Polylogarithm

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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

$\operatorname{Li}_{s+1}(z) = \int_0^z \frac {\operatorname{Li}_s(t)}{t}dt$

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.

1 Properties
2 Particular values
3 Alternate expressions
4 Relationship to other functions
5 Series representations
6 Limiting behavior
7 Dilogarithm
8 Polylogarithm ladders
9 Monodromy
10 References

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):

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

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:

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

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

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

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

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

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

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

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

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

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

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:

$\operatorname{Li}_{1}\left(1/2\right) = \textrm{Ln}(2)$

$\operatorname{Li}_{2}(1/2) = {1 \over 12}[\pi^2-6\textrm{Ln}^2(2)]$

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

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

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

$\operatorname{Li}_{s+1}(z) = {1 \over \Gamma(s+1)} \int_0^\infty {t^s \over e^t/z-1} dt.$

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:

$-\operatorname{Li}_{s+1}(-z) = {1 \over \Gamma(s+1)} \int_0^\infty {t^s \over e^t/z+1} dt.$

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:

$\operatorname{Li}_s(e^\mu)={{-\Gamma(1-s)}\over{2\pi i}}\oint_H {{(-t)^{s-1}}\over{e^{t-\mu}-1}}dt.$

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:

$\operatorname{Li}_s(e^\mu)=-{{\Gamma(1-s)}\over{2\pi i}}\oint_H {{(-t)^{s-1}}\over{e^{t-\mu}}-1}dt + 2\pi i R$

where R is the residue of the pole:

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

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

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

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

$\sum_{m=0}^{p-1}\operatorname{Li}_s(ze^{2\pi i m/p}) = p^{1-s}\,\operatorname{Li}_s(z^p)$

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

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

$\operatorname{Li}_s(1) = \zeta(s)~~~~~~~~~~~~~(\textrm{Re}(s)>1).$
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 (GSL)

$F_s(\mu)=-\operatorname{Li}_{s+1}(-e^\mu).\,$
The polylogarithm is a special case of the Lerch Transcendent (Erdélyi 1981 § 1.11-14)

$\operatorname{Li}_s(z)=z~\Phi(z,s,1).$
The polylogarithm is related to the Hurwitz zeta function by:

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

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

$\textrm{Re}(s)>1, \textrm{Im}(x)\ge 0, 0 \le \textrm{Re}(x) < 1$

and also for

$\textrm{Re}(s)>1, \textrm{Im}(x)\le 0, 0 < \textrm{Re}(x) \le 1.$

(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 $s \in \mathbb{C}$ and for all $z~\not\in~]1;+\infty[$ , the inversion formula is

$\operatorname{Li}_s(z)+(-1)^s \operatorname{Li}_s(1/z)={(2\pi i)^s \over \Gamma(s)}~\zeta\left (1\!-\!s,{\log z\over2i\pi}\right),$

and forall $s \in \mathbb{C}$ and forall $z \in ]1;+\infty[$

$\operatorname{Li}_s(z)+(-1)^s \operatorname{Li}_s(1/z)={(2\pi i)^s \over \Gamma(s)}~\zeta\left (1\!-\!s,{\log z\over2i\pi}\right)- 2i\pi\frac{(-\log z)^{s-1}}{\Gamma(s)},$

See below for a simplified formula when $s$ is an integer.
Using 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 can be seen that:

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

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):

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

More generally for $n=0,\pm1,\pm2,\pm3,\cdots$

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

$\operatorname{Li}_{n}(z)+ (-1)^n \operatorname{Li}_{n}\left(1/z\right)+\frac{(2i\pi)^n}{n!}\,B_n\left({\log z\over 2i\pi}\right)=\frac{2\pi\,(\log z)^{n-1}}{i\,(n\!-\!1)!} \qquad z~\in~]1;+\infty[.$
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)

$\operatorname{Li}_s(e^{\pm i \theta}) = Ci_s(\theta) \pm i Si_s(\theta).$
The Inverse tangent integral Ti_s(z) (Lewin, 1958 Ch. VII § 1.2) can be expressed in terms of polylogarithms:

$\operatorname{Li}_s(\pm iy)=2^{-s}\operatorname{Li}_s(-y^2)\pm i\,Ti_s(y).$
The Legendre chi function χ_s(z) (Lewin, 1958 Ch. VII § 1.1), (Boersma, 1992) can be expressed in terms of polylogarithms:

$\chi_s(z)={1 \over 2}~[\operatorname{Li}_s(z)-\operatorname{Li}_s(-z)].$
The polylogarithm may be expressed as a series of Debye functions Z_n(z) (Abramowitz & Stegun § 27.1, 27.7.7)

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

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

$Z_n(\mu)=\sum_{k=0}^{n-1}\operatorname{Li}_{n-k}(e^{-\mu}){\mu^k \over k!},~~~~~~n=1,2,3,\ldots$

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

$M_s(r) =\int_0^\infty \textrm{Li}_s(fe^{-u})u^{r-1}\,du ={1 \over \Gamma(s)}\int_0^\infty\int_0^\infty {t^{s-1}u^{r-1} \over e^{t+u}/f-1}~dt~du.$

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

$M_s(r)={1 \over \Gamma(s)}\int_0^1 b^{r-1} (1-b)^{s-1}db\int_0^\infty{a^{s+r-1} \over e^a/f-1}da = \Gamma(r)\textrm{Li}_{s+r}(f).$

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

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

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, …

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

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:

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

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

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

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 (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):

$\operatorname{Li}_s(e^\mu)=-{\Gamma(1-s) \over 2\pi i}\oint_H{(-t)^{s-1} \over e^{t-\mu}-1}dt$

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 $t - μ = 2 k π i$ 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 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

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

where $\left\langle{n\atop i}\right\rangle$ are Eulerian numbers:
Another explicit formula for negative integer s is (Wood 1992):

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

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

The following limits hold for the polylogarithm (Wood 1992):

$\lim_{|z|\rightarrow 0} \operatorname{Li}_s(z) = 0$

$\lim_{s \rightarrow \infty} \operatorname{Li}_s(z) = z$

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

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

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

The dilogarithm is just the polylogarithm with $s = 2$ . An alternate integral expression for the dilogarithm is: (Abramowitz & Stegun § 27.7)

$\operatorname{Li}_2 (z) = -\int_0^z{\ln (1-t) \over t} dt.$

The Abel identity for the dilogarithm is given by:

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

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

$L(x)+L(y)-L(xy)=L\left(\frac{x-xy}{1-xy}\right)+L\left(\frac{y-xy}{1-xy}\right)$

where

$L(x):=\mbox{Li}_2(x)+\frac{\ln(1-x)\ln(x)}{2}$

In terms of $L i 2 (x)$ , the identity is given by:

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

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

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 $\rho=\left(\sqrt{5}-1\right)/2$ 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 (Coxeter, 1935) and

$\operatorname{Li}_2(\rho)=\frac{\pi^2}{10} - \log^2\rho$

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.

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

$\langle m_0, m_1\vert w=m_0m_1m^{-1}_0m^{-1}_1,\, wm_1=m_1w\rangle$

For the special case of the dilogarithm, one also has that $w m 0 = m 0 w$ , and the monodromy group becomes the Heisenberg group (identifying $m 0, m 1$ and $w$ with $x, y, z$ ). (Vepstas, 2007)

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