Polylogarithm Explained

In mathematics, the polylogarithm (also known as Jonquière's function, for Alfred Jonquière) is a special function of order and argument . Only for special values of does the polylogarithm reduce to an elementary function such as the natural logarithm or a rational function. In quantum statistics, the polylogarithm function appears as the closed form of integrals of the Fermi–Dirac distribution and the Bose–Einstein distribution, and is also known as the Fermi–Dirac integral or the Bose–Einstein integral. In quantum electrodynamics, polylogarithms of positive integer order arise in the calculation of processes represented by higher-order Feynman diagrams.

The polylogarithm function is equivalent to the Hurwitz zeta function — either function can be expressed in terms of the other — and both functions are special cases of the Lerch transcendent. Polylogarithms should not be confused with polylogarithmic functions, nor with the offset logarithmic integral, which has the same notation without the subscript.

The polylogarithm function is defined by a power series in, which is also a Dirichlet series in :$$\backslash operatorname\_s(z)\; =\; \backslash sum\_^\backslash infty\; =\; z\; +\; +\; +\; \backslash cdots$$

This definition is valid for arbitrary complex order and for all complex arguments with ; it can be extended to by the process of analytic continuation. (Here the denominator is understood as). The special case involves the ordinary natural logarithm,, while the special cases and 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 also be defined as the repeated integral of itself:$$\backslash operatorname\_(z)\; =\; \backslash int\_0^z\; \backslash frac\; dt$$thus the dilogarithm is an integral of a function involving the logarithm, and so on. For nonpositive integer orders, the polylogarithm is a rational function.

Properties

In the case where the order

s is an integer, it will be represented by s = n (or s =- n when negative). It is often convenient to define \mu =ln (z) where ln (z) is the principal branch of the complex logarithm \operatorname{Ln}(z) so that - \pi< \operatorname{Im}(\mu) \le \pi. Also, all exponentiation will be assumed to be single-valued: z ^{s} = \exp(s ln (z)).

Depending on the order

s , the polylogarithm may be multi-valued. The principal branch of \operatorname{Li} _{s(z)} is taken to be 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 inf ty such that the axis is placed on the lower half plane of In terms of this amounts to - \pi < \arg( - \mu)\le \pi . The discontinuity of the polylogarithm in dependence on \mu can sometimes be confusing.

For real argument

z , the polylogarithm of real order s is real if and its imaginary part for z \ge 1 is :

$$\backslash operatorname\backslash left(\backslash operatorname\_s(z)\; \backslash right)\; =\; -.$$

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

$$\backslash operatorname\backslash left(\backslash operatorname\_s(z+i\backslash epsilon)\; \backslash right)\; =\; .$$

Both can be concluded from the series expansion (see below) of about

The derivatives of the polylogarithm follow from the defining power series:

$$z\; \backslash frac\; =\; \backslash operatorname\_(z)$$$$\backslash frac\; =\; \backslash operatorname\_(e^\backslash mu).$$

The square relationship is seen from the series definition, and is related to the duplication formula (see also,):

$$\backslash operatorname\_s(-z)\; +\; \backslash operatorname\_s(z)\; =\; 2^\; \backslash operatorname\_s(z^2).$$

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

$$\backslash sum\_^\; \backslash operatorname\_s(z\; e^)\; =\; p^\; \backslash operatorname\_s(z^p),$$

which can be proved using the series definition of the polylogarithm and the orthogonality of the exponential terms (see e.g. 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 also 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 Li₁(z):$$\backslash operatorname\_(z)\; =\; -\backslash ln(1-z)$$$$\backslash operatorname\_(z)\; =$$$$\backslash operatorname\_(z)\; =$$$$\backslash operatorname\_(z)\; =$$$$\backslash operatorname\_(z)\; =$$$$\backslash operatorname\_(z)\; =\; .$$

   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:$$\backslash operatorname\_(z)\; =\; \backslash left(z\; \backslash right)^n\; =\; \backslash sum\_^n\; k!\; S(n+1,\; k+1)\; \backslash left(\backslash right)^\; \backslash qquad\; (n=0,1,2,\backslash ldots),$$where S(n,k) are the Stirling numbers of the second kind. Equivalent formulae applicable to negative integer orders are :$$\backslash operatorname\_(z)\; =\; (-1)^\; \backslash sum\_^n\; k!\; S(n+1,\; k+1)\; \backslash left(\backslash right)^\; \backslash qquad\; (n=1,2,3,\backslash ldots),$$

   and:$$\backslash operatorname\_(z)\; =\; \backslash sum\_^\; \backslash left\backslash langle\; \backslash right\backslash rangle\; z^\; \backslash qquad\; (n=1,2,3,\backslash ldots),$$

   where

   \scriptstyle \left\langle {n \atop k} \right\rangle 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 ; 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 half-integer values of the argument z are:$$\backslash operatorname\_1(\backslash tfrac12)\; =\; \backslash ln\; 2$$$$\backslash operatorname\_2(\backslash tfrac12)\; =\; \backslash tfrac1\; \backslash pi^2\; -\; \backslash tfrac12\; (\backslash ln\; 2)^2$$$$\backslash operatorname\_3(\backslash tfrac12)\; =\; \backslash tfrac16\; (\backslash ln\; 2)^3\; -\; \backslash tfrac1\; \backslash pi^2\; \backslash ln\; 2\; +\; \backslash tfrac78\; \backslash zeta(3),$$

   where ζ is the Riemann zeta function. No formulae of this type are known for higher integer orders, but one has for instance :$$\backslash operatorname\_4(\backslash tfrac12)\; =\; \backslash tfrac\; 1\; \backslash pi^4\; -\; \backslash tfrac\; 1(\backslash ln\; 2)^4\; +\; \backslash tfrac1\; \backslash pi^2\; (\backslash ln\; 2)^2\; -\; \backslash tfrac\; 1\; 2\; \backslash zeta(\backslash bar3,\; \backslash bar1),$$

   which involves the alternating double sum $$\backslash zeta(\backslash bar3,\; \backslash bar1)=\backslash sum\_\; (-1)^\; m^\; n^.$$

   In general one has for integer orders n ≥ 2 :$$\backslash operatorname\_n(\backslash tfrac12)\; =\; -\backslash zeta(\backslash bar1,\; \backslash bar1,\; \backslash left\backslash ^),$$

   where ζ(s₁, …, s_k) is the multiple zeta function; for example:$$\backslash operatorname\_5(\backslash tfrac12)\; =\; -\backslash zeta(\backslash bar1,\; \backslash bar1,\; 1,1,1).$$

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:$$\backslash operatorname\_s(e^)\; =\; p^\; \backslash sum\_^p\; e^\; \backslash zeta(s,\; \backslash tfrac)\; \backslash qquad\; (m\; =\; 1,\; 2,\; \backslash dots,\; p-1),$$

   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 non-negative 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, the polylogarithm reduces to the Riemann zeta function $$\backslash operatorname\_s(1)\; =\; \backslash zeta(s)\; \backslash qquad\; (\backslash operatorname(s)>1).$$
• The polylogarithm is related to Dirichlet eta function and the Dirichlet beta function: $$\backslash operatorname\_s(-1)\; =\; -\backslash eta(s),$$ where is the Dirichlet eta function. For pure imaginary arguments, we have: $$\backslash operatorname\_s(\backslash pm\; i)\; =\; -2^\; \backslash eta(s)\; \backslash pm\; i\backslash beta(s),$$ where is the Dirichlet beta function.
• The polylogarithm is related to the complete Fermi–Dirac integral as: $$F\_s(\backslash mu)\; =\; -\backslash operatorname\_(-e^\backslash mu).$$
• The polylogarithm is related to the complete Bose–Einstein integral as: $$G\_s(\backslash mu)\; =\; \backslash operatorname\_(e^\backslash mu).$$
• The polylogarithm is a special case of the incomplete polylogarithm function $$\backslash operatorname\_s(z)\; =\; \backslash operatorname\_s(0,z)\; .$$
• The polylogarithm is a special case of the Lerch transcendent $$\backslash operatorname\_s(z)\; =\; z\backslash Phi(z,s,1).$$
• The polylogarithm is related to the Hurwitz zeta function by: $$\backslash operatorname\_s(z)\; =\; \backslash left[i^\{1-s\}\; \backslash zeta\; \backslash left(1\; -\; s,\; \backslash frac\{1\}\{2\}\; +\; \{\backslash ln(-z)\; \backslash over\; \{2\backslash pi\; i\}\}\; \backslash right)\; +\; i^\{s-1\}\; ~\backslash zeta\; \backslash left(1\; -\; s,\; \backslash frac\{1\}\{2\}\; -\; \{\backslash ln(-z)\; \backslash over\; \{2\backslash pi\; i\}\}\; \backslash right)\; \backslash right],$$ which relation, however, is invalidated at positive integer s by poles of the gamma function, and at by a pole of both zeta functions; a derivation of this formula is given under series representations below. With a little help from a functional equation for the Hurwitz zeta function, the polylogarithm is consequently also related to that function via : $$$$

i^ \operatorname_s(e^) + i^s \operatorname_s(e^) = \frac \zeta(1 - s, x), which relation holds for if, and for if . Equivalently, for all complex s and for complex, the inversion formula reads $$\backslash operatorname\_s(z)\; +\; (-1)^s\; \backslash operatorname\_s(1/z)\; =\; \backslash frac\; ~\backslash zeta\; \backslash left(1\; -\; s,\; ~\backslash frac\; +\; \backslash right),$$ and for all complex s and for complex $$\backslash operatorname\_s(z)\; +\; (-1)^s\; \backslash operatorname\_s(1/z)\; =\; ~\backslash zeta\; \backslash left(1\; -\; s,\; ~\backslash frac\; -\; \backslash right)\; .$$ For, one has, and both expressions agree. These relations furnish the analytic continuation of the polylogarithm beyond the circle of convergence |z| = 1 of the defining power series. (The corresponding equation of and is not correct if one assumes that the principal branches of the polylogarithm and the logarithm are used simultaneously.) See the next item 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,, and Jonquière's inversion formula for n = 1, 2, 3, … becomes: $$\backslash operatorname\_(e^)\; +\; (-1)^n\; \backslash operatorname\_(e^)\; =\; -\; B\_n(x),$$ 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 : $$\backslash operatorname\_(z)\; +\; (-1)^n\; \backslash operatorname\_(1/z)\; =\; 0\; \backslash qquad\; (n\; =\; 1,2,3,\backslash ldots).$$

More generally, one has for : Ci_s(θ) and Si_s(θ), and vice versa (;):$$\backslash operatorname\_s(e^)\; =\; Ci\_s(\backslash theta)\; \backslash pm\; i\; Si\_s(\backslash theta).$$

• The inverse tangent integral can be expressed in terms of polylogarithms: $$\backslash operatorname\_s(z)\; =\; \backslash left[\backslash operatorname\{Li\}\_s(i\; z)\; -\; \backslash operatorname\{Li\}\_s(-i\; z)\; \backslash right].$$ The relation in particular implies: $$\backslash operatorname\_0(z)\; =,\; \backslash quad\; \backslash operatorname\_1(z)\; =\; \backslash arctan\; z,\; \backslash quad\; \backslash operatorname\_2(z)\; =\; \backslash int\_0^z\; dt,\; \backslash quad\; \backslash ldots~\; \backslash quad\; \backslash operatorname\_(z)\; =\; \backslash int\_0^z\; \backslash frac\; dt,$$ which explains the function name.
• The Legendre chi function χ_s(z) (;) can be expressed in terms of polylogarithms: $$$$

\chi_s(z) = \tfrac \left[\operatorname{Li}_s(z) - \operatorname{Li}_s(-z) \right].

• The polylogarithm of integer order can be expressed as a generalized hypergeometric function: $$\backslash begin$$

\operatorname_n(z) &= z _F_ (1,1,\dots,1; 2,2,\dots,2; z) & (n = 0,1,2,\ldots), \\\operatorname_(z) &= z _F_ (2,2,\dots,2; 1,1,\dots,1; z) & (n = 1,2,3,\ldots) ~.\end

• In terms of the incomplete zeta functions or "Debye functions" : $$$$

Z_n(z) = \int_z^\infty dt \qquad (n = 1,2,3,\ldots), the polylogarithm Li_n(z) for positive integer n may be expressed as the finite sum : $$\backslash operatorname\_(e^\backslash mu)\; =\; \backslash sum\_^\; Z\_(-\backslash mu)\; \backslash qquad\; (n\; =\; 1,2,3,\backslash ldots)\; .$$ A remarkably similar expression relates the "Debye functions" Z_n(z) to the polylogarithm: $$Z\_n(z)\; =\; \backslash sum\_^\; \backslash operatorname\_(e^)\; \backslash qquad\; (n\; =\; 1,2,3,\backslash ldots)\; .$$

• Using Lambert series, if

J _{s(n)} is Jordan's totient function, then $$\backslash sum\_^\backslash infty\backslash frac=\backslash operatorname\_(z).$$

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.

1. The polylogarithm can be expressed in terms of the integral of the Bose–Einstein distribution:$$\backslash operatorname\_(z)\; =\; \backslash int\_0^\backslash infty\; dt\; .$$

   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 but more commonly as a Bose–Einstein integral .^[1] Similarly, the polylogarithm can be expressed in terms of the integral of the Fermi–Dirac distribution:$$-\backslash operatorname\_(-z)\; =\; \backslash frac\; \backslash int\_0^\backslash infty\; dt\; .$$

   This converges for and all except for z real and ≤ −1. The polylogarithm in this context is sometimes referred to as a Fermi integral or a Fermi–Dirac integral . These representations are readily verified by Taylor expansion of the integrand with respect to z and termwise integration. The papers of Dingle contain detailed investigations of both types of integrals.

   The polylogarithm is also related to the integral of the Maxwell–Boltzmann distribution:$$\backslash lim\_\; \backslash frac\; =\; \backslash int\_0^\backslash infty\; dt\; =\; 1\; .$$

   This also gives the asymptotic behavior of polylogarithm at the vicinity of origin.

2. A complementary integral representation applies to Re(s) < 0 and to all z except to z real and ≥ 0:$$\backslash operatorname\_(z)\; =\backslash int\_0^\backslash infty\; dt\; .$$

   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.

3. The polylogarithm may be quite generally represented by a Hankel contour integral, which extends the Bose–Einstein representation to negative orders s. 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:$$\backslash operatorname\_s(e^\backslash mu)\; =\; -\; \backslash oint\_H\; dt$$

   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 non-negative, we can simply subtract the contribution of the enclosed t = μ pole:$$\backslash operatorname\_s(e^\backslash mu)\; =\; -\; \backslash oint\_H\; dt\; -\; 2\backslash pi\; i\; R$$

   where R is the residue of the pole:$$R\; =\; \backslash Gamma(1\; -\; s)\; (-\backslash mu)^\; .$$

4. When the Abel–Plana formula is applied to the defining series of the polylogarithm, a Hermite-type integral representation results that is valid for all complex z and for all complex s:$$\backslash operatorname\_s(z)\; =\; \backslash tfracz\; +\; +\; 2z\; \backslash int\_0^\backslash infty\; \backslash frac\; dt$$

   where Γ is the upper incomplete gamma-function. 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,$$\backslash operatorname\_s(z)\; =\; \backslash tfracz\; +\; z\; \backslash int\_0^\backslash infty\; \backslash frac\; dt,$$

   avoids the use of the incomplete gamma function, but this integral fails for z on the positive real axis if Re(s) ≤ 0. This expression is found by writing 2^s Li_s(−z) / (−z) = Φ(z², s, ¹⁄₂) − z Φ(z², s, 1), where Φ is the Lerch transcendent, and applying the Abel–Plana formula to the first Φ series and a complementary formula that involves 1 / (e^2πt + 1) in place of 1 / (e^2πt − 1) to the second Φ series.

5. We can express an integral for the polylogarithm by integrating the ordinary geometric series termwise for
   s \in N as $$\backslash operatorname\_(z)\; =\; \backslash frac\; \backslash int\_0^1\; \backslash frac\; dt.$$

Series representations

1. As noted under integral representations above, the Bose–Einstein integral representation of the polylogarithm may be extended to negative orders s by means of Hankel contour integration:$$\backslash operatorname\_s(e^\backslash mu)\; =\; -\; \backslash oint\_H\; dt,$$

   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 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 (;):$$\backslash operatorname\_s(e^\backslash mu)\; =\; \backslash Gamma(1\; -\; s)\; \backslash sum\_^\backslash infty\; (2k\; \backslash pi\; i\; -\; \backslash mu)^.$$

   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:$$\backslash operatorname\_s(e^\backslash mu)\; =\; \backslash Gamma(1-s)\; \backslash left[(-2\backslash pi\; i)^\{s-1\}\; \backslash sum\_\{k=0\}^\backslash infty\; \backslash left(k\; +\; \{\backslash mu\; \backslash over\; \{2\backslash pi\; i\}\}\; \backslash right)^\{s-1\}\; +\; (2\backslash pi\; i)^\{s-1\}\; \backslash sum\_\{k=0\}^\backslash infty\; \backslash left(k+1-\; \{\backslash mu\; \backslash over\; \{2\backslash pi\; i\}\}\; \backslash right)^\{s-1\}\; \backslash right],$$

   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 ≠ 0, 1, 2, 3, … and was first derived in .

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:$$\backslash operatorname\_s(e^\backslash mu)\; =\; \backslash Gamma(1\; -\; s)\; (-\backslash mu)^\; +\; \backslash Gamma(1\; -\; s)\; \backslash sum\_^\backslash infty\; \backslash left[(-2\; h\; \backslash pi\; i\; -\; \backslash mu)^\{s-1\}\; +\; (2\; h\; \backslash pi\; i\; -\; \backslash mu)^\{s-1\}\; \backslash right]\; .$$

   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:$$\backslash operatorname\_s(e^\backslash mu)\; =\; \backslash Gamma(1\; -\; s)\; (-\backslash mu)^\; +\; \backslash sum\_^\backslash infty\; \backslash mu^k\; .$$

   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 is a positive integer, s = n, both the term with k = n − 1 and the gamma function become infinite, although their sum does not. One obtains (;):$$\backslash lim\_\; \backslash left[\{\backslash zeta(s-k)\; \backslash over\; k!\}\; \backslash mu^k\; +\; \backslash Gamma(1\; -\; s)\; (-\backslash mu)^\{s-1\}\; \backslash right]\; =\; \backslash left[\backslash sum\_\{h=1\}^k\; \{1\; \backslash over\; h\}\; -\; \backslash ln(-\backslash mu)\; \backslash right],$$

   where the sum over h vanishes if k = 0. So, for positive integer orders and for |μ| < 2π we have the series:$$\backslash operatorname\_(e^\backslash mu)\; =\; \backslash left[H\_\{n-1\}\; -\; \backslash ln(-\backslash mu)\; \backslash right]\; +\; \backslash sum\_^\backslash infty\; \backslash mu^k,$$

   where H_n denotes the nth harmonic number:$$H\_n\; =\; \backslash sum\_^n,\; \backslash qquad\; H\_0\; =\; 0.$$

   The problem terms now contain −ln(−μ) which, when multiplied by μⁿ⁻¹, will tend to zero as μ → 0, except for n = 1. This reflects the fact that Li_s(z) exhibits a true logarithmic singularity at s = 1 and z = 1 since:$$\backslash lim\_\; \backslash Gamma(1-s)(-\backslash mu)^\; =\; 0\; \backslash qquad\; (\backslash operatorname(s)\; >\; 1).$$

   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 . Erdélyi's corresponding expansion 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$$1\; =\; \backslash int\_0^\backslash infty\; e^\; t^\; dt\; \backslash qquad\; (\backslash operatorname(s)\; >\; 0),$$

   the Bose–Einstein integral representation of the polylogarithm (see above) may be cast in the form:$$\backslash operatorname\_s(z)\; =\; \backslash tfracz\; +\; \backslash int\_0^\backslash infty\; e^\; t^\; \backslash coth\; dt\; \backslash qquad\; (\backslash operatorname(s)\; >\; 0).$$

   Replacing the hyperbolic cotangent with a bilateral series,$$\backslash coth\; =\; 2\; \backslash sum\_^\backslash infty,$$

   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:$$\backslash operatorname\_s(z)\; =\; \backslash tfracz\; +\; \backslash sum\_^\backslash infty\; .$$

   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:$$\backslash operatorname\_(z)\; =\; \backslash sum\_^n\; \backslash left(\backslash right)^\; \backslash sum\_^k\; (-1)^\; (j+1)^n\; \backslash qquad\; (n=0,1,2,\backslash ldots).$$

   The infinite series obtained by simply extending the outer summation to ∞ :$$\backslash operatorname\_s(z)\; =\; \backslash sum\_^\backslash infty\; \backslash left(\backslash right)^\; ~\backslash sum\_^k\; (-1)^\; (j+1)^,$$

   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:$$\backslash sum\_^\backslash infty\; \backslash left(\backslash right)^\; =\; \backslash left[\backslash left(\{-z\; \backslash over\; 1-z\}\; \backslash right)^\{-1\}\; -1\; \backslash right]^\; =\; (-z)^.$$

   The inner coefficients of these series can be expressed by Stirling-number-related formulas involving the generalized harmonic numbers. For example, see generating function transformations to find proofs (references to proofs) of the following identities:

   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):

$$\backslash operatorname\_s(z)\; =\; [\backslash ln(-z)\; \backslash pm\; i\backslash pi]^\; -\; \backslash sum\_^\backslash infty\; (-1)^k\; (2\backslash pi)^,$$$$\backslash operatorname\_s(z)\; =\; \backslash sum\_^\backslash infty\; (-1)^k\; (1-2^)\; (2\backslash pi)^,$$

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 non-negative integer s, they break off after a finite number of terms. describes a method for obtaining these series from the Bose–Einstein integral representation (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 :

$$\backslash operatorname\_s(z)\; \backslash sim\_$$

\to 0

z$$\backslash operatorname\_s(e^\backslash mu)\; \backslash sim\_$$

\to 0

\Gamma(1 - s) (-\mu)^ \qquad (\operatorname(s) < 1)$$\backslash operatorname\_s(\backslash pm\; e^\backslash mu)\; \backslash sim\_\; -\backslash qquad\; (s\; \backslash ne\; -1,\; -2,\; -3,\; \backslash ldots)$$$$\backslash operatorname\_(e^\backslash mu)\; \backslash sim\_\; -(-1)^n\; e^\; \backslash qquad\; (n\; =\; 1,\; 2,\; 3,\; \backslash ldots)$$$$\backslash operatorname\_s(z)\; \backslash sim\_\; z$$$$\backslash operatorname\_s(-e^\backslash mu)\; \backslash sim\_\; \backslash Gamma(1\; -\; s)\; \backslash left[(-\backslash mu\; -\; i\backslash pi)^\{s-1\}\; +\; (-\backslash mu\; +\; i\backslash pi)^\{s-1\}\; \backslash right]\; \backslash qquad\; (\backslash operatorname(\backslash mu)\; =\; 0)$$

Wood's first limit for has been corrected in accordance with his equation 11.3. The limit for follows from the general relation of the polylogarithm with the Hurwitz zeta function (see above).

Dilogarithm

See main article: Dilogarithm.

The dilogarithm is the polylogarithm of order s = 2. An alternate integral expression of the dilogarithm for arbitrary complex argument z is :$$\backslash operatorname\_2\; (z)\; =\; -\backslash int\_0^z\; dt\; =\; -\backslash int\_0^1\; dt.$$

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

In the case of real z ≥ 1 the first integral expression for the dilogarithm can be written as$$\backslash operatorname\_2(z)\; =\; \backslash frac\; -\; \backslash int\_1^z\; dt\; -\; i\backslash pi\; \backslash ln\; z$$

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

$$\backslash operatorname\_2(z)\; =\; \backslash frac\; -\; \backslash frac(\backslash ln\; z)^2\; -\; \backslash sum\_^\backslash infty\; -\; i\backslash pi\; \backslash ln\; z\; \backslash qquad\; (z\; \backslash ge\; 1).$$

The Abel identity for the dilogarithm is given by

$$\backslash operatorname\_2\; \backslash left(\backslash frac\; \backslash right)\; +\; \backslash operatorname\_2\; \backslash left(\backslash frac\; \backslash right)\; -\; \backslash operatorname\_2\; \backslash left(\backslash frac\; \backslash right)\; =\; \backslash operatorname\_2(x)\; +\; \backslash operatorname\_2(y)\; +\; \backslash ln(1-x)\; \backslash ln(1-y)$$

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$$\backslash operatorname\_2\; \backslash left(x\; \backslash right)\; +\; \backslash operatorname\_2\; \backslash left(1-x\backslash right)\; =\; \backslash frac\; \backslash pi^2\; -\; \backslash ln(x)\backslash ln(1-x),$$where Li₂(1) = ζ(2) = ¹⁄₆ π² has been used and x may take any complex value.

In terms of the new variables u = x/(1−y), v = y/(1−x) the Abel identity reads$$\backslash operatorname\_2(u)\; +\; \backslash operatorname\_2(v)\; -\; \backslash operatorname\_2(uv)\; =\; \backslash operatorname\_2\; \backslash left(\backslash frac\; \backslash right)\; +\; \backslash operatorname\_2\; \backslash left(\backslash frac\; \backslash right)\; +\; \backslash ln\; \backslash left(\backslash frac\; \backslash right)\; \backslash ln\backslash left(\backslash frac\; \backslash right),$$which corresponds to the pentagon identity given in .

From the Abel identity for x = y = 1−z and the square relationship we have Landen's identity$$\backslash operatorname\_2(1-z)\; +\; \backslash operatorname\_2\; \backslash left(1-\backslash frac\; \backslash right)\; =\; -\; \backslash frac\; (\backslash ln\; z)^2\; \backslash qquad\; (z\; \backslash not\; \backslash in\; ~]-\backslash infty;\; 0]),$$and applying the reflection formula to each dilogarithm we find the inversion formulabilogarithmische Function was introduced by Carl Johan Danielsson Hill (professor in Lund, Sweden) in 1828 . has remarked that the dilogarithm is the only mathematical function possessing a sense of humor.

Special values of the dilogarithm

x	\operatorname{Li} _{2(x)}	x	\operatorname{Li} _{2(x)}
-1	- \tfrac { 1 }{ 12 } \pi ^{2}	- \phi	- \tfrac { 1 }{ 10 } \pi ^{2} - ln ^{2} \phi
0	0	-1 / \phi	- \tfrac { 1 }{ 15 } \pi ^{2} + \tfrac { 1 }{ 2 } ln ^{2} \phi
\tfrac { 1 }{ 2 }	\tfrac { 1 }{ 12 } \pi ^{2} - \tfrac { 1 }{ 2 } ln ^{2} 2	1 / \phi ^{2}	\tfrac { 1 }{ 15 } \pi ^{2} - ln ^{2} \phi
1	\tfrac { 1 }{ 6 } \pi ^{2}	1 / \phi	\tfrac { 1 }{ 10 } \pi ^{2} - ln ^{2} \phi
2	\tfrac { 1 }{ 4 } \pi ^{2} - \pi i ln 2	\phi	\tfrac { 11 }{ 15 } \pi ^{2} + \tfrac { 1 }{ 2 } ln ^{2(-1} / \phi)
		\phi ^{2}	- \tfrac { 11 }{ 15 } \pi ^{2} - ln ^{2(-\phi)}

Here

\phi = \tfrac{ 1 }{ 2 } (\sqrt{ 5 } +1 ) denotes the golden ratio.

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 dilogarithm ladders are

$$\backslash operatorname\_2(\backslash rho^6)\; =\; 4\; \backslash operatorname\_2(\backslash rho^3)\; +\; 3\; \backslash operatorname\_2(\backslash rho^2)\; -\; 6\; \backslash operatorname\_2(\backslash rho)\; +\; \backslash tfrac\; \backslash pi^2$$

given by and

$$\backslash operatorname\_2(\backslash rho)\; =\; \backslash tfrac\; \backslash pi^2\; -\; \backslash ln^2\backslash rho$$

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 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 m₀ and m₁, the monodromy group has the group presentation

$$\backslash langle\; m\_0,\; m\_1\; \backslash vert\; w\; =\; m\_0\; m\_1\; m^\_0\; m^\_1,\; w\; m\_1\; =\; m\_1\; w\; \backslash rangle.$$

For the special case of the dilogarithm, one also has that wm₀ = m₀w, and the monodromy group becomes the Heisenberg group (identifying m₀, m₁ and w with x, y, z) .

References

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External links

• • • Algorithms in Analytic Number Theory provides an arbitrary-precision, GMP-based, GPL-licensed implementation.

Notes and References

1. Bose integral is result of multiplication between Gamma function and Zeta function.

   One can begin with equation for Bose integral, then use series equation.

   $$\backslash int\_^\backslash fracdx\; =\; \backslash int\_^x^s\backslash fracdx\; =\; \backslash int\_^\backslash frac\backslash frac\; dx\backslash quad\; \backslash wedge\; \backslash quad\; \backslash frac\; =\; \backslash sum\_^r^n$$

   $$\backslash int\_^\backslash frac\backslash sum\_^\; \backslash left(\backslash frac\backslash right)^n\; dx\; =\; \backslash int\_^\backslash frac\backslash sum\_^e^dx\; =\; \backslash sum\_^\backslash int\_^x^s\; e^\; e^dx$$

   Secondly, regroup expressions.

   $$\backslash sum\_^\backslash int\_^x^s\; e^dx\backslash quad\backslash wedge\backslash quad\; u=(n+1)x,du=(n+1)dx\; \backslash Rightarrow\; dx=\backslash frac$$

   $$\backslash sum\_^\backslash int\_^\backslash left(\backslash frac\backslash right)^s\; e^\backslash frac\; =\; \backslash sum\_^\backslash int\_^\backslash fracu^s\; e^du$$

   $$\backslash sum\_^\backslash frac\; \backslash left(\backslash int\_^\; u^s\; e^\; du\; \backslash right)\; =\; \backslash left(\backslash int\_^u^s\; e^du\; \backslash right)\; \backslash left(\backslash sum\_^\backslash frac\backslash right)=$$

   $$\backslash left(\backslash int\_^u^e^du\backslash right)\; \backslash left(\backslash sum\_^\backslash frac\backslash right)\; =\; \backslash Gamma(s+1)\backslash zeta\; (s+1).$$