Polylogarithm

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

Failed to parse (Missing texvc executable; please see math/README to configure.): \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.

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

Failed to parse (Missing texvc executable; please see math/README to configure.): \operatorname{Li}_{s+1}(z) = \int_0^z \frac {\operatorname{Li}_s(t)}{t}dt

so that 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. (e.g. 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 (Wood 1992):

Failed to parse (Missing texvc executable; please see math/README to configure.): \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:

Failed to parse (Missing texvc executable; please see math/README to configure.): \textrm{Im}(\operatorname{Li}_s(z+i\delta)) = {{\pi \mu^{s-1}}\over{\Gamma(s)}}.

The derivatives of the polylogarithm are:

Failed to parse (Missing texvc executable; please see math/README to configure.): z{\partial \operatorname{Li}_s(z) \over \partial z} = \operatorname{Li}_{s-1}(z)

Failed to parse (Missing texvc executable; please see math/README to configure.): {\partial \operatorname{Li}_s(e^\mu) \over \partial \mu} = \operatorname{Li}_{s-1}(e^\mu).

Particular values

See also the "Relationship to other functions" section below.

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

Failed to parse (Missing texvc executable; please see math/README to configure.): \operatorname{Li}_{1}(z) = -\textrm{Ln}\left(1-z\right)

Failed to parse (Missing texvc executable; please see math/README to configure.): \operatorname{Li}_{0}(z) = {z \over 1-z}

Failed to parse (Missing texvc executable; please see math/README to configure.): \operatorname{Li}_{-1}(z) = {z \over (1-z)^2}

Failed to parse (Missing texvc executable; please see math/README to configure.): \operatorname{Li}_{-2}(z) = {z(1+z) \over (1-z)^3}

Failed to parse (Missing texvc executable; please see math/README to configure.): \operatorname{Li}_{-3}(z) = {z(1+4z+z^2) \over (1-z)^4}.

Failed to parse (Missing texvc executable; please see math/README to configure.): \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:

Failed to parse (Missing texvc executable; please see math/README to configure.): \operatorname{Li}_{1}\left(1/2\right) = \textrm{Ln}(2)

Failed to parse (Missing texvc executable; please see math/README to configure.): \operatorname{Li}_{2}(1/2) = {1 \over 12}[\pi^2-6\textrm{Ln}^2(2)]

Failed to parse (Missing texvc executable; please see math/README to configure.): \operatorname{Li}_{3}(1/2) = {1 \over 24}[4\textrm{Ln}^3(2)-2\pi^2\textrm{Ln} (2)+21\,\zeta(3)]

where ζ is the Riemann zeta function. No similar formulas of this type are known for higher orders (Lewin, 1991 p2.)

Alternate expressions

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

Failed to parse (Missing texvc executable; please see math/README to configure.): \operatorname{Li}_{s+1}(z) = {1 \over \Gamma(s+1)} \int_0^\infty {t^s \over e^t/z-1} dt.

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:

Failed to parse (Missing texvc executable; please see math/README to configure.): -\operatorname{Li}_{s+1}(-z) = {1 \over \Gamma(s+1)} \int_0^\infty {t^s \over e^t/z+1} dt.

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. (GSL)

• The polylogarithm may be rather generally represented by a Hankel contour integral (Whittaker & Watson § 12.22, § 13.13). 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:

Failed to parse (Missing texvc executable; please see math/README to configure.): \operatorname{Li}_s(e^\mu)={{-\Gamma(1-s)}\over{2\pi i}}\oint_H {{(-t)^{s-1}}\over{e^{t-\mu}-1}}dt.

where H represents the Hankel contour. The integrand has a cut along the real axis from zero to infinity, with the real axis being on the lower half of the sheet (Im(t) ≤ 0). For the case where μ is real and non-negative, we can simply add the limiting contribution of the pole:

Failed to parse (Missing texvc executable; please see math/README to configure.): \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

where R is the residue of the pole:

Failed to parse (Missing texvc executable; please see math/README to configure.): R = {{\Gamma(1-s)(-\mu)^{s-1}}\over{2\pi}}.

• The square relationship is easily seen from the duplication formula (see also (Clunie), (Schrödinger)):

Failed to parse (Missing texvc executable; please see math/README to configure.): \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:

Failed to parse (Missing texvc executable; please see math/README to configure.): \sum_{m=0}^{p-1}\operatorname{Li}_s(ze^{2\pi i m/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).

Relationship to other functions

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

  Failed to parse (Missing texvc executable; please see math/README to configure.): \operatorname{Li}_s(1) = \zeta(s)~~~~~~~~~~~~~(\textrm{Re}(s)>1).

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

  Failed to parse (Missing texvc executable; please see math/README to configure.): \operatorname{Li}_s(-1) = -\eta\left(s\right)

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

  Failed to parse (Missing texvc executable; please see math/README to configure.): \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 (GSL)

  Failed to parse (Missing texvc executable; please see math/README to configure.): F_s(\mu)=-\operatorname{Li}_{s+1}(-e^\mu).\,

• The polylogarithm is a special case of the Lerch Transcendent (Erdélyi 1981 § 1.11-14)

  Failed to parse (Missing texvc executable; please see math/README to configure.): \operatorname{Li}_s(z)=z~\Phi(z,s,1).

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

  Failed to parse (Missing texvc executable; please see math/README to configure.): \operatorname{Li}_s(e^{2\pi i x})+(-1)^s \operatorname{Li}_s(e^{-2\pi i x})={(2\pi i)^s \over \Gamma(s)}~\zeta\left (1-s,x\right)

  where Γ(s) is the gamma function. This holds for

  Failed to parse (Missing texvc executable; please see math/README to configure.): \textrm{Re}(s)>1, \textrm{Im}(x)\ge 0, 0 \le \textrm{Re}(x) < 1

  and also for

  Failed to parse (Missing texvc executable; please see math/README to configure.): \textrm{Re}(s)>1, \textrm{Im}(x)\le 0, 0 < \textrm{Re}(x) \le 1.

  (Note that Erdélyi's equivalent Equation (Erdélyi 1981 § 1.11-16) is not correct if we assume that the principal branches of the polylogarithm and the logarithm are used simultaneously.) This equation furnishes the analytical continuation of the series representation of the polylogarithm beyond its circle of convergence|z|= 1. Alternatively, for all Failed to parse (Missing texvc executable; please see math/README to configure.): s \in \mathbb{C} and for all Failed to parse (Missing texvc executable; please see math/README to configure.): z~\not\in~]1;+\infty[ , the inversion formula is

  Failed to parse (Missing texvc executable; please see math/README to configure.): \operatorname{Li}_s(z)+(-1)^s \operatorname{Li}_s(1/z)={(2\pi i)^s \over \Gamma(s)}~\zeta\left (1\!-\!s,{\log z\over2i\pi}\right),

  and forall Failed to parse (Missing texvc executable; please see math/README to configure.): s \in \mathbb{C} and forall Failed to parse (Missing texvc executable; please see math/README to configure.): z \in ]1;+\infty[

  Failed to parse (Missing texvc executable; please see math/README to configure.): \operatorname{Li}_s(z)+(-1)^s \operatorname{Li}_s(1/z)={(2\pi i)^s \over \Gamma(s)}~\zeta\left (1\!-\!s,{\log z\over2i\pi}\right)- 2i\pi\frac{(-\log z)^{s-1}}{\Gamma(s)},

  See below for a simplified formula when Failed to parse (Missing texvc executable; please see math/README to configure.): s is an integer.
• Using the relationship between the Hurwitz zeta function and the Bernoulli polynomials:

  Failed to parse (Missing texvc executable; please see math/README to configure.): \zeta(-n,x)=-{B_{n+1}(x) \over n+1}

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

  Failed to parse (Missing texvc executable; please see math/README to configure.): \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\left({x}\right)

  under the same constraints on s and x as above. (Note that the corresponding equation (Erdélyi 1981 § 1.11-18) is not correct) For negative integer values of the parameter, we have for all z (Erdélyi 1981 § 1.11-17):

  Failed to parse (Missing texvc executable; please see math/README to configure.): \operatorname{Li}_{-n}(z)+ (-1)^n \operatorname{Li}_{-n}\left(1/z\right)=0,~~~~~n=1,2,3,\ldots

  More generally for Failed to parse (Missing texvc executable; please see math/README to configure.): n=0,\pm1,\pm2,\pm3,\cdots

  Failed to parse (Missing texvc executable; please see math/README to configure.): \operatorname{Li}_{n}(z)+ (-1)^n \operatorname{Li}_{n}\left(1/z\right)+\frac{(2i\pi)^n}{n!}\,B_n\left({\log z\over 2i\pi}\right)=0 \qquad z~\not\in~]1;+\infty[,

  Failed to parse (Missing texvc executable; please see math/README to configure.): \operatorname{Li}_{n}(z)+ (-1)^n \operatorname{Li}_{n}\left(1/z\right)+\frac{(2i\pi)^n}{n!}\,B_n\left({\log z\over 2i\pi}\right)=\frac{2\pi\,(\log z)^{n-1}}{i\,(n\!-\!1)!} \qquad z~\in~]1;+\infty[.

• The polylogarithm with pure imaginary μ may be expressed in terms of Clausen functions Ci_s(θ) and Si_s(θ) (Lewin (1958) Ch. VII § 1.4), (Abramowitz & Stegun § 27.8)

  Failed to parse (Missing texvc executable; please see math/README to configure.): \operatorname{Li}_s(e^{\pm i \theta}) = Ci_s(\theta) \pm i Si_s(\theta).

• The Inverse tangent integral Ti_s(z) (Lewin, 1958 Ch. VII § 1.2) can be expressed in terms of polylogarithms:

  Failed to parse (Missing texvc executable; please see math/README to configure.): \operatorname{Li}_s(\pm iy)=2^{-s}\operatorname{Li}_s(-y^2)\pm i\,Ti_s(y).

• The Legendre chi function χ_s(z) (Lewin, 1958 Ch. VII § 1.1), (Boersma, 1992) can be expressed in terms of polylogarithms:

  Failed to parse (Missing texvc executable; please see math/README to configure.): \chi_s(z)={1 \over 2}~[\operatorname{Li}_s(z)-\operatorname{Li}_s(-z)].

• The polylogarithm may be expressed as a series of Debye functions Z_n(z) (Abramowitz & Stegun § 27.1, 27.7.7)

  Failed to parse (Missing texvc executable; please see math/README to configure.): \operatorname{Li}_{n}(e^\mu)=\sum_{k=0}^{n-1}Z_{n-k}(-\mu){\mu^k \over k!},~~~~~~n=1,2,3,\ldots

  A remarkably similar expression relates the Debye function to the polylogarithm:

  Failed to parse (Missing texvc executable; please see math/README to configure.): Z_n(\mu)=\sum_{k=0}^{n-1}\operatorname{Li}_{n-k}(e^{-\mu}){\mu^k \over k!},~~~~~~n=1,2,3,\ldots

Series representations

• We may represent the polylogarithm as a power series about μ = 0 as follows: (Robinson, 1951) Consider the Mellin transform:

  Failed to parse (Missing texvc executable; please see math/README to configure.): M_s(r) =\int_0^\infty \textrm{Li}_s(fe^{-u})u^{r-1}\,du ={1 \over \Gamma(s)}\int_0^\infty\int_0^\infty {t^{s-1}u^{r-1} \over e^{t+u}/f-1}~dt~du.

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

  Failed to parse (Missing texvc executable; please see math/README to configure.): M_s(r)={1 \over \Gamma(s)}\int_0^1 b^{r-1} (1-b)^{s-1}db\int_0^\infty{a^{s+r-1} \over e^a/f-1}da = \Gamma(r)\textrm{Li}_{s+r}(f).

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

  Failed to parse (Missing texvc executable; please see math/README to configure.): \operatorname{Li}_{s}(e^{-u})={1 \over 2\pi i}\int_{c-i\infty}^{c+i\infty}\Gamma(r) \zeta(s+r)u^{-r}dr

  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, …

  Failed to parse (Missing texvc executable; please see math/README to configure.): \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:

  Failed to parse (Missing texvc executable; please see math/README to configure.): \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:

  Failed to parse (Missing texvc executable; please see math/README to configure.): \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:

  Failed to parse (Missing texvc executable; please see math/README to configure.): \operatorname{Li}_{n}(e^\mu) = {\mu^{n-1} \over (n-1)!}\left(H_{n-1}-\ln(-\mu)\right) +

  Failed to parse (Missing texvc executable; please see math/README to configure.): \sum_{k=0,k\ne n-1}^\infty {\zeta(n-k) \over k!}~\mu^k, ~~~~~~~~~~~~~~~~~~~~~~n=2,3,4,\ldots

  Failed to parse (Missing texvc executable; please see math/README to configure.): \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:

  Failed to parse (Missing texvc executable; please see math/README to configure.): 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:

  Failed to parse (Missing texvc executable; please see math/README to configure.): \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:

  Failed to parse (Missing texvc executable; please see math/README to configure.): \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:

  Failed to parse (Missing texvc executable; please see math/README to configure.): \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 (Erdélyi 1981 § 1.11-15) 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 (Wood 1992) (Gradshteyn & Ryzhik § 9.553):

  Failed to parse (Missing texvc executable; please see math/README to configure.): \operatorname{Li}_s(e^\mu)=-{\Gamma(1-s) \over 2\pi i}\oint_H{(-t)^{s-1} \over e^{t-\mu}-1}dt

  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 Failed to parse (Missing texvc executable; please see math/README to configure.): t-\mu=2k\pi i and the integral can be evaluated as the sum of the residues:

  Failed to parse (Missing texvc executable; please see math/README to configure.): \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

  Failed to parse (Missing texvc executable; please see math/README to configure.): \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 compact written in terms of the Dirichlet eta function.
• For negative integer s, the polylogarithm may be expressed as a series involving the Eulerian numbers

  Failed to parse (Missing texvc executable; please see math/README to configure.): \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 Failed to parse (Missing texvc executable; please see math/README to configure.): \left\langle{n\atop i}\right\rangle are Eulerian numbers:
• Another explicit formula for negative integer s is (Wood 1992):

  Failed to parse (Missing texvc executable; please see math/README to configure.): \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 (Wood 1992):

Failed to parse (Missing texvc executable; please see math/README to configure.): \lim_{|z|\rightarrow 0} \operatorname{Li}_s(z) = 0

Failed to parse (Missing texvc executable; please see math/README to configure.): \lim_{s \rightarrow \infty} \operatorname{Li}_s(z) = z

Failed to parse (Missing texvc executable; please see math/README to configure.): \lim_{\mathrm{Re}(\mu) \rightarrow \infty} \operatorname{Li}_s(e^\mu) = -{\mu^s \over \Gamma(s+1)} ~~~~~~(s\ne -1, -2,-3,\ldots)

Failed to parse (Missing texvc executable; please see math/README to configure.): \lim_{\mathrm{Re}(\mu) \rightarrow \infty} \operatorname{Li}_{-n}(e^\mu) = -(-1)^ne^{-\mu} ~~~~~~(n=1,2,3,\ldots)

Failed to parse (Missing texvc executable; please see math/README to configure.): \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 Failed to parse (Missing texvc executable; please see math/README to configure.): s=2 . An alternate integral expression for the dilogarithm is: (Abramowitz & Stegun § 27.7)

Failed to parse (Missing texvc executable; please see math/README to configure.): \operatorname{Li}_2 (z) = -\int_0^z{\ln (1-t) \over t} dt.

The Abel identity for the dilogarithm is given by:

Failed to parse (Missing texvc executable; please see math/README to configure.): \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)

Another similar identity is the Pentagon Identity proved by (Rogers):

Failed to parse (Missing texvc executable; please see math/README to configure.): L(x)+L(y)-L(xy)=L\left(\frac{x-xy}{1-xy}\right)+L\left(\frac{y-xy}{1-xy}\right)

where

Failed to parse (Missing texvc executable; please see math/README to configure.): L(x):=\mbox{Li}_2(x)+\frac{\ln(1-x)\ln(x)}{2}

In terms of Failed to parse (Missing texvc executable; please see math/README to configure.): Li_2(x) , the identity is given by:

Failed to parse (Missing texvc executable; please see math/README to configure.): \mbox{Li}_2(x)+\mbox{Li}_2(y)-\mbox{Li}_2(xy)=\mbox{Li}_2 \left( \frac{x-xy}{1-xy} \right)+\mbox{Li}_2 \left( \frac{y-xy}{1-xy} \right)+\ln \left( \frac{1-x}{1-xy} \right)\ln\left( \frac{1-y}{1-xy} \right).

History 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 Failed to parse (Missing texvc executable; please see math/README to configure.): \rho=\left(\sqrt{5}-1\right)/2

as the reciprocal of the golden ratio. Then two simple examples of results from ladders include

Failed to parse (Missing texvc executable; please see math/README to configure.): \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 (Coxeter, 1935) and

Failed to parse (Missing texvc executable; please see math/README to configure.): \operatorname{Li}_2(\rho)=\frac{\pi^2}{10} - \log^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 Failed to parse (Missing texvc executable; please see math/README to configure.): z=1

and another at Failed to parse (Missing texvc executable; please see math/README to configure.): z=0

. The second branch point, at Failed to parse (Missing texvc executable; please see math/README to configure.): 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 Failed to parse (Missing texvc executable; please see math/README to configure.): m_0

and Failed to parse (Missing texvc executable; please see math/README to configure.): m_1

, the monodromy group has the group presentation

Failed to parse (Missing texvc executable; please see math/README to configure.): \langle m_0, m_1\vert w=m_0m_1m^{-1}_0m^{-1}_1,\, wm_1=m_1w\rangle

For the special case of the dilogarithm, one also has that Failed to parse (Missing texvc executable; please see math/README to configure.): wm_0=m_0w , and the monodromy group becomes the Heisenberg group (identifying Failed to parse (Missing texvc executable; please see math/README to configure.): m_0,m_1

and Failed to parse (Missing texvc executable; please see math/README to configure.): w
with Failed to parse (Missing texvc executable; please see math/README to configure.): x,y,z

). (Vepstas, 2007)

References

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