polylogarithm examples

A cluster (polylogarithm) function f [35] of (transcen-dental) weight w has the defining property that its differ-ential has the form dfðwÞ ¼ X i f i dloga i; ð7Þ where the f i are again cluster functions, of weight (w−1). The first integral that we will evaluate in this post is the following: I_1 = \int_0^1 \frac{\log^2(x) \arctan(x)}{1+x^2}dx Of course, one can use brute force methods to find a closed form anti-derivative in terms of polylogarithms. PolyLog[n, p, z] gives the Nielsen generalized polylogarithm function S n, p (z). Graph of y = x2, y = p x, y = ex, y = ln(x). Share. polylog(a, z) a-expression z-expression. Complex Numbers. The polylogarithm … Matlab and Octave have the following primitives for complex numbers: octave:1> help j j is a built-in constant - Built-in Variable: I - Built-in Variable: J - Built-in Variable: i - Built-in Variable: j A pure imaginary number, defined as `sqrt (-1)'. Naturally, as curious mathematicians, we ask whether the polylogarithm can be found in all places that logarithms appear. The default value is 100 * eps and the resulting tolerance for a given complex pair is tol * abs (z(i))) . It is defined as. Add a comment. Polylogarithm functions, symbols, and the coproduct We begin by recalling some elementary mathematical facts about polylogarithm functions from [28, 29] (see[2, 30–32] for recent reviews written for physicists). 0. These are sufficient to evaluate it numerically, with reasonable efficiency, in all cases. The Polylogarithm is a very simple Taylor series, a generalisation of the widely used logarithm function. The polylogarithm function Lis(x) is Lis(x) = X1 k=1 k¡sxk: (1.1) If s is a negative integer, say s = ¡r, then the polylogarithm function converges for jxj < 1 and equals Li¡r(x) = Pr j=0 D r j E x ¡j (1¡x)r+1; (1.2) where the D r j E are the Eulerian numbers. For example, Li 0 (z) = 1/(1−z), thus (2) implies that for all α ∈ Z − , Li α (z) ∈ Q(z) is a rational function with a single singularity at z = 1. Otherwise they are different and your other examples are all polylogarithmic. The polylogarithm function or Jonquiegraveres function of index and argument is a special function defined in the complex plane for and by analytic continuation otherwise It can be plotted for complex values for example along the celebrated critical line for Riemanns zeta function 1 The polylogarithm function appears in the FermindashDirac and BosendashEinstein distributions and also in quantum e What does polylogarithms mean? 3.17.4. The generalized polylogarithm G(a1,…,an;x) diverges whenever x=a1. Naturally, as curious mathematicians, we ask whether the polylogarithm can be found in all places that logarithms appear. The polylogarithm has occurred in situations analagous to other situations involving the logarithm. Polylogarithm ladders provide the basis for the rapid computations of various mathematical constants by means of the BBP algorithm (Bailey, Borwein & Plouffe 1997). generalized polylogarithms in perturbative quantum field theory a dissertation submitted to the department of physics and the committee on graduate studies Definite integral for ζ(3) By making use of Mathematica, I detected the following integral expression for zeta (3): ∫1 0log(x) 1 + xlog(2 + x 1 + x)dx = − 5 12ζ(3). • In [BK10a], Bannai and Kings determined the Eisenstein classes associated to the syntomic polylogarithm of a modular curve in terms of p-adic Eisenstein series. Any proof of it would be ... integration polylogarithm. A number of examples are given, together with a representative sample of lad- The polylogarithm functions above give rise to certain so-called "Eisenstein classes" which have turned out to be useful tools in proofs. Special function Lis(z) of order s and argument z. These are now called polylogarithm ladders. The polylogarithm function is related to other special functions. Contextual translation of "legendresche" into English. Polylogarithm. The polynomial is f (n) = n^2. It also arises in the closed form of the integral of the Fermi-Dirac and the Bose-Einstein distributions. When the order is 2 or 3, the polylogarithm simplifies to a numerical value if z is a real floating point number or the numer evaluation flag is present. It is defined as. ( March 2018) In mathematics, a generating function is a way of encoding an infinite sequence of numbers ( an) by treating them as the coefficients of a formal power series. Currently, mainly the case of negative integer sis well supported,as that is used for some of the Archimedean copula densities. jv (v, z). Abstract. Suppose we have a function G({a1(t),...,an(t)},z), we want to rewrite it into a sum ofconstants and G functions with the fromG({b1,...,bn},t),where bi is free of t. Then we can calcluate the 1d integral from the definition of G function. existence of relations among polylogarithm values, which was the original motivation for the conjectures in [Z].) The Polylogarithm is also known as Jonquiere's function. We give a few concrete examples of exponential motives M such that at least some of the periods of M are special values of E-functions. 2. jve (v, z). Please discuss this issue on the article's talk page. 2 bronze badges. But these three classes of functions tend to 1at di erent rates. The"default" method uses the dilog orcomplex_dilog function from package gsl,res… Details. Here are some examples: Definite integration. The relative sizes are di erent for x near 0 and for large x. x f(x) Figure 1. Several examples in the case of the dilogarithm are discussed in §2, and for higher polylogarithms in §3. In this work, we demonstrate the existence of a privileged choice of delta in the sense that it is continuous, invertible, maximal and it is the solution of a simple functional equation. One of them … polylogarithm (1.2) often arises in physical problems via the multiple integration of rational forms, one might expect that the more general multiple polylogarithm (1.1) would likewise find application in a wide variety of physical contexts. li [1] is - log (1 - z).li [2] and li [3] are the dilogarithm and trilogarithm functions, respectively. This series is called the generating function of the sequence. For s = 2, Li_2(z) is also called‘dilogarithm’ or “Spence's function”. Examples¶ Some of these examples show how to use the delta function definition of the functional derivative in equation . Nevertheless, lest it be suspected that the authors have embarked on a program of generaliza- A q-analogue of the polylogarithm function is introduced via a consideration of the spectral zeta-function of the quantum group SU q (2). The length of ~a =: w is the weight of the polylogarithm. Suppose you wanted to implicitize x = a + b t and y = t 2. Of or pertaining to a polylogarithm. This paper develops an approach to evaluation of Euler sums and integrals of polylogarithm functions. We also introduce an algorithm that can be used to numerically calculate this map in polylogarithm time, proving the computability of the epsilon--delta relation. Available for free under the MIT/X11 License . &*p x d$(x) $ The polylogarithm function, Li p(z), is defined, and a number of algorithms are derived for its computation, valid in different ranges of its real parameter p and complex argument z.

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