Polylogarithm

Not to be confused with polylogarithmic.

The polylogarithm (also known as de Jonquière's function) is a special function Li_s(z) that is defined by the sum

$\operatorname {Li} _{s}(z)=\sum _{k=1}^{\infty }{z^{k} \over k^{s}}.$

It is in general not an elementary function, unlike the related logarithm function. The above definition is valid for all complex numbers s and z where |z| < 1. The polylogarithm is defined over a larger range of z than the above definition allows by the process of analytic continuation.

Different polylogarithm functions in the complex plane

$\operatorname {Li} _{-3}(z)$	$\operatorname {Li} _{-2}(z)$	$\operatorname {Li} _{-1}(z)$	$\operatorname {Li} _{0}(z)$	$\operatorname {Li} _{1}(z)$	$\operatorname {Li} _{2}(z)$	$\operatorname {Li} _{3}(z)$

The special case s = 1 involves the ordinary natural logarithm (Li₁(z) = −ln(1−z)) while the special cases s = 2 and 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 alternatively be defined as the repeated integral of itself, namely that

$\operatorname {Li} _{s+1}(z)=\int _{0}^{z}{\frac {\operatorname {Li} _{s}(t)}{t}}\,\mathrm {d} t.$

Thus the dilogarithm is an integral of the logarithm, and so on. For negative integer values of s, the polylogarithm is a rational function.

The polylogarithm also arises in the closed form of the integral of the Fermi–Dirac distribution and the Bose–Einstein distribution and is sometimes known as the Fermi–Dirac integral or the Bose–Einstein integral. Polylogarithms should not be confused with polylogarithmic functions nor with the offset logarithmic integral which has a similar notation.

Properties

In the important case where the parameter s is an integer, it will be represented by n (or −n when negative). It is often convenient to define μ = ln(z) where ln(z) is the principal branch of the complex logarithm Ln(z) so that −π < Im(μ) ≤ π. Also, all exponentiation will be assumed to be single valued: z^s = exp(s ln(z)).

Depending on the parameter s, the polylogarithm may be multi-valued. The principal branch of the polylogarithm is chosen to be that for which Li_s(z) is real for z real, 0 ≤ z ≤ 1 and is continuous except on the positive real axis, where a cut is made from z = 1 to ∞ such that the cut puts the real axis on the lower half plane of z. In terms of μ, this amounts to −π < arg(−μ) ≤ π. The fact that the polylogarithm may be discontinuous in μ can cause some confusion.

For z real and z ≥ 1 the imaginary part of the polylogarithm is Template:Ref harvard:

${\textrm {Im}}(\operatorname {Li} _{s}(z))=-{{\pi \mu ^{s-1}} \over {\Gamma (s)}}.$

Going across the cut, if δ is an infinitesimally small positive real number, then:

${\textrm {Im}}(\operatorname {Li} _{s}(z+i\delta ))={{\pi \mu ^{s-1}} \over {\Gamma (s)}}.$

The derivatives of the polylogarithm are:

$z{\partial \operatorname {Li} _{s}(z) \over \partial z}=\operatorname {Li} _{s-1}(z)$

${\partial \operatorname {Li} _{s}(e^{\mu }) \over \partial \mu }=\operatorname {Li} _{s-1}(e^{\mu }).$

The square relationship is easily seen from the duplication formula (see also Template:Ref harvard, Template:Ref harvard):

$\operatorname {Li} _{s}(-z)+\operatorname {Li} _{s}(z)=2^{1-s}~\operatorname {Li} _{s}(z^{2}).$

Note that Kummer's function obeys a very similar duplication formula. This is a special case of the multiplication formula, for any integer p:

$\sum _{m=0}^{p-1}\operatorname {Li} _{s}(ze^{2\pi im/p})=p^{1-s}\,\operatorname {Li} _{s}(z^{p})$

which can be proven using the summation definition of the polylogarithm and the orthogonality of the exponential terms (e.g. see Discrete Fourier transform).

Particular values

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

For integer values of s, we have the following explicit expressions:

$\operatorname {Li} _{1}(z)=-\ln \left(1-z\right)$

$\operatorname {Li} _{0}(z)={z \over 1-z}$

$\operatorname {Li} _{-1}(z)={z \over (1-z)^{2}}$

$\operatorname {Li} _{-2}(z)={z(1+z) \over (1-z)^{3}}$

$\operatorname {Li} _{-3}(z)={z(1+4z+z^{2}) \over (1-z)^{4}}$

$\operatorname {Li} _{-4}(z)={z(1+z)(1+10z+z^{2}) \over (1-z)^{5}}.$

The polylogarithm for all negative integer values of s can be expressed as a ratio of polynomials in z and are therefore rational functions (See series representations below). Some particular expressions for half-integer values of the argument are:

$\operatorname {Li} _{1}({\tfrac {1}{2}})=\ln 2$

$\operatorname {Li} _{2}({\tfrac {1}{2}})={\tfrac {1}{12}}[\pi ^{2}-6(\ln 2)^{2}]$

$\operatorname {Li} _{3}({\tfrac {1}{2}})={\tfrac {1}{24}}[4(\ln 2)^{3}-2\pi ^{2}\ln 2+21\,\zeta (3)]$

where ζ is the Riemann zeta function. No similar formulas of this type are known for higher orders Template:Ref harvard

Integral representations

• The integral of the Bose–Einstein distribution is expressed in terms of a polylogarithm:

$\operatorname {Li} _{s+1}(z)={1 \over \Gamma (s+1)}\int _{0}^{\infty }{t^{s} \over e^{t}/z-1}\,\mathrm {d} t.$

This converges for Re(s) > 0 and all z except for z real and ≥ 1. The polylogarithm in this context is sometimes referred to as a Bose integral or a Bose-Einstein integral.

• The integral of the Fermi-Dirac distribution is also expressed in terms of a polylogarithm:

$-\operatorname {Li} _{s+1}(-z)={1 \over \Gamma (s+1)}\int _{0}^{\infty }{t^{s} \over e^{t}/z+1}\,\mathrm {d} t.$

This converges for Re(s) > 0 and all z except for z real and < (−1). The polylogarithm in this context is sometimes referred to as a Fermi integral or a Fermi–Dirac integral. Template:Ref harvard

• The polylogarithm may be rather generally represented by a Hankel contour integral Template:Ref harvard. As long as the t = μ pole of the integrand does not lie on the non-negative real axis, and s ≠ 1, 2, 3, …, we have:

$\operatorname {Li} _{s}(e^{\mu })={{-\Gamma (1-s)} \over {2\pi i}}\oint _{H}{{(-t)^{s-1}} \over {e^{t-\mu }-1}}\,\mathrm {d} t$

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 of the sheet (Im(t) < 0). The integration starts at +∞ on the upper half of the sheet, circles the origin without enclosing any of the poles t = µ + 2 k π i, and terminates at +∞ on the lower half of the sheet. For the case where µ is real and non-negative, we can simply subtract the contribution of the enclosed t = µ pole:

$\operatorname {Li} _{s}(e^{\mu })=-{{\Gamma (1-s)} \over {2\pi i}}\oint _{H}{{(-t)^{s-1}} \over {e^{t-\mu }}-1}\,\mathrm {d} t-2\pi iR$

where R is the residue of the pole:

$R={i \over 2\pi }\Gamma (1-s)(-\mu )^{s-1}.$

• A Hermite-like integral representation valid for all complex z and for all complex s is:

$\operatorname {Li} _{s}(z)={z \over 2}+(-\ln z)^{s-1}\,\Gamma (1-s,-\ln z)+2z\int _{0}^{\infty }{\frac {\sin(s\arctan t\,-\,t\ln z)}{(1+t^{2})^{s/2}\,(e^{2\pi t}-1)}}\,\mathrm {d} t$

where Γ is the upper incomplete gamma-function. Note that all (but not part) of the ln(z) in this expression can be replaced by -ln(1/z).

Relationship to other functions

• For z = 1 the polylogarithm reduces to the Riemann zeta function

$\operatorname {Li} _{s}(1)=\zeta (s)\qquad ({\textrm {Re}}(s)>1).$

• The polylogarithm is related to Dirichlet eta function and the Dirichlet beta function:

$\operatorname {Li} _{s}(-1)=-\eta \left(s\right)$

where η(s) is the Dirichlet eta function. For pure imaginary arguments, we have:

$\operatorname {Li} _{s}(\pm i)=2^{-s}~\eta (s)\pm i~\beta (s)\,$

where β(s) is the Dirichlet beta function.

• The polylogarithm is equivalent to the Fermi-Dirac integral Template:Ref harvard

$F_{s}(\mu )=-\operatorname {Li} _{s+1}(-e^{\mu }).\,$

• The polylogarithm is a special case of the incomplete polylogarithm function

$\operatorname {Li} _{s}(z)=\operatorname {Li} _{s}(0,z).$

• The polylogarithm is a special case of the Lerch Transcendent Template:Ref harvard

$\operatorname {Li} _{s}(z)=z~\Phi (z,s,1).$

• The polylogarithm is related to the Hurwitz zeta function by:

$\operatorname {Li} _{s}(z)={\Gamma (1-s) \over (2\pi )^{1-s}}\left[i^{1-s}~\zeta \left(1\!-\!s,~{\frac {1}{2}}+{\ln(-z) \over {2\pi i}}\right)+i^{s-1}~\zeta \left(1\!-\!s,~{\frac {1}{2}}-{\ln(-z) \over {2\pi i}}\right)\right]$

where Γ(1-s) is the gamma function, which causes the relation to fail for positive integer s. It is also related to the Hurwitz zeta function by Jonquière's identity:

$i^{-s}~\operatorname {Li} _{s}(e^{2\pi ix})+i^{s}~\operatorname {Li} _{s}(e^{-2\pi ix})={(2\pi )^{s} \over \Gamma (s)}~\zeta \left(1\!-\!s,\,x\right)$

which holds for 0 ≤ Re(x) < 1 if Im(x) ≥ 0, and for 0 < Re(x) ≤ 1 if Im(x) < 0. (Note that Erdélyi's equivalent equation Template:Ref harvard is not correct if we assume that the principal branches of the polylogarithm and the logarithm are used simultaneously.) Alternatively, for all $s\in \mathbb {C}$ and for all $~z\not \in ~]0;1]$, the inversion formula reads

$\operatorname {Li} _{s}(z)+(-1)^{s}~\operatorname {Li} _{s}(1/z)={(2\pi i)^{s} \over \Gamma (s)}~\zeta \left(1\!-\!s,~{\frac {1}{2}}+{\ln(-z) \over {2\pi i}}\right),$

and for all $s\in \mathbb {C}$ and for all $z~\not \in ~]1;\infty [$

$\operatorname {Li} _{s}(z)+(-1)^{s}~\operatorname {Li} _{s}(1/z)={(2\pi i)^{s} \over \Gamma (s)}~\zeta \left(1\!-\!s,~{\frac {1}{2}}-{\ln(-1/z) \over {2\pi i}}\right).$

Note that this furnishes the analytic continuation of the series representation of the polylogarithm beyond its circle of convergence |z| = 1. See below for a simplified formula when $s$ is an integer.

• From the relationship between the Hurwitz zeta function and the Bernoulli polynomials:

$\zeta (-n,x)=-{B_{n+1}(x) \over n+1}$

which holds for all x and n = 0, 1, 2, 3, … it follows that:

$\operatorname {Li} _{n}(e^{2\pi ix})+(-1)^{n}~\operatorname {Li} _{n}(e^{-2\pi ix})=-{(2\pi i)^{n} \over n!}B_{n}\left({x}\right)$

under the same constraints on x as above. (Note that the corresponding equationTemplate:Ref harvard is not correct.) For negative integer values of the parameter, we have for all z Template:Ref harvard:

$\operatorname {Li} _{-n}(z)+(-1)^{n}~\operatorname {Li} _{-n}\left(1/z\right)=0\qquad (n=1,2,3,\ldots )$

More generally for $n=0,\pm 1,\pm 2,\pm 3,\cdots$

$\operatorname {Li} _{n}(z)+(-1)^{n}~\operatorname {Li} _{n}\left(1/z\right)=-{\frac {(2\pi i)^{n}}{n!}}\,B_{n}\left({\frac {1}{2}}+{\ln(-z) \over {2\pi i}}\right)\qquad (z~\not \in ~]0;1]),$

$\operatorname {Li} _{n}(z)+(-1)^{n}~\operatorname {Li} _{n}\left(1/z\right)=-{\frac {(2\pi i)^{n}}{n!}}\,B_{n}\left({\frac {1}{2}}-{\ln(-1/z) \over {2\pi i}}\right)\qquad (z~\not \in ~]1;\infty [).$

• The polylogarithm with pure imaginary μ may be expressed in terms of Clausen functions Ci_s(θ)and Si_s(θ) Template:Ref harvard, Template:Ref harvard

$\operatorname {Li} _{s}(e^{\pm i\theta })=Ci_{s}(\theta )\pm i~Si_{s}(\theta ).$

• The Inverse tangent integral Ti_s(z) Template:Ref harvard can be expressed in terms of polylogarithms:

$\operatorname {Li} _{s}(\pm iy)=2^{-s}~\operatorname {Li} _{s}(-y^{2})\pm i~Ti_{s}(y).$

• The Legendre chi function χ_s(z) Template:Ref harvard, Template:Ref harvard can be expressed in terms of polylogarithms:

$\chi _{s}(z)={1 \over 2}~[\operatorname {Li} _{s}(z)-\operatorname {Li} _{s}(-z)].$

• In terms of the "Debye functions":

$Z_{n}(z)={1 \over (n-1)!}\int _{z}^{\infty }{t^{n-1} \over e^{t}-1}\,\mathrm {d} t\qquad (n=1,2,3,\ldots )$

the polylogarithm Li_n(z) for positive integer n may be expressed as the finite sum Template:Ref harvard:

$\operatorname {Li} _{n}(e^{\mu })=\sum _{k=0}^{n-1}Z_{n-k}(-\mu ){\mu ^{k} \over k!}\qquad (n=1,2,3,\ldots ).$

A remarkably similar expression relates the function Z_n(z) to the polylogarithm:

$Z_{n}(\mu )=\sum _{k=0}^{n-1}\operatorname {Li} _{n-k}(e^{-\mu }){\mu ^{k} \over k!}\qquad (n=1,2,3,\ldots ).$

Series representations

We may represent the polylogarithm as a power series about μ = 0 as follows: Template:Ref harvard Consider the Mellin transform:

$M_{s}(r)=\int _{0}^{\infty }{\textrm {Li}}_{s}(fe^{-u})u^{r-1}\,\mathrm {d} u={1 \over \Gamma (s)}\int _{0}^{\infty }\int _{0}^{\infty }{t^{s-1}u^{r-1} \over e^{t+u}/f-1}\,\mathrm {d} t\,\mathrm {d} u.$

The change of variables t = ab, u = a(1 - b) allows the integrals to be separated:

$M_{s}(r)={1 \over \Gamma (s)}\int _{0}^{1}b^{r-1}(1-b)^{s-1}\,\mathrm {d} b\int _{0}^{\infty }{a^{s+r-1} \over e^{a}/f-1}\,\mathrm {d} a=\Gamma (r){\textrm {Li}}_{s+r}(f).$

For f = 1 we have, through the inverse Mellin transform:

$\operatorname {Li} _{s}(e^{-u})={1 \over 2\pi i}\int _{c-i\infty }^{c+i\infty }\Gamma (r)\zeta (s+r)u^{-r}\,\mathrm {d} r$

where c is a constant to the right of the poles of the integrand. The path of integration may be converted into a closed contour, and the poles of the integrand are those of Γ(r) at r = 0, −1, −2, …, and of ζ(s + r) at r = 1 - s. Summing the residues gives, for |μ| < 2π and s ≠ 1, 2, 3, …

$\operatorname {Li} _{s}(e^{\mu })=\Gamma (1-s)(-\mu )^{s-1}+\sum _{k=0}^{\infty }{\zeta (s-k) \over k!}~\mu ^{k}.$

If the parameter s is a positive integer, n, both the k = n - 1 term and the gamma function become infinite, although their sum does not. For integers k > 0 we have:

$\lim _{s\rightarrow k+1}\left[{\zeta (s-k)\mu ^{k} \over k!}+\Gamma (1-s)(-\mu )^{s-1}\right]={\mu ^{k} \over k!}\left(\sum _{m=1}^{k}{1 \over m}-\ln(-\mu )\right)$

and for k = 0:

$\lim _{s\rightarrow 1}\left[\zeta (s)+\Gamma (1-s)(-\mu )^{s-1}\right]=-\ln(-\mu ).$

So, for s = n where n is a positive integer and |μ| < 2π we have the following:

$\operatorname {Li} _{n}(e^{\mu })={\mu ^{n-1} \over (n-1)!}\left(H_{n-1}-\ln(-\mu )\right)+$

$\sum _{k=0,k\neq n-1}^{\infty }{\zeta (n-k) \over k!}~\mu ^{k},~~~~~~~~~~~~~~~~~~~~~~n=2,3,4,\ldots$

$\operatorname {Li} _{1}(e^{\mu })=-\ln(-\mu )+\sum _{k=1}^{\infty }{\zeta (1-k) \over k!}~\mu ^{k},~~~~~~~~~~(n=1)$

where H_n is a harmonic number:

$H_{n}=\sum _{k=1}^{n}{1 \over k}.$

The problem terms now contain −ln(−μ) which, when multiplied by μ^k will tend to zero as μ tends to zero, except for k = 0. This reflects the fact that there is a true logarithmic singularity in Li_s(z) at s = 1 and z = 1 since:

$\lim _{\mu \rightarrow 0}\Gamma (1-s)(-\mu )^{s-1}=0~~~~~({\textrm {Re}}(s)>1)$

Using the relationship between the Riemann zeta function and the Bernoulli numbers B_k:

$\zeta (-n)=(-1)^{n}{B_{n+1} \over n+1},~~~~~~~~~~~n=0,1,2,3,\ldots$

we obtain for negative integer values of s and |μ| < 2π:

$\operatorname {Li} _{-n}(z)={n! \over (-\mu )^{n+1}}-\sum _{k=0}^{\infty }{B_{k+n+1} \over k!~(k+n+1)}~\mu ^{k},~~~~~~~~~~~n=1,2,3,\ldots$

since, except for B₁, all odd Bernoulli numbers are zero. We obtain the n = 0 term using ζ(0) = B₁ = −¹⁄₂. Note again that Erdélyi's equivalent Equation Template:Ref harvard is not correct if we assume that the principal branches of the polylogarithm and the logarithm are used simultaneously, since ln(¹⁄_z) is not uniformly equal to −ln(z).

As noted above, the polylogarithm may be extended to negative values of the parameter s using a Hankel contour integral Template:Ref harvard Template:Ref harvard:

$\operatorname {Li} _{s}(e^{\mu })=-{\Gamma (1-s) \over 2\pi i}\oint _{H}{(-t)^{s-1} \over e^{t-\mu }-1}\,\mathrm {d} t$

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 Hankel contour can be modified so that it encloses the poles of the integrand, at $t-\mu =2k\pi i$ and the integral can be evaluated as the sum of the residues:

$\operatorname {Li} _{s}(e^{\mu })=\Gamma (1-s)\sum _{k=-\infty }^{\infty }(2k\pi i-\mu )^{s-1}.$

This will hold for Re(s) < 0 and all μ except where e^μ = 1. Summing the series, one obtains

$\operatorname {Li} _{s}(e^{\mu })=-\sum _{k=0}^{\infty }{\frac {1}{k!}}\left[1-{\frac {2}{2^{s-k}}}\right]\zeta (s-k)(\mu -\pi i)^{k}$

Note that this sum can be more compactly written in terms of the Dirichlet eta function.

For negative integer s, the polylogarithm may be expressed as a series involving the Eulerian numbers

$\operatorname {Li} _{-n}(z)={1 \over (1-z)^{n+1}}\sum _{i=0}^{n-1}\left\langle {n \atop i}\right\rangle z^{n-i},~~~~~~~~~~~~~n=1,2,3,\ldots$

where $\left\langle {n \atop i}\right\rangle$ are Eulerian numbers:

Another explicit formula for negative integer s is Template:Ref harvard:

$\operatorname {Li} _{-n}(z)=\sum _{k=1}^{n+1}{(-1)^{n+k+1}(k-1)!S(n+1,k) \over (1-z)^{k}}~~~~~~~~~~(n=1,2,3,\ldots )$

where S(n,k) are Stirling numbers of the second kind.

Limiting behavior

The following limits hold for the polylogarithm Template:Ref harvard:

$\lim _{|z|\rightarrow 0}\operatorname {Li} _{s}(z)=0$

$\lim _{s\rightarrow \infty }\operatorname {Li} _{s}(z)=z$

$\lim _{\mathrm {Re} (\mu )\rightarrow \infty }\operatorname {Li} _{s}(-e^{\mu })=-{\mu ^{s} \over \Gamma (s+1)}~~~~~~(s\neq -1,-2,-3,\ldots )$

$\lim _{\mathrm {Re} (\mu )\rightarrow \infty }\operatorname {Li} _{-n}(e^{\mu })=-(-1)^{n}e^{-\mu }~~~~~~(n=1,2,3,\ldots )$

$\lim _{|\mu |\rightarrow 0}\operatorname {Li} _{s}(e^{\mu })=\Gamma (1-s)(-\mu )^{s-1}~~~~~~(s<1)$

Dilogarithm

The dilogarithm is just the polylogarithm with $s=2$. An alternate integral expression for the dilogarithm for arbitrary z is: Template:Ref harvard

$\operatorname {Li} _{2}(z)=-\int _{0}^{z}{\ln(1-t) \over t}\,\mathrm {d} t.$

A source of confusion is that some computer algebra systems define the dilogarithm as dilog($z$) = Li₂($1-z$).

In the case of z ≥ 1 the integral expression for the dilogarithm can be written as

${\mbox{Li}}_{2}(z)={\frac {\pi ^{2}}{6}}-\int _{1}^{z}{\ln(t-1) \over t}\,\mathrm {d} t-i\pi \ln(z)$

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

${\mbox{Li}}_{2}(z)={\frac {\pi ^{2}}{3}}-{\frac {1}{2}}(\ln(z))^{2}-\sum _{k=1}^{\infty }{1 \over k^{2}z^{k}}-i\pi \ln(z)$

The Abel identity for the dilogarithm is given by

$\ln(1-x)\ln(1-y)={\mbox{Li}}_{2}\left({\frac {x}{1-y}}\right)+{\mbox{Li}}_{2}\left({\frac {y}{1-x}}\right)-{\mbox{Li}}_{2}\left(x\right)-{\mbox{Li}}_{2}\left(y\right)-{\mbox{Li}}_{2}\left({\frac {xy}{(1-x)(1-y)}}\right)$

for $~x\not \in ~]1;\infty [,~y\not \in ~]1;\infty [.$

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 Euler's reflection formula

${\mbox{Li}}_{2}\left(x\right)+{\mbox{Li}}_{2}\left(1-x\right)={\frac {\pi ^{2}}{6}}-\ln(x)\ln(1-x)$

where Li₂(1) = ¹⁄₆ π² has been used.

In terms of the new variables $u=x/(1-y)$, $v=y/(1-x)$ the Abel identity reads

${\mbox{Li}}_{2}(u)+{\mbox{Li}}_{2}(v)-{\mbox{Li}}_{2}(uv)={\mbox{Li}}_{2}\left({\frac {u-uv}{1-uv}}\right)+{\mbox{Li}}_{2}\left({\frac {v-uv}{1-uv}}\right)+\ln \left({\frac {1-u}{1-uv}}\right)\ln \left({\frac {1-v}{1-uv}}\right),$

which corresponds to the Pentagon Identity given by Template:Ref harvard.

From the Abel identity for $y=x$ and the square relationship we have Landen's Identity

${\mbox{Li}}_{2}(-x)=-{\mbox{Li}}_{2}\left({\frac {x}{1+x}}\right)-{\frac {1}{2}}(\ln(x+1))^{2}\qquad (x\not \in ~]-\infty ;-1[),$

and the inversion formula for the dilogarithm reads

${\mbox{Li}}_{2}(x)+{\mbox{Li}}_{2}\left({\frac {1}{x}}\right)=-{\frac {\pi ^{2}}{6}}-{\frac {1}{2}}(\ln(-x))^{2}\qquad (x\not \in ~]0;1[),$

and for x ≥ 1 also

${\mbox{Li}}_{2}(x)+{\mbox{Li}}_{2}\left({\frac {1}{x}}\right)={\frac {\pi ^{2}}{3}}-{\frac {1}{2}}(\ln(x))^{2}-i\pi \ln(x).$

Historical note: Don Zagier remarked that "The dilogarithm is the only mathematical function with a sense of humor".

Polylogarithm ladders

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 $\rho ={\tfrac {1}{2}}({\sqrt {5}}-1)$ as the reciprocal of the golden ratio. Then two simple examples of results from ladders include

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

given by Template:Ref harvard and

$\operatorname {Li} _{2}(\rho )={\frac {\pi ^{2}}{10}}-\ln ^{2}\rho$

given by 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.

Monodromy

The polylogarithm has two branch points; 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 polylog 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 $m_{0}$ and $m_{1}$, the monodromy group has the group presentation

$\langle m_{0},m_{1}\vert w=m_{0}m_{1}m_{0}^{-1}m_{1}^{-1},\,wm_{1}=m_{1}w\rangle .$

For the special case of the dilogarithm, one also has that $wm_{0}=m_{0}w$, and the monodromy group becomes the Heisenberg group (identifying $m_{0},m_{1}$ and $w$ with $x,y,z$). Template:Ref harvard

References

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• Bailey, David (1997). "On the Rapid Computation of Various Polylogarithmic Constants" (PDF). Mathematics of Computation. 66 (218): 903–913. doi:10.1090/S0025-5718-97-00856-9. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help); Unknown parameter |month= ignored (help)
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• Coxeter, H.S.M. (1935). "The functions of Schlafli and Lobatschefsky". Quarterly Journal of Mathematics (Oxford). 6: 13–29. doi:10.1093/qmath/os-6.1.13.
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