Not to be confused with polylogarithmic function or logarithmic integral function.
In mathematics, the polylogarithm (also known as Jonquière's function, for Alfred Jonquière) is a special function Lis(z) of order s and argument z. Only for special values of s 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 Li(z), which has the same notation without the subscript.
Different polylogarithm functions in the complex plane
Li –3(z)
Li –2(z)
Li –1(z)
Li0(z)
Li1(z)
Li2(z)
Li3(z)
The polylogarithm function is defined by a power series in z, which is also a Dirichlet series in s:
This definition is valid for arbitrary complex order s and for all complex arguments z with |z| < 1; it can be extended to |z| ≥ 1 by the process of analytic continuation. (Here the denominator ks is understood as exp(s ln k)). 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 also be defined as the repeated integral of itself:
thus the dilogarithm is an integral of a function involving the logarithm, and so on. For nonpositive integer orders s, the polylogarithm is a rational function.
In mathematics, the polylogarithm (also known as Jonquière's function, for Alfred Jonquière) is a special function Lis(z) of order s and argument z. Only...
In mathematics, the Incomplete Polylogarithm function is related to the polylogarithm function. It is sometimes known as the incomplete Fermi–Dirac integral...
Spence's function), denoted as Li2(z), is a particular case of the polylogarithm. Two related special functions are referred to as Spence's function...
Polylogarithm and related functions: Incomplete polylogarithm Clausen function Complete Fermi–Dirac integral, an alternate form of the polylogarithm....
special function that generalizes the Hurwitz zeta function and the polylogarithm. It is named after Czech mathematician Mathias Lerch, who published...
series, and various other forms. It is intimately connected with the polylogarithm, inverse tangent integral, polygamma function, Riemann zeta function...
polygamma function. Lis(z){\displaystyle \operatorname {Li} _{s}(z)} is a polylogarithm. (nk){\displaystyle n \choose k} is binomial coefficient exp(x){\displaystyle...
resembles the Dirichlet series for the polylogarithm, and, indeed, is trivially expressible in terms of the polylogarithm as χ ν ( z ) = 1 2 [ Li ν ( z )...
(disambiguation), rivers in China and Thailand Long Island, New York Li, the polylogarithm function Li, the logarithmic integral function <li></li>, indicating...
imq}}{m^{s}}}=\operatorname {Li} _{s}\left(e^{2\pi iq}\right)} where Lis(z) is the polylogarithm. It obeys the duplication formula 2 1 − s F ( s ; q ) = F ( s , q 2...
Pollard's kangaroo algorithm Pollard's rho algorithm for logarithms Polylogarithm Polylogarithmic function Prime number theorem Richter magnitude scale...
related functions see the articles zeta function and L-function. The polylogarithm is given by Li s ( z ) = ∑ k = 1 ∞ z k k s {\displaystyle \operatorname...
Bk appearing in the series for tanh x are the Bernoulli numbers. The polylogarithms have these defining identities: Li 2 ( x ) = ∑ n = 1 ∞ 1 n 2 x n Li...
whose field of study is hyperbolic geometry, geometric group theory and polylogarithm identities. As a child, she went to a gymnasium in Basel and then studied...
algebraic geometry as differential forms with logarithmic poles. The polylogarithm is the function defined by Li s ( z ) = ∑ k = 1 ∞ z k k s . {\displaystyle...
n ( 1 − p ) {\displaystyle \operatorname {Li} _{-n}(1-p)} is the polylogarithm function. The probability-generating functions of X and Y are, respectively...
function Nicholas Mercator – first to use the term natural logarithm Polylogarithm Von Mangoldt function Including C, C++, SAS, MATLAB, Mathematica, Fortran...
{\frac {1}{\operatorname {Li} _{s}(x)}}} for s > 1 where Lis(x) is the polylogarithm. For x = 1 the product above is just 1/ζ(s). Many well known constants...
+1}(z)}{\left(\beta E_{c}\right)^{\alpha }}},} where Lis(x) is the polylogarithm function. The problem with this continuum approximation for a Bose gas...
s. The Dirichlet beta function can also be written in terms of the polylogarithm function: β ( s ) = i 2 ( Li s ( − i ) − Li s ( i ) ) . {\displaystyle...
formulas for special values of Dedekind zeta functions in terms of polylogarithm functions. He discovered a short and elementary proof of Fermat's theorem...
3/2), z is exp(μ/kBT) where μ is the chemical potential, Li is the polylogarithm, ζ is the Riemann zeta function, and Tc is the critical temperature...
Global Information
Li –3(z)
Li –2(z)
Li –1(z)
Li0(z)
Li1(z)
Li2(z)
Li3(z)
