k)}{k!}{\mu }^k+\mathrm{\Gamma }(1-s)(-\mu {)}^{s-1}\right]=\frac{{\mu }^k}{k!}\left[\sum _{h=1}^k\frac{1}{h}-\ln (-\mu )\right],}$$where the sum overh vanishes ifk = 0. So, for positive integer orders and for |μ| < 2π we have the series:$${\displaystyle {\mathrm{Li}}_n(e^{\mu })=\frac{{\mu }^{n-1}}{(n-1)!}\left[H_{n-1}-\ln (-\mu )\right]+\sum _{k=0,k\ne n-1}^{\mathrm{\infty }}\frac{\zeta (n-k)}{k!}{\mu }^k,}$$whereH_n denotes thenthharmonic number:$${\displaystyle H_n=\sum _{h=1}^n\frac{1}{h},H_0=0.}$$The problem terms now contain −ln(−μ) which, when multiplied byμⁿ⁻¹, will tend to zero asμ → 0, except forn = 1. This reflects the fact that Li_s(z) exhibits a truelogarithmic singularity ats = 1 andz = 1 since:$${\displaystyle \underset{\mu \to 0}{lim}\mathrm{\Gamma }(1-s)(-\mu {)}^{s-1}=0(\mathrm{Re}(s)>1).}$$Fors close, but not equal, to a positive integer, the divergent terms in the expansion aboutμ = 0 can be expected to cause computational difficulties (Wood 1992, § 9). Erdélyi's corresponding expansion (Erdélyi et al. 1981, § 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(¹⁄_z) is not uniformly equal to −ln(z).For nonpositive integer values ofs, the zeta function ζ(s −k) in the expansion aboutμ = 0 reduces toBernoulli numbers: ζ(−n −k) = −B_1+n+k / (1 +n +k). Numerical evaluation of Li_−n(z) by this series does not suffer from the cancellation effects that the finite rational expressions given underparticular values above exhibit for largen.
3. By use of the identity$${\displaystyle 1=\frac{1}{\mathrm{\Gamma }(s)}{\int }_0^{\mathrm{\infty }}{e}^{-t}{t}^{s-1}dt(\mathrm{Re}(s)>0),}$$the Bose–Einstein integral representation of the polylogarithm (see above) may be cast in the form:$${\displaystyle {\mathrm{Li}}_s(z)=\frac{1}{2}z+\frac{z}{2\mathrm{\Gamma }(s)}{\int }_0^{\mathrm{\infty }}{e}^{-t}{t}^{s-1}\coth \frac{t-\ln z}{2}dt(\mathrm{Re}(s)>0).}$$Replacing the hyperbolic cotangent with a bilateral series,$${\displaystyle \coth \frac{t-\ln z}{2}=2\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{1}{2k\pi i+t-\ln z},}$$then reversing the order of integral and sum, and finally identifying the summands with an integral representation of theupper incomplete gamma function, one obtains:$${\displaystyle {\mathrm{Li}}_s(z)=\frac{1}{2}z+\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{\mathrm{\Gamma }(1-s,2k\pi i-\ln z)}{(2k\pi i-\ln z{)}^{1-s}}.}$$For both the bilateral series of this result and that for the hyperbolic cotangent, symmetric partial sums from −k_max tok_max converge unconditionally ask_max → ∞. Provided the summation is performed symmetrically, this series for Li_s(z) thus holds for all complexs as well as all complexz.
4. Introducing an explicit expression for theStirling numbers of the second kind into the finite sum for the polylogarithm of nonpositive integer order (see above) one may write:$${\displaystyle {\mathrm{Li}}_{-n}(z)=\sum _{k=0}^n{\left(\frac{-z}{1-z}\right)}^{k+1}\sum _{j=0}^k(-1{)}^{j+1}(\genfrac{}{}{0ex}{}{k}{j})(j+1{)}^n(n=0,1,2,\dots ).}$$The infinite series obtained by simply extending the outer summation to ∞ (Guillera & Sondow 2008, Theorem 2.1):$${\displaystyle {\mathrm{Li}}_s(z)=\sum _{k=0}^{\mathrm{\infty }}{\left(\frac{-z}{1-z}\right)}^{k+1}\sum _{j=0}^k(-1{)}^{j+1}(\genfrac{}{}{0ex}{}{k}{j})(j+1{)}^{-s},}$$turns out to converge to the polylogarithm for all complexs and for complexz with Re(z) <¹⁄₂, as can be verified for |^−z⁄_(1−z)| <¹⁄₂ by reversing the order of summation and using:$${\displaystyle \sum _{k=j}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{k}{j}){\left(\frac{-z}{1-z}\right)}^{k+1}={\left[{\left(\frac{-z}{1-z}\right)}^{-1}-1\right]}^{-j-1}=(-z{)}^{j+1}.}$$The inner coefficients of these series can be expressed byStirling-number-related formulas involving the generalizedharmonic numbers. For example, seegenerating function transformations to find proofs (references to proofs) of the following identities:For the other arguments with Re(z) <¹⁄₂ the result follows byanalytic continuation. This procedure is equivalent to applyingEuler's transformation to the series inz that defines the polylogarithm.

For |z| ≫ 1, the polylogarithm can be expanded intoasymptotic series in terms of ln(−z):

$${\displaystyle {\mathrm{Li}}_s(z)=\frac{\pm i\pi }{\mathrm{\Gamma }(s)}[\ln (-z)\pm i\pi {]}^{s-1}-\sum _{k=0}^{\mathrm{\infty }}(-1{)}^k(2\pi {)}^{2k}\frac{B_{2k}}{(2k)!}\frac{[\ln (-z)\pm i\pi {]}^{s-2k}}{\mathrm{\Gamma }(s+1-2k)},}$$$${\displaystyle {\mathrm{Li}}_s(z)=\sum _{k=0}^{\mathrm{\infty }}(-1{)}^k(1-2^{1-2k})(2\pi {)}^{2k}\frac{B_{2k}}{(2k)!}\frac{[\ln (-z){]}^{s-2k}}{\mathrm{\Gamma }(s+1-2k)},}$$

whereB_2k are theBernoulli numbers. Both versions hold for alls and for any arg(z). As usual, the summation should be terminated when the terms start growing in magnitude. For negative integers, the expansions vanish entirely; for non-negative integers, they break off after a finite number of terms.Wood (1992, § 11) describes a method for obtaining these series from the Bose–Einstein integral representation (his equation 11.2 for Li_s(e^μ) requires −2π < Im(μ) ≤ 0).

The followinglimits result from the various representations of the polylogarithm (Wood 1992, § 22):

$${\displaystyle {\mathrm{Li}}_s(z){\sim }_{|z|\to 0}z}$$$${\displaystyle {\mathrm{Li}}_s(\pm e^{\mu }){\sim }_{\mathrm{Re}(\mu )\to \mathrm{\infty }}-\frac{{\mu }^s}{\mathrm{\Gamma }(s+1)}(s\ne -1,-2,-3,\dots )}$$$${\displaystyle {\mathrm{Li}}_{-n}(e^{\mu }){\sim }_{\mathrm{Re}(\mu )\to \mathrm{\infty }}-(-1{)}^n{e}^{-\mu }(n=1,2,3,\dots )}$$$${\displaystyle {\mathrm{Li}}_s(z){\sim }_{\mathrm{Re}(s)\to \mathrm{\infty }}z}$$$${\displaystyle {\mathrm{Li}}_s(-e^{\mu }){\sim }_{\mathrm{Re}(s)\to -\mathrm{\infty }}\mathrm{\Gamma }(1-s)\left[(-\mu -i\pi {)}^{s-1}+(-\mu +i\pi {)}^{s-1}\right](\mathrm{Im}(\mu )=0)}$$

Wood's first limit forRe(μ) → ∞ has been corrected in accordance with his equation 11.3. The limit forRe(s) → −∞ follows from the general relation of the polylogarithm with theHurwitz zeta function (see above).

The dilogarithm is the polylogarithm of orders = 2. An alternate integral expression of the dilogarithm for arbitrary complex argumentz is (Abramowitz & Stegun 1972, § 27.7):$${\displaystyle {\mathrm{Li}}_2(z)=-{\int }_0^z\frac{\ln (1-t)}{t}dt=-{\int }_0^1\frac{\ln (1-zt)}{t}dt.}$$

A source of confusion is that somecomputer algebra systems define the dilogarithm as dilog(z) = Li₂(1−z).

In the case of realz ≥ 1 the first integral expression for the dilogarithm can be written as$${\displaystyle {\mathrm{Li}}_2(z)=\frac{{\pi }^2}{6}-{\int }_1^z\frac{\ln (t-1)}{t}dt-i\pi \ln z}$$

from which expanding ln(t−1) and integrating term by term we obtain

$${\displaystyle {\mathrm{Li}}_2(z)=\frac{{\pi }^2}{3}-\frac{1}{2}(\ln z{)}^2-\sum _{k=1}^{\mathrm{\infty }}\frac{1}{k^2{z}^k}-i\pi \ln z(z\ge 1).}$$

TheAbel identity for the dilogarithm is given by (Abel 1881)

$${\displaystyle {\mathrm{Li}}_2\left(\frac{x}{1-y}\right)+{\mathrm{Li}}_2\left(\frac{y}{1-x}\right)-{\mathrm{Li}}_2\left(\frac{xy}{(1-x)(1-y)}\right)={\mathrm{Li}}_2(x)+{\mathrm{Li}}_2(y)+\ln (1-x)\ln (1-y)}$$

This is immediately seen to hold for eitherx = 0 ory = 0, and for general arguments is then easily verified by differentiation ∂/∂x ∂/∂y. Fory = 1−x the identity reduces toEuler'sreflection formula$${\displaystyle {\mathrm{Li}}_2\left(x\right)+{\mathrm{Li}}_2\left(1-x\right)=\frac{1}{6}{\pi }^2-\ln (x)\ln (1-x),}$$where Li₂(1) = ζ(2) =¹⁄₆π² has been used andx may take any complex value.

In terms of the new variablesu =x/(1−y),v =y/(1−x) the Abel identity reads$${\displaystyle {\mathrm{Li}}_2(u)+{\mathrm{Li}}_2(v)-{\mathrm{Li}}_2(uv)={\mathrm{Li}}_2\left(\frac{u-uv}{1-uv}\right)+{\mathrm{Li}}_2\left(\frac{v-uv}{1-uv}\right)+\ln \left(\frac{1-u}{1-uv}\right)\ln \left(\frac{1-v}{1-uv}\right),}$$which corresponds to thepentagon identity given in (Rogers 1907).

From the Abel identity forx =y = 1−z and the square relationship we haveLanden's identity$${\displaystyle {\mathrm{Li}}_2(1-z)+{\mathrm{Li}}_2\left(1-\frac{1}{z}\right)=-\frac{1}{2}(\ln z{)}^2(z\notin ]-\mathrm{\infty };0]),}$$and applying the reflection formula to each dilogarithm we find the inversion formula$${\displaystyle {\mathrm{Li}}_2(z)+{\mathrm{Li}}_2(1/z)=-\frac{1}{6}{\pi }^2-\frac{1}{2}[\ln (-z){]}^2(z\notin [0;1[),}$$

and for realz ≥ 1 also$${\displaystyle {\mathrm{Li}}_2(z)+{\mathrm{Li}}_2(1/z)=\frac{1}{3}{\pi }^2-\frac{1}{2}(\ln z{)}^2-i\pi \ln z.}$$

Known closed-form evaluations of the dilogarithm at special arguments are collected in the table below. Arguments in the first column are related by reflectionx ↔ 1−x or inversionx ↔¹⁄_x to eitherx = 0 orx = −1; arguments in the third column are all interrelated by these operations.

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 (Maximon 2003, § 10); an equivalent toAbel's identity was already published bySpence in 1809, before Abel wrote his manuscript in 1826 (Zagier 1989, § 2). The designationbilogarithmische Function was introduced byCarl Johan Danielsson Hill (professor in Lund, Sweden) in 1828 (Maximon 2003, § 10).Don Zagier (1989) has remarked that the dilogarithm is the only mathematical function possessing a sense of humor.

Special values of the dilogarithm

${\displaystyle x}$	${\displaystyle {\mathrm{Li}}_2(x)}$	${\displaystyle x}$	${\displaystyle {\mathrm{Li}}_2(x)}$
${\displaystyle -1}$	${\displaystyle -\frac{1}{12}{\pi }^2}$	${\displaystyle -\varphi }$	${\displaystyle -\frac{1}{10}{\pi }^2-{\ln }^2\varphi }$
${\displaystyle 0}$	${\displaystyle 0}$	${\displaystyle -1/\varphi }$	${\displaystyle -\frac{1}{15}{\pi }^2+\frac{1}{2}{\ln }^2\varphi }$
${\displaystyle \frac{1}{2}}$	${\displaystyle \frac{1}{12}{\pi }^2-\frac{1}{2}{\ln }^{2}2}$	${\displaystyle 1/{\varphi }^2}$	${\displaystyle \frac{1}{15}{\pi }^2-{\ln }^2\varphi }$
${\displaystyle 1}$	${\displaystyle \frac{1}{6}{\pi }^2}$	${\displaystyle 1/\varphi }$	${\displaystyle \frac{1}{10}{\pi }^2-{\ln }^2\varphi }$
${\displaystyle 2}$	${\displaystyle \frac{1}{4}{\pi }^2-\pi i\ln 2}$	${\displaystyle \varphi }$	${\displaystyle \frac{11}{15}{\pi }^2+\frac{1}{2}{\ln }^2(-1/\varphi )}$
		${\displaystyle {\varphi }^2}$	${\displaystyle -\frac{11}{15}{\pi }^2-{\ln }^2(-\varphi )}$

Here${\displaystyle \varphi =\frac{1}{2}(\sqrt{5}+1)}$ denotes thegolden ratio.

Leonard Lewin discovered a remarkable and broad generalization of a number of classical relationships on the polylogarithm for special values. These are now calledpolylogarithm ladders. Define${\displaystyle \rho =\frac{1}{2}(\sqrt{5}-1)}$ as the reciprocal of thegolden ratio. Then two simple examples of dilogarithm ladders are

$${\displaystyle {\mathrm{Li}}_2({\rho }^6)=4{\mathrm{Li}}_2({\rho }^3)+3{\mathrm{Li}}_2({\rho }^2)-6{\mathrm{Li}}_2(\rho )+\frac{7}{30}{\pi }^2}$$

given byCoxeter (1935) and

$${\displaystyle {\mathrm{Li}}_2(\rho )=\frac{1}{10}{\pi }^2-{\ln }^2\rho }$$

given byLanden. Polylogarithm ladders occur naturally and deeply inK-theory andalgebraic geometry. Polylogarithm ladders provide the basis for the rapid computations of various mathematical constants by means of theBBP algorithm (Bailey, Borwein & Plouffe 1997).

The polylogarithm has twobranch points; one atz = 1 and another atz = 0. The second branch point, atz = 0, is not visible on the main sheet of the polylogarithm; it becomes visible only when the function isanalytically continued to its other sheets. Themonodromy group for the polylogarithm consists of thehomotopy classes of loops that wind around the two branch points. Denoting these two bym₀ andm₁, the monodromy group has thegroup presentation

$${\displaystyle ⟨m_0,m_1|w=m_0{m}_1{m}_0^{-1}{m}_1^{-1},w{m}_1=m_{1}w⟩.}$$

For the special case of the dilogarithm, one also has thatwm₀ =m₀w, and the monodromy group becomes theHeisenberg group (identifyingm₀,m₁ andw withx,y,z) (Vepstas 2008).