Polylogarithm

Not to be confused with polylogarithmic.
In mathematics, the polylogarithm (also known as Jonquière's function) is a special function Li_{s}(z) that is defined by the infinite sum, or power series:
It is in general not an elementary function, unlike the related logarithm function. The above definition is valid for all complex values of the order s and the argument 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 The special case s = 1 involves the ordinary natural logarithm (Li_{1}(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
Thus the dilogarithm is an integral of the logarithm, and so on. For nonpositive integer orders 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.
Contents
Properties
Preliminary note: In the important case where the polylogarithm order 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 singlevalued: z^{s} = exp(s ln(z)).
Depending on the order s, the polylogarithm may be multivalued. The principal branch of Li_{s}(z) is taken to be that given for z < 1 by the above series definition and taken to be continuous except on the positive real axis, where a cut is made from z = 1 to ∞ such that axis is placed on the lower half plane of z. In terms of μ, this amounts to −π < arg(−μ) ≤ π. The discontinuity of the polylogarithm in dependence on μ can sometimes be confusing.
For real argument z, the polylogarithm of real order s is real if z < 1, and its imaginary part for z ≥ 1 is (Wood 1992, § 3):
Going across the cut, if δ is an infinitesimally small positive real number, then:
Both can be concluded from the series expansion (see below) of Li_{s}(e^{µ}) about µ = 0.
The derivatives of the polylogarithm follow from the defining power series:
The square relationship is easily seen from the duplication formula (see also (Clunie 1954), (Schrödinger 1952)):
Note that Kummer's function obeys a very similar duplication formula. This is a special case of the multiplication formula, for any integer p:
which can be proved using the series definition of the polylogarithm and the orthogonality of the exponential terms (e.g. see discrete Fourier transform).
Another important property, the inversion formula, involves the Hurwitz zeta function or the Bernoulli polynomials and is found under relationship to other functions below.
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.
1. For integer values of the polylogarithm order, the following explicit expressions are obtained by repeated application of z·∂/∂z to Li_{1}(z):
Accordingly the polylogarithm reduces to a ratio of polynomials in z, and is therefore a rational function of z, for all nonpositive integer orders. The general case may be expressed as a finite sum:
where S(n,k) are the Stirling numbers of the second kind. Equivalent formulae applicable to negative integer orders are (Wood 1992, § 6):
and:
where are the Eulerian numbers. All roots of Li_{−n}(z) are distinct and real; they include z = 0, while the remainder is negative and centered about z = −1 on a logarithmic scale. As n 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 halfinteger values of the argument z are:
where ζ is the Riemann zeta function. No formulae of this type are known for higher integer orders (Lewin 1991, p. 2). However, one has for instance:
involving the alternating double sum (CITATION REQUIRED!).
3. As a straightforward consequence of the series definition, values of the polylogarithm at the pth complex roots of unity are given by the Fourier sum:
where ζ is the Hurwitz zeta function. For Re(s) > 1, where Li_{s}(1) is finite, the relation also holds with m = 0 or m = p. While this formula is not as simple as that implied by the more general relation with the Hurwitz zeta function listed under relationship to other functions below, it has the advantage of applying to positive integer values of s as well. As usual, the relation may be inverted to express ζ(s, ^{m}⁄_{p}) for any m = 1, ..., p as a Fourier sum of Li_{s}(exp(2πi ^{k}⁄_{p})) over k = 1, ..., p.
Relationship to other functions
 For z = 1 the polylogarithm reduces to the Riemann zeta function
 The polylogarithm is related to Dirichlet eta function and the Dirichlet beta function:
 where η(s) is the Dirichlet eta function. For pure imaginary arguments, we have:
 where β(s) is the Dirichlet beta function.
 The polylogarithm is related to the complete Fermi–Dirac integral as:
 The polylogarithm is a special case of the incomplete polylogarithm function
 The polylogarithm is a special case of the Lerch Transcendent (Erdélyi 1981, § 1.1114)
 The polylogarithm is related to the Hurwitz zeta function by:
 where Γ(1−s) is the gamma function, which causes the relation to fail for positive integer s; a derivation of this formula is given under series representations below. The polylogarithm is consequently also related to the Hurwitz zeta function by (Jonquière 1889):
 which holds for 0 ≤ Re(x) < 1 if Im(x) ≥ 0, and for 0 < Re(x) ≤ 1 if Im(x) < 0. Equivalently, for all complex s and for complex z ∉ ]0;1], the inversion formula reads
 and for all complex s and for complex z ∉ ]1;∞[
 These relations furnish the analytic continuation of the polylogarithm beyond the circle of convergence z = 1 of the defining power series. (Note that Erdélyi's corresponding equation (Erdélyi 1981, § 1.1116) is not correct if one assumes that the principal branches of the polylogarithm and the logarithm are used simultaneously.) See below for a simplified formula when s is an integer.
 For positive integer polylogarithm orders s, the Hurwitz zeta function ζ(1−s, x) reduces to Bernoulli polynomials, ζ(1−n, x) = −B_{n}(x) / n, and Jonquière's inversion formula for n = 1, 2, 3, ... becomes:
 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 the unit circle, Im(x) = 0, the left hand side of this formula simplifies to 2 Re(Li_{n}(e^{2πix})) if n is even, and to 2i Im(Li_{n}(e^{2πix})) if n is odd. For negative integer orders, on the other hand, the divergence of Γ(s) implies for all z that (Erdélyi 1981, § 1.1117):
 More generally one has for n = 0, ±1, ±2, ±3, ... :
 (Note that the corresponding equation (Erdélyi 1981, § 1.1118) is again not correct.)
 The polylogarithm with pure imaginary μ may be expressed in terms of the Clausen functions Ci_{s}(θ) and Si_{s}(θ), and vice versa (Lewin 1958, Ch. VII § 1.4), (Abramowitz & Stegun 1972, § 27.8):
 The inverse tangent integral Ti_{s}(z) (Lewin 1958, Ch. VII § 1.2) can be expressed in terms of polylogarithms:
 The relation in particular implies:
 which explains the function name.
 The Legendre chi function χ_{s}(z) (Lewin 1958, Ch. VII § 1.1), (Boersma 1992) can be expressed in terms of polylogarithms:
 The polylogarithm of integer order can be expressed as a generalized hypergeometric function:
 In terms of the incomplete zeta functions or "Debye functions" (Abramowitz & Stegun 1972, § 27.1):
 the polylogarithm Li_{n}(z) for positive integer n may be expressed as the finite sum (Wood 1992, § 16):
 A remarkably similar expression relates the function Z_{n}(z) to the polylogarithm:
Integral representations
Any of the following integral representations furnishes the analytic continuation of the polylogarithm beyond the circle of convergence z = 1 of the defining power series.
 The integral of the Bose–Einstein distribution is expressed in terms of a polylogarithm:
 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 BoseEinstein integral.
 The integral of the FermiDirac distribution is also expressed in terms of a polylogarithm:
 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 2010).
 A complementary integral representation applies to Re(s) < 0 and to all z except to z real and ≥ 0:
 This integral follows from the general relation of the polylogarithm with the Hurwitz zeta function (see above) and a familiar integral representation of the latter.
 The polylogarithm may be rather generally represented by a Hankel contour integral (Whittaker & Watson 1952, § 12.22, § 13.13). As long as the t = μ pole of the integrand does not lie on the nonnegative real axis, and s ≠ 1, 2, 3, ..., we have:
 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 plane of t. The integration starts at +∞ on the upper half plane (Im(t) > 0), circles the origin without enclosing any of the poles t = µ + 2kπi, and terminates at +∞ on the lower half plane (Im(t) < 0). For the case where µ is real and nonnegative, we can simply subtract the contribution of the enclosed t = µ pole:
 where R is the residue of the pole:
 A Hermitelike integral representation valid for all complex z and for all complex s is:
 where Γ is the upper incomplete gammafunction. Note that 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 complex s,
 avoids the use of the incomplete gamma function, but this integral fails for z on the positive real axis.
Series representations
1. As noted under integral representations above, the BoseEinstein integral representation of the polylogarithm may be extended to negative orders s by means of Hankel contour integration:
where H is the Hankel contour, s ≠ 1, 2, 3, ..., and the t = μ pole of the integrand does not lie on the nonnegative real axis. The contour 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 (Wood 1992, § 12, 13), (Gradshteyn & Ryzhik 1980, § 9.553):
This will hold for Re(s) < 0 and all μ except where e^{μ} = 1. For 0 < Im(µ) ≤ 2π the sum can be split as:
where the two series can now be identified with the Hurwitz zeta function:
This relation, which has already been given under relationship to other functions above, holds for all complex s ≠ 1, 2, 3, ... and was first derived in (Jonquière 1889).
2. In order to represent the polylogarithm as a power series about µ = 0, we write the series derived from the Hankel contour integral as:
When the binomial powers in the sum are expanded about µ = 0 and the order of summation is reversed, the sum over h can be expressed in closed form:
This result holds for µ < 2π and, thanks to the analytic continuation provided by the zeta functions, for all s ≠ 1, 2, 3, ... . If the order s is a positive integer, n, both the term with k = n − 1 and the gamma function become infinite, although their sum does not. One obtains (Wood 1992, § 9), (Gradshteyn & Ryzhik 1980, § 9.554):
where the sum over h vanishes if k = 0. So, for s = n where n is a positive integer and for μ < 2π we have the series:
where H_{n} denotes the nth harmonic number:
The problem terms now contain −ln(−μ) which, when multiplied by μ^{n−1}, will tend to zero as μ → 0, except for n = 1. This reflects the fact that there is a true logarithmic singularity in Li_{s}(z) at s = 1 and z = 1 since:
For s close, but not equal, to a positive integer, the divergent terms in the expansion about µ = 0 can be expected to cause computational difficulties (Wood 1992, § 9). Note also that Erdélyi's corresponding expansion (Erdélyi 1981, § 1.1115) 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 of s, the zeta function ζ(s − k) in the expansion about µ = 0 reduces to Bernoulli 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 under particular values above exhibit for large n.
3. By use of the identity
the BoseEinstein integral representation of the polylogarithm (see above) may be cast in the form:
Replacing the hyperbolic cotangent with a bilateral series,
then reversing the order of integral and sum, and finally identifying the summands with an integral representation of the upper incomplete gamma function, one obtains:
For both the bilateral series of this result and that for the hyperbolic cotangent, symmetric partial sums from −k_{max} to k_{max} converge unconditionally as k_{max} → ∞. Provided the summation is performed symmetrically, this series for Li_{s}(z) thus holds for all complex s as well as all complex z.
4. Introducing an explicit expression for the Stirling numbers of the second kind into the finite sum for the polylogarithm of nonpositive integer order (see above) one may write:
The infinite series obtained by simply extending the outer summation to ∞ (Guillera & Sondow 2008, Theorem 2.1):
turns out to converge to the polylogarithm for all complex s and for complex z with Re(z) < ^{1}⁄_{2}, as can be verified for ^{−z}⁄_{(1−z)} < ^{1}⁄_{2} by reversing the order of summation and using:
For the other arguments with Re(z) < ^{1}⁄_{2} the result follows by analytic continuation. This procedure is equivalent to applying Euler's transformation to the series in z that defines the polylogarithm.
Asymptotic expansions
For z ≫ 1, the polylogarithm can be expanded into asymptotic series in terms of ln(−z):
where B_{2k} are the Bernoulli numbers. Both versions hold for all s and for any arg(z). As usual, the summation should be terminated when the terms start growing in magnitude. For negative integer s, the expansions vanish entirely; for nonnegative integer s, they break off after a finite number of terms. Wood (1992, § 11) describes a method for obtaining these series from the BoseEinstein integral representation (note that his equation 11.2 for Li_{s}(e^{µ}) requires −2π < Im(µ) ≤ 0).
Limiting behavior
The following limits result from the various representations of the polylogarithm (Wood 1992, § 22):
Dilogarithm
See also: Spence's functionThe dilogarithm is just the polylogarithm with s = 2. An alternate integral expression for the dilogarithm for arbitrary complex z is (Abramowitz & Stegun 1972, § 27.7):
A source of confusion is that some computer algebra systems define the dilogarithm as dilog(z) = Li_{2}(1−z).
In the case of real z ≥ 1 the first integral expression for the dilogarithm can be written as
from which expanding ln(t−1) and integrating term by term we obtain
The Abel identity for the dilogarithm is given by (Abel 1881)
for x ∉ ]1;∞[ and y ∉ ]1;∞[.
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
where Li_{2}(1) = ζ(2) = ^{1}⁄_{6} π^{2} has been used.
In terms of the new variables u = x/(1−y), v = y/(1−x) the Abel identity reads
which corresponds to the pentagon identity given in (Rogers 1907).
From the Abel identity for y = x and the square relationship we have Landen's identity
and the inversion formula for the dilogarithm reads
and for x ≥ 1 also
Known closedform evaluations of the dilogarithm at special arguments are collected in the table below. Arguments in the first column are related by reflection x ↔ 1−x or inversion x ↔ ^{1}⁄_{x} to either x = 0 or x = −1; arguments in the third column are all interrelated by these operations.
Historical note: Don Zagier (1989) remarked that "The dilogarithm is the only mathematical function with a sense of humor".

Special values of the dilogarithm
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 as the reciprocal of the golden ratio. Then two simple examples of results from ladders are
given by Landen. Polylogarithm ladders occur naturally and deeply in Ktheory and algebraic geometry. Polylogarithm ladders provide the basis for the rapid computations of various mathematical constants by means of the BBP algorithm (Bailey, Borwein & Plouffe 1997).
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 function 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
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) (Vepstas 2007).
References
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 Rogers, L.J. (1907). "On function sum theorems connected with the series ". Proceedings of the London Mathematical Society (2) 4 (1): 169–189. doi:10.1112/plms/s24.1.169.
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 Truesdell, C. (1945). "On a function which occurs in the theory of the structure of polymers". Annals of Mathematics, Series 2 46 (1): 144–157. doi:10.2307/1969153. JSTOR 1969153.
 Vepstas, L. (February 2007). "An efficient algorithm for accelerating the convergence of oscillatory series, useful for computing the polylogarithm and Hurwitz zeta functions". Numerical Algorithms 47 (3): 211. arXiv:math.CA/0702243. doi:10.1007/s1107500791538. (see also Numerical Algorithms 47 (2008), pp. 211–252.)
 Whittaker, E.T.; Watson, G.N. (1952). A Course of Modern Analysis (4th ed.). Cambridge, UK: Cambridge University Press.
 Wood, D.C. (June 1992). "The Computation of Polylogarithms. Technical Report 1592*" (PS). Canterbury, UK: University of Kent Computing Laboratory. http://www.cs.kent.ac.uk/pubs/1992/110. Retrieved 20051101.
 Zagier, D. (1989). "The dilogarithm function in geometry and number theory". Number Theory and Related Topics: papers presented at the Ramanujan Colloquium, Bombay, 1988. Studies in Mathematics. 12. Bombay: Tata Institute of Fundamental Research and Oxford University Press. pp. 231–249. ISBN 0195623673. (also appeared as "The remarkable dilogarithm" in Journal of Mathematical and Physical Sciences 22 (1988), pp. 131–145, and as Chapter I of (Zagier 2007).)
 Zagier, D. (2007). "The Dilogarithm Function". In Cartier, P.; Julia, B.; Moussa, P. et al. (PDF). Frontiers in Number Theory, Physics, and Geometry II – On Conformal Field Theories, Discrete Groups and Renormalization. Berlin: SpringerVerlag. pp. 3–65. ISBN 9783540303077. http://mathlab.snu.ac.kr/~top/articles/zagier.pdf.
External links
 Weisstein, Eric W., "Polylogarithm" from MathWorld.
 Weisstein, Eric W., "Dilogarithm" from MathWorld.
Categories: Special functions
 Zeta and Lfunctions
 Rational functions
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Gas in a box — In quantum mechanics, the results of the quantum particle in a box can be used to look at the equilibrium situation for a quantum ideal gas in a box which is a box containing a large number of molecules which do not interact with each other… … Wikipedia
Dilogarithmus — Der Polylogarithmus ist eine spezielle Funktion, die durch die Reihe definiert ist. Für s = 1 geht der Polylogarithmus in den gewöhnlichen Logarithmus über: Li1(z) = − ln(1 − z). Im Fall s = 2 oder s = 3 spricht man entsprechend von Dilogarithmus … Deutsch Wikipedia
Trilogarithmus — Der Polylogarithmus ist eine spezielle Funktion, die durch die Reihe definiert ist. Für s = 1 geht der Polylogarithmus in den gewöhnlichen Logarithmus über: Li1(z) = − ln(1 − z). Im Fall s = 2 oder s = 3 spricht man entsprechend von Dilogarithmus … Deutsch Wikipedia
List of mathematical functions — In mathematics, several functions or groups of functions are important enough to deserve their own names. This is a listing of pointers to those articles which explain these functions in more detail. There is a large theory of special functions… … Wikipedia
Clausen function — In mathematics, the Clausen function is defined by the following integral: It was introduced by Thomas Clausen (1832). The Lobachevsky function Λ or Л is essentially the same function with a change of variable: though the name Lobachevsky… … Wikipedia