NationMaster - Encyclopedia: Polylogarithm

The polylogarithm (also known as de Jonquière's function) is a special function Li_s(z) that is defined by the sum In mathematics, several functions are important enough to deserve their own name. ...

$operatorname{Li}_s(z) = sum_{k=1}^infty {z^k over k^s}.$

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. In mathematics, several functions are important enough to deserve their own name. ... Look up logarithm in Wiktionary, the free dictionary. ... The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ... In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. ...

**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 The natural logarithm, formerly known as the hyperbolic logarithm, is the logarithm to the base e, where e is an irrational constant approximately equal to 2. ... There are two related special functions of mathematics that are referred to as Spences function: the Dilogarithm. ...

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

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. In mathematics, a rational function in algebra is a function defined as a ratio of polynomials. ...

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. In statistical mechanics, Fermi-Dirac statistics determines the statistical distribution of fermions over the energy states for a system in thermal equilibrium. ... For other topics related to Einstein see Einstein (disambig) In statistical mechanics, Bose-Einstein statistics determines the statistical distribution of identical indistinguishable bosons over the energy states in thermal equilibrium. ... A polylogarithmic function in n is a polynomial in the logarithm of n, In computer science, polylogarithmic functions occur as the order of some algorithms (eg. ... The offset logarithmic integral, or European logarithmic integral, is a non-elementary function Li(x) differing by a constant from the logarithmic integral function li(x), defined such that: Explicitly, this means where ln is the natural logarithm. ...

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

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))). In mathematics, a principal branch is a function which selects one branch, or slice, of a multi-valued function. ... The natural logarithm is the logarithm to the base e, where e is equal to 2. ...

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:

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

The derivatives of the polylogarithm are:

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

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

Particular values

See also the "Relationship to other functions" section below. Image File history File links This is a lossless scalable vector image. ...

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

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

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

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

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

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

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

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: In mathematics, a rational function in algebra is a function defined as a ratio of polynomials. ...

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

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

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

where ζ is the Riemann zeta function. No similar formulas of this type are known for higher orders (Lewin, 1991 p2.) In mathematics, the Riemann zeta function, named after German mathematician Bernhard Riemann, is a function of significant importance in number theory, because of its relation to the distribution of prime numbers. ...

Alternate expressions

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:

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

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:

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

where R is the residue of the pole:

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

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:

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

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 other topics related to Einstein see Einstein (disambig) In statistical mechanics, Bose-Einstein statistics determines the statistical distribution of identical indistinguishable bosons over the energy states in thermal equilibrium. ... In statistical mechanics, Fermi-Dirac statistics determines the statistical distribution of fermions over the energy states for a system in thermal equilibrium. ... Categories: Complex analysis | Special functions | Stub ... A residue, broadly, is anything left behind by a reaction or event. ... In mathematics, the multiplication theorem is a certain type of identity obeyed by many special functions related to the gamma function. ... In mathematics, there are several functions known as Kummers function. ... In mathematics, the multiplication theorem is a certain type of identity obeyed by many special functions related to the gamma function. ... In mathematics, the discrete Fourier transform (DFT), occasionally called the finite Fourier transform, is a transform for Fourier analysis of finite-domain discrete-time signals. ...

Relationship to other functions

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:

$operatorname{Li}_s(-1) = -etaleft(sright)$

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:

$operatorname{Li}_s(e^{2pi i x})+(-1)^s operatorname{Li}_s(e^{-2pi i x})={(2pi i)^s over Gamma(s)}~zetaleft (1-s,xright)$

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 $z~notin~]1;+infty[$ , the inversion formula is

$operatorname{Li}_s(z)+(-1)^s operatorname{Li}_s(1/z)={(2pi i)^s over Gamma(s)}~zetaleft (1!-!s,{log zover2ipi}right),$

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

$operatorname{Li}_s(z)+(-1)^s operatorname{Li}_s(1/z)={(2pi i)^s over Gamma(s)}~zetaleft (1!-!s,{log zover2ipi}right)- 2ipifrac{(-log z)^{s-1}}{Gamma(s)},$

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:

$operatorname{Li}_{n}(e^{2pi i x})+ (-1)^n operatorname{Li}_{n}(e^{-2pi i x}) = -{(2 pi i)^nover n!} B_nleft({x}right)$

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):

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

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

$operatorname{Li}_{n}(z)+ (-1)^n operatorname{Li}_{n}left(1/zright)+frac{(2ipi)^n}{n!},B_nleft({log zover 2ipi}right)=0 qquad z~notin~]1;+infty[,$

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

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

In mathematics, the Riemann zeta function, named after German mathematician Bernhard Riemann, is a function of significant importance in number theory, because of its relation to the distribution of prime numbers. ... The Dirichlet eta function can be defined as where ζ is Riemanns zeta function. ... In mathematics, the Dirichlet beta function (also known as the Catalan beta function) is a special function, closely related to the Riemann zeta function. ... In mathematics, the Lerch zeta function is a special function that generalizes the Hurwitz zeta function and the polylogarithm. ... In mathematics, the Hurwitz zeta function is one of the many zeta functions. ... The Gamma function along part of the real axis In mathematics, the Gamma function (represented by the capitalized Greek letter Î“) is an extension of the factorial function to real and complex numbers. ... In mathematics, the Bernoulli polynomials occur in the study of many special functions and in particular the Riemann zeta function and the Hurwitz zeta function. ... In mathematics, the Clausen function is defined by the following integral: More generally, one defines . It is related to the polylogarithm by . Ernst Kummer and Rogers give the relation valid for . ... The polylogarithm (also known as de JonquiÃ¨res function) is a special function Lis(z) that is defined by the sum The above definition is valid for all complex numbers s and z where |z|< 1. ... In mathematics, the Legendre chi function is defined as The discrete fourier transform of the Legendre chi function with respect to the order n is the Hurwitz zeta function (Cvijovic). ... In mathematics, the family of Debye functions is defined by Categories: Math stubs ...

Series representations

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

$M_s(r) =int_0^infty textrm{Li}_s(fe^{-u})u^{r-1},du ={1 over Gamma(s)}int_0^inftyint_0^infty {t^{s-1}u^{r-1} over e^{t+u}/f-1}~dt~du.$

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

$M_s(r)={1 over Gamma(s)}int_0^1 b^{r-1} (1-b)^{s-1}dbint_0^infty{a^{s+r-1} over e^a/f-1}da = Gamma(r)textrm{Li}_{s+r}(f).$

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

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

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:

$lim_{srightarrow k+1}left[ {zeta(s-k)mu^k over k!}+Gamma(1-s)(-mu)^{s-1}right] = {mu^k over k!}left(sum_{m=1}^k{1 over m}-ln(-mu)right)$

and for k = 0:

$lim_{srightarrow 1}left[ zeta(s)+Gamma(1-s)(-mu)^{s-1}right] = -ln(-mu).$

So, for s = n where n is a positive integer and|μ|< 2π we have the following:

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

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

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

where H_n is a harmonic number:

$H_n = sum_{k=1}^n{1over k}.$

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:

$lim_{murightarrow 0}Gamma(1-s)(-mu)^{s-1}=0~~~~~(textrm{Re}(s)>1)$

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):

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

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:

$operatorname{Li}_s(e^mu)=Gamma(1-s)sum_{k=-infty}^infty (2kpi i-mu)^{s-1}.$

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

$operatorname{Li}_{-n}(z) = {1 over (1-z)^{n+1}} sum_{i=0}^{n-1}leftlangle{natop i}rightrangle z^{n-i}, ~~~~~~~~~~~~~n=1,2,3,ldots$

where $leftlangle{natop i}rightrangle$ 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.

In mathematics, the Mellin transform is an integral transform that may be regarded as the multiplicative version of the two-sided Laplace transform. ... The Comet Nucleus Tour (CONTOUR) was a Discovery-class space mission. ... In complex analysis, the residue is a complex number which describes the behavior of line integrals of a meromorphic function around a singularity. ... The harmonic number with (red line) with its asymptotic limit (blue line). ... In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability. ... In mathematics, the Bernoulli numbers Bn were first discovered in connection with the closed forms of the sums for various fixed values of n. ... Categories: Complex analysis | Special functions | Stub ... In complex analysis, the residue is a complex number which describes the behavior of line integrals of a meromorphic function around a singularity. ... The Dirichlet eta function can be defined as where ζ is Riemanns zeta function. ... In mathematics, Stirling numbers of the second kind, together with Stirling numbers of the first kind, are one of the two types of Stirling numbers. ...

Limiting behavior

The following limits hold for the polylogarithm (Wood 1992): Wikibooks Calculus has a page on the topic of Limits In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements as...

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

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

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

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

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

Dilogarithm

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):

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

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:

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

History note: Don Zagier remarked that "The dilogarithm is the only mathematical function with a sense of humor." Don Bernhard Zagier (1951 - ) is an American mathematician. ...

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 $rho=left(sqrt{5}-1right)/2$ as the reciprocal of the golden ratio. Then two simple examples of results from ladders include Not to be confused with Golden mean (philosophy), the felicitous middle between two extremes, Golden numbers, an indicator of years in astronomy and calendar studies, or the Golden Rule. ...

$operatorname{Li}_2(rho^6)=4operatorname{Li}_2(rho^3)+3operatorname{Li}_2(rho^2)-6operatorname{Li}_2(rho)+frac{7pi^2}{30}$

given by (Coxeter, 1935) and

$operatorname{Li}_2(rho)=frac{pi^2}{15} - log^2rho$

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. In mathematics, K-theory is, firstly, an extraordinary cohomology theory which consists of topological K-theory. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines techniques of abstract algebra, especially commutative algebra, with the language and the problematics of geometry. ... In mathematics, the Bailey-Borwein-Plouffe formula (BBP formula) originally referred to the π summation formula discovered in 1995 by Simon Plouffe. ...

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 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 In complex analysis, a branch point may be thought of informally as a point z0 at which a multiple_valued function changes values when one winds once around z0. ... In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. ... In mathematics, monodromy is the study of how objects from mathematical analysis, algebraic topology and differential geometry behave as they run round a singularity. ... The two bold paths shown above are homotopic relative to their endpoints. ... In mathematics, one method of defining a group is by a presentation. ...

$langle m_0, m_1vert w=m_0m_1m^{-1}_0m^{-1}_1,, wm_1=m_1wrangle$

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) In mathematics, the Heisenberg group is a group of 3×3 upper triangular matrices of the form Elements a,b,c can be taken from some (arbitrary) commutative ring. ...

References

Milton Abramowitz and Irene A. Stegun, eds. (1972). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover. ISBN 0-486-61272-4.
Bailey, David; Borwein, Peter B., and Plouffe, Simon (April 1997). "On the Rapid Computation of Various Polylogarithmic Constants". Mathematics of Computation 66 (218): 903-913.
Bailey, D. H. and Broadhurst, D. J. (June 20, 1999). A Seventeenth-Order Polylogarithm Ladder (PDF). Retrieved on November 1, 2005.
Boersma, J.; Dempsey, J. P. (1992). "On the evaluation of Legendre's chi-function". Mathematics of Computation 59 (199): 157-163.
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Categories: Special functions | Zeta and L-functions | Rational functions

Peter B. Borwein is a Canadian mathematician, co-developer of an algorithm for calculating Ï€ to the nth digit, co-discoverer of the billionth, four billionth, 40th billionth, and quadrillionth digits of Ï€, and professor at Simon Fraser University. ... Simon Plouffe is a Quebec mathematician born on June 11, 1956 in St-Jovite. ... Events of 2008: (EMILY) Me Lesley and MIley are going to China! This article is about the year. ... Year 2005 (MMV) was a common year starting on Saturday (link displays full calendar) of the Gregorian calendar. ... H.S.M. Coxeter. ... Year 2006 (MMVI) was a common year starting on Sunday of the Gregorian calendar. ... Year 2005 (MMV) was a common year starting on Saturday (link displays full calendar) of the Gregorian calendar. ... Year 2005 (MMV) was a common year starting on Saturday (link displays full calendar) of the Gregorian calendar. ...

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Polylogarithmic Extensions on Mixed Shimura varieties, by Joerg Wildeshaus (495 words)
	This paper is a slightly revised version of the author's thesis (November 1993) and is identical to the Heft 12 of the Schriftenreihe des Mathematischen Instituts Muenster.
	Chapter I (chapter1.dvi) gives partial results on what we call the "generic relatively unipotent sheaf", which is defined for any sufficiently nice morphism with a section of schemes over C (in the Hodge theoretic context) or a number field (in the context of l-adic sheaves, or systems of smooth sheaves).
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NC (284 words)
	In complexity theory, the class NC ("Nick's Class") is the set of decision problems decidable in polylogarithmic time on a parallel computer with a polynomial number of processors.
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