Polylogarithm

Articles about Polylogarithm

Information About Polylogarithm

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:$$ \operatorname{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 . 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. ''z''''s'' = Exp(''s'' ln(''z''))). Depending on the parameter ''s'', the polylogarithm may be multi-valued. The principal branch of the polylogarithm is chosen to be that for which Li''s''(''z'') is real for ''z'' real, 0 ≤ ''z'' ≤ 1 and is continuous except on the positive real axis, where a cut is made from ''z'' = 1 to ∞ such that the cut puts the real axis on the lower half plane of ''z''. In terms of μ, this amounts to -π < arg(-μ) ≤ π. The fact that the polylogarithm may be discontinuous in μ can cause some confusion. For ''z'' real and ''z'' ≥ 1 the imaginary part of the polylogarithm is : :$extrm\{Im\}(\backslash operatorname\{Li\}\_s(z))\; =\; -\backslash over\{\backslash Gamma(s)\}\}.$ Going across the cut, if δ is an infinitesimally small positive real number, then: :$extrm\{Im\}(\backslash operatorname\{Li\}\_s(z+i\backslash delta))\; =\; \backslash over\{\backslash Gamma(s)\}\}.$ The derivatives of the polylogarithm are: :$z\{\backslash partial\; \backslash operatorname\{Li\}\_s(z)\; \backslash over\; \backslash partial\; z\}\; =\; \backslash operatorname\{Li\}\_\{s-1\}(z)$ :$\{\backslash partial\; \backslash operatorname\{Li\}\_s(e^\backslash mu)\; \backslash over\; \backslash partial\; \backslash mu\}\; =\; \backslash operatorname\{Li\}\_\{s-1\}(e^\backslash mu).$ PARTICULAR VALUES See also the " Relationship To Other Functions " section below. For integer values of ''s'', we have the following explicit expressions: :$\backslash operatorname\{Li\}\_\{1\}(z)\; =\; -\; extrm\{Ln\}\backslash left(1-z\; ight)$ :$\backslash operatorname\{Li\}\_\{0\}(z)\; =\; \{z\; \backslash over\; 1-z\}$ :$\backslash operatorname\{Li\}\_\{-1\}(z)\; =\; \{z\; \backslash over\; (1-z)^2\}$ :$\backslash operatorname\{Li\}\_\{-2\}(z)\; =\; \{z(1+z)\; \backslash over\; (1-z)^3\}$ :$\backslash operatorname\{Li\}\_\{-3\}(z)\; =\; \{z(1+4z+z^2)\; \backslash 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: :$\backslash operatorname\{Li\}\_\{1\}\backslash left(1/2\; ight)\; =\; extrm\{Ln\}(2)$ :$\backslash operatorname\{Li\}\_\{2\}(1/2)\; =\; \{1\; \backslash over\; 12\}[\backslash pi^2-6\; extrm\{Ln\}^2(2)]$ :$\backslash operatorname\{Li\}\_\{3\}(1/2)\; =\; \{1\; \backslash over\; 24\}[4\; extrm\{Ln\}^3(2)-2\backslash pi^2\; extrm\{Ln\}\; (2)+21\backslash ,\backslash zeta(3)]$ where ζ is the Riemann Zeta Function . No similar formulas of this type are known for higher orders ALTERNATE EXPRESSIONS The integral of the Bose-Einstein Distribution is expressed in terms of a polylogarithm: ::$$ \operatorname{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. The integral of the Fermi-Dirac Distribution is also expressed in terms of a polylogarithm: ::$$ -\operatorname{Li}_{s+1}(-z) \equiv {1 \over \Gamma(s+1)} \int_0^\infty {t^s \over e^t/z+1} dt. :This converges for Re(s) > 0 and all ''z'' except for ''z'' real and < (−1). The polylogarithm in this context is sometimes referred to as a Fermi integral or a Fermi-Dirac integral. The polylogarithm may be rather generally represented by a Hankel Contour integral . As long as the t = μ pole of the integrand does not lie on the non-negative real axis, and ''s'' ≠ 1, 2, 3, …, we have: ::$$ \operatorname{Li}_s(e^\mu)=\oint_H \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: ::$$ \operatorname{Li}_s(e^\mu)=-\oint_H \over{e^{t-\mu}}-1}dt + 2\pi i R :where ''R'' is the Residue of the pole: ::$$ R = \over{2\pi}}. The square relationship is easily seen from the defining equation (see also , : ::$$ \operatorname{Li}_s(-z) + \operatorname{Li}_s(z) = 2^{1-s} ~ \operatorname{Li}_s(z^2). :Note that Kummer's Function obeys a very similar duplication formula. RELATIONSHIP TO OTHER FUNCTIONS For ''z'' = 1 the polylogarithm reduces to the Riemann Zeta Function :$\backslash operatorname\{Li\}\_s(1)\; =\; \backslash zeta(s)~~~~~~~~~~~~~(\; extrm\{Re\}(s)>1).$ The polylogarithm is related to Dirichlet Eta Function and the Dirichlet Beta Function : :$$ \operatorname{Li}_s(-1) = -\eta\left(s ight) where η(''s'') is the Dirichlet eta function. For pure imaginary arguments, we have: :$$ \operatorname{Li}_s(\pm i) = 2^{-s}\eta(s)\pm i \beta(s)\, where β(''s'') is the Dirichlet beta function. The polylogarithm is equivalent to the Fermi-Dirac integral :$$ F_s(\mu)=-\operatorname{Li}_{s+1}(-e^\mu).\, The polylogarithm is a special case of the Lerch Transcendent :$\backslash operatorname\{Li\}\_s(z)=z~\backslash Phi(z,s,1).$ The polylogarithm is related to the Hurwitz Zeta Function by: :$$ \operatorname{Li}_s(e^{2\pi i x})+(-1)^s \operatorname{Li}_s(e^{-2\pi i x})={(2\pi i)^s \over \Gamma(s)}~\zeta\left (1-s,x ight) where Γ(''s'') is the Gamma Function . This holds for : and also for : (Note that Erdélyi's equivalent Equation 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

See below for a simplified formula when $s$ is an interger.

Using the relationship between the Hurwitz zeta function and the Bernoulli Polynomials :

:$$
\zeta(-n,x)=-{B_{n+1}(x) \over n+1}


which holds for all ''x'' and ''n'' = 0, 1, 2, 3, … it can be seen that:

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


under the same constraints on ''s'' and ''x'' as above. (Note that the corresponding equation is not correct) For negative integer values of the parameter, we have for all ''z''
:

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


More generally for $n=0,\backslash pm1,\backslash pm2,\backslash pm3,\backslash cdots$
:$$
\operatorname{Li}_{n}(z)+ (-1)^n \operatorname{Li}_{n}\left(1/z ight)+rac{(2i\pi)^n}{n!}\,B_n\left({\log
z\over 2i\pi} ight)=0 \qquad z~
ot\in~]1;+\infty[,

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

The polylogarithm with pure imaginary μ may be expressed in terms of Clausen Function s Ci_''s''(θ) and Si_''s''(θ)
,

:$$
\operatorname{Li}_s(e^{\pm i heta}) = Ci_s( heta) \pm i Si_s( heta).

The Inverse Tangent Integral Ti_''s''(''z'')
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'')
,

can be expressed in terms of polylogarithms:

:$$
\chi_s(z)={1 \over 2}~ {Link without Title} .

The polylogarithm may be expressed as a series of Debye Function s Z_''n''(''z'')

:$$
\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





SERIES REPRESENTATIONS

• We may represent the polylogarithm as a power series about μ = 0 as follows:
  
  Consider the Mellin Transform :
  
  :$$
  M_s(r)
  =\int_0^\infty extrm{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) extrm{Li}_{s+r}(f).
  
  
  For ''f'' = 1 we have, through the inverse Mellin transform:
  
  :$$
  \operatorname{Li}_{s}(e^{-u})={1 \over 2\pi i}\int_{c-i\infty}^{c+i\infty}\Gamma(r)
  \zeta(s+r)u^{-r}dr
  
  
    So, For ''s'' ''n'' where ''n'' is a positive integer and&mu< 2&pi we have the following:
  
  
  \sum_{k=0}^{\infty} { B_{k+n+1}\over k!~(k+n+1)}~\mu^k,
  ~~~~~~~~~~~n=1,2,3,\ldots
  
  
  since, except for ''B''₁, all odd Bernoulli numbers are zero. We obtain the ''n'' = 0 term using ζ(0) = ''B''₁ = −¹⁄₂. Note again that Erdélyi's equivalent Equation 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
  
  :
  
  :$$
  \operatorname{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 = 2''k''πi and the integral can be evaluated as the sum of the Residues :
  
  :$$
  \operatorname{Li}_s(e^\mu)=\Gamma(1-s)\sum_{k=-\infty}^\infty (2k\pi i-\mu)^{s-1}.
  
  
  This will hold for Re(''s'') < 0 and all μ except where ''e''^μ = 1. Summing the series, one obtains
  
  :$$
  \operatorname{Li}_s(e^\mu)=-\sum_{k=0}^\infty rac{1}{k!}
  \left[1-rac{2}{2^{s-k}} ight]\zeta(s-k) (\mu-\pi i)^k
  
  
  Note that this sum can be more compact written in terms of the Dirichlet Eta Function .
  
  
• For negative integer ''s'', the polylogarithm may be expressed as a series involving the Eulerian Numbers
  
  :$$
  \operatorname{Li}_{-n}(z) =
  {1 \over (1-z)^{n+1}} \sum_{i=0}^{n-1}\left\langle{n\atop i} ight angle
  z^{n-i}, ~~~~~~~~~~~~~n=1,2,3,\ldots
  
  
  where $\backslash left\backslash langle\{n\backslash atop\; i\}\; ight\; angle$ are Eulerian numbers:
  
  
• Another explicit formula for negative integer ''s'' is
  :
  
  :$$
  \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 Number s of the second kind.
  
  

LIMITING BEHAVIOR


The following Limits hold for the polylogarithm
:

:$$

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