Polylogarithm

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The polylogarithm (also known as Jonquiére's function) is a special function that is defined for all complex numbers s and z where |z| < 1 by:

Li_s(z) \equiv \sum_{k=1}^\infty {z^k \over k^s}.

The special cases s = 2 and s = 3 are called the dilogarithm (also referred to as Spence's function) and trilogarithm respectively. 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.

The polylogarithm is actually defined over a larger range of z than the above definition allows by the process of analytic continuation.

Contents

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 natural logarithm Ln(z) so that -π < Im(μ) ≤ π. Also, all exponentiation will be assumed to be single valued. (e.g. zs = 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 Lis(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):

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

The derivatives of the polylogarithm are:

z{\partial Li_s(z) \over \partial z} = Li_{s-1}(z)
{\partial Li_s(e^\mu) \over \partial \mu} = 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:

Li_{1}(z) = -\textrm{Ln}\left(1-z\right)
Li_{0}(z) = {z \over 1-z}
Li_{-1}(z) = {z \over (1-z)^2}
Li_{-2}(z) = {z(1+z) \over (1-z)^3}
Li_{-3}(z) = {z(1+4z+z^2) \over (1-z)^4}.

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

Li_{1}\left(1/2\right) = \textrm{Ln}(2)
Li_{2}(1/2) = {1 \over 12}[\pi^2-6\textrm{Ln}^2(2)]
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

Li_{s+1}(z) \equiv {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.
-Li_{s+1}(-z) \equiv {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. (GNU)
  • 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:
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:
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 defining equation (see also Clunie, Schrödinger):
Li_s(-z) + Li_s(z) = 2^{1-s} ~ Li_s(z^2).
Note that Kummer's function obeys a very similar duplication formula.

Relationship to other functions

  • For z = 1 the polylogarithm reduces to the Riemann zeta function
    Li_s(1) = \zeta(s)~~~~~~~~~~~~~(\textrm{Re}(s)>1).
  • The polylogarithm is related to Dirichlet eta function and the Dirichlet beta function:
    Li_s(-1) = \eta\left(s\right)
    where η(s) is the Dirichlet eta function. For pure imaginary arguments, we have:
    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 (GNU)
    F_s(\mu)=-Li_{s+1}(-e^\mu).\,
  • The polylogarithm is a special case of the Lerch Transcendent (Erdélyi § 1.11-14)
    Li_s(z)=z~\Phi(z,s,1).
  • The polylogarithm is related to the Hurwitz zeta function by:
    Li_s(e^{2\pi i x})+(-1)^s 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 § 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.
  • 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:
    Li_{n}(e^{2\pi i x})+ (-1)^n 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 § 1.11-18 is not correct) For negative integer values of the parameter, we have for all z (Erdélyi § 1.11-17):
    Li_{-n}(z)+ (-1)^n Li_{-n}\left(1/z\right)=0,~~~~~n=1,2,3,\ldots
  • The polylogarithm with pure imaginary μ may be expressed in terms of Clausen functions Cis(θ) and Sis(θ) (Lewin, 1958 Ch. VII § 1.4, Abramowitz & Stegun § 27.8)
    Li_s(e^{\pm i \theta}) = Ci_s(\theta) \pm i Si_s(\theta).
  • The Inverse Tangent Integral Tis(z) (Lewin, 1958 Ch. VII § 1.2) can be expressed in terms of polylogarithms:
    Li_s(\pm iy)=2^{-s}Li_s(-y^2)\pm i\,Ti_s(y).
  • The Legendre chi function χs(z) (Lewin, 1958 Ch. VII § 1.1, Boersma) can be expressed in terms of polylogarithms:
    \chi_s(z)={1 \over 2}~[Li_s(z)-Li_s(-z)].
  • The polylogarithm may be expressed as a series of Debye functions Zn(z) (Abramowitz & Stegun § 27.1)
    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}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) 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:
    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, …
    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:
    Li_{n}(e^\mu) = {\mu^{n-1} \over (n-1)!}\left(H_n-\ln(-\mu)\right) +
    \sum_{k=0,k\ne n-1}^\infty {\zeta(n-k) \over k!}~\mu^k, ~~~~~~~~~~~~~~~~~~~~~~n=2,3,4,\ldots
    Li_{1}(e^\mu) =-\ln(-\mu)+ \sum_{k=1}^\infty {\zeta(1-k) \over k!}~\mu^k, ~~~~~~~~~~(n=1)
    where Hn is a harmonic number:
    H_n\equiv \sum_{k=1}^n{1\over k}.
    The problem terms now contain −ln(−μ) which, when multiplied by μk 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 Lis(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 Bk:
    \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:
    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 § 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).
  • The defining equation may be extended to negative values of the parameter s using a Hankel contour integral (Wood, Gradshteyn & Ryzhik § 9.553):
    Li_s(e^\mu)=-{\Gamma(1-p) \over 2\pi i}\oint_H{(-t)^{s-1} \over e^{t-\mu}-1}dt
    where H is the Hankel contour which can be modified so that it encloses the poles of the integrand, at t - μ = 2kπi and the integral can be evaluated as the sum of the residues:
    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 eμ = 1. Summing the series, one obtains
    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
    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):
    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):

\lim_{|z|\rightarrow 0} Li_s(z) = 0
\lim_{s \rightarrow \infty} Li_s(z) = z
\lim_{\mathrm{Re}(\mu) \rightarrow \infty} Li_s(e^\mu) = -{\mu^s \over \Gamma(s+1)} ~~~~~~(s\ne -1, -2,-3,\ldots)
\lim_{\mathrm{Re}(\mu) \rightarrow \infty} Li_{n}(e^\mu) = -(-1)^ne^{-\mu} ~~~~~~(n=1,2,3,\ldots)
\lim_{|\mu|\rightarrow 0} Li_s(e^\mu) = \Gamma(1-s)(-\mu)^s~~~~~~(s<1)

The dilogarithm

The dilogarithm is just the polylogarithm with s = 2. An alternate integral expression for the dilogarithm is:

Li_{2}(z) \equiv -\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)

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

Li_2(\rho^6)=4Li_2(\rho^3)+3Li_2(\rho^2)-6Li_2(\rho)+\frac{7\pi^2}{30}

given by Coxeter in 1935, and

Li_2(\rho)=\frac{\pi^2}{10} - \log^2\rho

given by Landen. Polylogarithm ladders occur naturally and deeply in K-theory.

References

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