These polylogarithms are widely used in the calculation of Feynman integrals and amplitudes. It has a long history, and many connections to other special functions and series, and many applications, for instance in statistical physics. Documenta Mathematica 3: 297-299, 1998. HPL, a Mathematica implementation of the harmonic polylogarithms. We found the identity (1.2) by a combination of inspired guessing and extensive searching using the PSLQ integer relation algorithm [3],[12]. We give an explicit description of the syntomic elliptic polylogarithm on the universal elliptic curve over the It is built with a focus on extensibility and ease of use, through both interactive and programmatic applications. N[Erf[28/33], 25] 0.7698368826185349656257148 But: Exact or arbitrary-precision arithmetic is fairly slow! It contains an implementation of the product algebra, the derivative properties, series expansion and numerical evaluation. It returns the expression unevaluated. In this paper, we presented an implementation of the harmonic polylogarithm of Remiddi and Vermaseren [1] for Mathematica. (2) For , ... “On the asymptotic expansion of the logarithm of Barnes triple gamma function,” Mathematica Scandinavica, vol. Classical motivic polylogarithm according to Beilinson and Deligne. of the polylogarithm functions, once the chemical potential is known. 23, No. A 15 (2000) 725, hep-ph/9905237] for Mathematica. 2.1 How to use it from within C++ The GiNaC open framework for symbolic computation within the C++ programming language does not try to define a language of its own as conventional CAS do. The Mathematica routines for the series allow calculation to arbitrary order. Various results for the particular values of these parameters are deduced and compared with the existing literature. ), and special functions, including particle physics. L s ( z) = ∑ n = 1 ∞ z n n s. for the polylogarithm is valid only in the open disk | z | < 1. A 15 (2000) 725, hep-ph/9905237] for Mathematica. The dimension k of the vector a is called the weight of the HPL. Documenta Mathematica 3: 297-299, 1998. … See also: SimplifyPolyLog. As it turns out, these other formulas for π can all be written as formula (1.2) plus a rational multiple of the identity 0= ∞ i=0 1 16i −8 8i+1 8 8i+2 4 8i+3 8 8i+4 2 8i+5 2 8i+6 − 1 8i+7 The proof of The Analytical Evaluation of the Half-Order Fermi-Dirac Integrals Article Polylogarithm, 62.180.184.13 has smiled at you!Smiles promote WikiLove and hopefully this one has made your day better. These functions are useful to define because some identities connect them with the Barnes. This reduction can be done if z-a1, ai-ai+1, an, an-z are all linear reducible in t,i… 136 Asifa Tassaddiq, Rana Alabdan Therefore, Polylogarithm functions were first known to C. Truesdell when Mr. H. Jacobson informed him that these function play an important role in his researches on the theory of structure of polymers [7]. It contains an implementation of the product algebra, the derivative properties, series expansion and numerical evaluation. Its definition on the whole complex plane then follows uniquely via analytic continuation. The purpose of this paper is to 17 poz. Abstract In this paper, we present an implementation of the harmonic polylogarithm of Remiddi and Vermaseren [1] for Mathematica. Loading the Mathematica implementation of Rubi defines an extensive system of rules capable of integrating large classes of mathematical expressions. We give an explicit description of the syntomic elliptic polylogarithm on the universal elliptic curve over the The polylogarithm, also known as the Jonquière's function, is the function (1) defined in the complex plane over the open unit disk. Polylogarithm function Duo Tao California Institute of Technology November 25, 2018 Background Today I am trying to do an integral Z 2ˇ 0 t 1 + et dt: (1) I do not know how to do it so I tried to use Mathematica, which Abstract. Although, depending on your needs, it might be faster to just go with a numerical integration (or pre-calculated table lookup) rather than puyting all of this together, and converting it to python. Outside this region, the polylogarithm is defined through analytical continuation. We discuss the link with recent works on the subject, and show that the combinations of umbral and other techniques (such as the Laplace and other types of integral transforms) yield a very efficient tool to explore the properties of these numbers. In this paper, we present an implementation of the harmonic polylogarithm of Remiddi and Vermaseren for Mathematica. > > Mathematica processes the new cg definition into ... > > Clear[ "Global`\*"] > > which is not what I expected, either. The polylogarithm, also known as the Jonquière's function, is the function (1) defined in the complex plane over the open unit disk. The harmonic polylogarithms (HPL) H ( a 1, …, a k; x) are functions of one variable x labeled by a vector a = ( a 1, …, a k). PolyLog [ nu, z] (224 formulas) This can be achieved with a Polylogarithm. Stack Exchange network consists of 177 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. 3.Mathematica refuses to compute (m)(0) for m >= 4. Stack Exchange network consists of 177 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. > > Tests of both definitions for cg yield failures with error Contiguous polylogs have the ladder operators Lis + 1(z) = ∫z 0Lis(t) t dt, Lis − 1(z) = z∂Lis(z) ∂z, and the sequence can be started with either Li1(z) = − ln(1 − z), Li0(z) = z 1 − z. done in Maple or Mathematica. Plot level curves (contour lines) of the matrix z and fill the region between lines with colors from the current colormap. We give an explicit description of the syntomic elliptic polylogarithm on the universal elliptic curve over the The Mathematica routines for the series allow calculation to arbitrary order. A numerical realization of multiple polylogarithm (or Goncharov polylogarithm, generalized polylogarithm) in pure Mathematica based on the algorithm given in this paper 0410259. We can avoid the need for complex arithmetic in this case by substituting the expression: This is an improvement because with polylogarithm arguments in [0, 1], the results are purely real values. gives the Nielsen generalized polylogarithm function . Mathematical function, suitable for both symbolic and numerical manipulation. . . . PolyLog [ n, z] has a branch cut discontinuity in the complex plane running from 1 to . For certain special arguments, PolyLog automatically evaluates to exact values. Shortly after the authors originally 1. rasterContourPlot and rasterListContourPlot replace ContourPlot and ListContourPlot. Elliptic Polylogarithm via the Poincaré Bundle Johannes Sprang Received: October4,2018 Revised: April12,2019 CommunicatedbyOtmarVenjakob Abstract. In this paper we provide Mathematica functions to obtain the chemical potential. However, the practical aspects of its numerical evaluation have not received the type of comprehensive treatments lavished on its siblings. Contiguous polylogs have the ladder operators Lis + 1(z) = ∫z 0Lis(t) t dt, Lis − 1(z) = z∂Lis(z) ∂z, and the sequence can be started with either Li1(z) = − ln(1 − z), Li0(z) = z 1 − z. NUMBER THEORY IN PHYSICS MATILDE MARCOLLI Several fields of mathematics have been closely associated to physics: this has always been the case for the theory of differential equations. This page is motivated by the discussion of Mathematica's ContourPlot shading here. Note that the notation is unfortunately similar to that for the logarithmic integral. j0 (x) Bessel function of the first kind of order 0. j1 (x) Bessel function of the first kind of order 1. y0 (x) Bessel function of the second kind of order 0. y1 (x) Bessel function of the second kind of order 1. i0 (x) Modified Bessel function of order 0. i0e (x) Exponentially The library should be written in Fortran or C/C++. We can avoid the need for complex arithmetic in this case by substituting the expression: This is an improvement because with polylogarithm arguments in [0, 1], the results are purely real values. J. It contains an implementation of the product algebra, the derivative properties, series expansion and numerical evaluation. I suppose > that's because the linear term is 0 (the Wikipedia article claims that > the constant needs to be 0 and the linear term needs to be nonzero). However, the authors will illustrate that the application of real convolution allows for the complete analytical evaluation of the integrals in ( 1 ) and ( 2 ) for a wide range of functions, . Some of these properties are equivalent to theorems of [BL] but I will 4.Mathematica’s(m)( n) to For refer-ence we include Involving two polyilogarithms. Introduction The properties of ideal Fermi and Bose gases are the starting points for the understanding of the low-temperature behaviour of a broad range of physical systems, including electrons in metals [ 1 , 2 ], the helium liquids [ 3 , 4 ] and systems of trapped gases [ 5 ]. The book gives the most elementary arguments possible and uses Mathematica® to verify the formulas. Instead, it extends the capabilities of C++ by symbolic manipulations. Vermaseren, Int. (Mathematica) where the following example is done for calculating THD by the two equations (15.a) and (21) and calculating time consumed in seconds for executing these calculation as following Mathematica commands {1277.04,0.644025} {25.569,0.644025} This proof entirely conceals the route to discovery. Note the proper result when x = 0 is zero, and this is achieved by cancellation between the … j0 (x) Bessel function of the first kind of order 0. j1 (x) Bessel function of the first kind of order 1. y0 (x) Bessel function of the second kind of order 0. y1 (x) Bessel function of the second kind of order 1. i0 (x) Modified Bessel function of order 0. i0e (x) Exponentially : Special Integrals of Gradshetyn and Ryzhik: The Proofs provides self-contained proofs of a variety of entries in the frequently used table of integrals by I.S. Correction to the paper: "Classical motivic polylogarithm according to Beilinson and Deligne". Shortly after the authors originally HPL, a Mathematica implementation of the harmonic polylogarithms General cases. Spread the WikiLove by smiling at someone else, whether it be someone you have had disagreements with in the past or a good friend. Documenta Math. PolyLog[n, p, z] gives the Nielsen generalized polylogarithm function S n, p (z). Please have a look at part 1 and part 2 before reading this post. While all the series converge in the interior of the unit disk, the behavior on the boundary depends on the value of . We define the functions (1) f 1 ( x) = 1 1 − x, f 0 ( x) = 1 x, f −1 ( x) = 1 1 + x. Vermaseren, Int. Hermite polynomials, q-analogue of poly-Bernoulli polynomials, q-analogue of Hermite poly-Bernoulli polynomials, Stirling numbers of the second kind, q-polylogarithm function, Symmetric identities. Then Truesdell [35 With respect to your recent addition to the Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals. Ryzhik. The Analytical Evaluation of the Half-Order Fermi-Dirac Integrals Article It contains an implementation of the product algebra, the derivative properties, series expansion and numerical evaluation. For refer-ence we include PolyLog[n, z] gives the polylogarithm function Lin (z). NUMBER THEORY IN PHYSICS MATILDE MARCOLLI Several fields of mathematics have been closely associated to physics: this has always been the case for the theory of differential equations. 136 Asifa Tassaddiq, Rana Alabdan Therefore, Polylogarithm functions were first known to C. Truesdell when Mr. H. Jacobson informed him that these function play an important role in his researches on the theory of structure of polymers [7]. So, Newton method should also work for complex numbers. A Guide to the Evaluation of IntegralsSpecial Integrals of Gradshetyn and Ryzhik: the Proofs provides self-contained proofs of a variety of entries in the frequently used table of integrals by I.S. The notion of the elliptic polylogarithm functions as a natural generalization of the usual polylogarithms was introduced in [BL, 4.8]. Strictly speaking x*sin(x) is not invertible near x=0 (it's an Wolfram Alpha produces analytic expressions for these, but will only give numerical to a few digits unless one signs up for Wolfram Alpha Pro. 287–306, 2009. In this paper, we present an implementation of the harmonic polylogarithm of Remiddi and Vermaseren [E. Remiddi, J.A.M. Involving several polylogarithms. Elliptic Polylogarithm via the Poincaré Bundle Johannes Sprang Received: October4,2018 Revised: April12,2019 CommunicatedbyOtmarVenjakob Abstract. 3: 27-133, 1998. They can also be used to evaluate some divergent Fourier series [3] and in the computation of singular integrals in quantum field theory [4]. As it turns out, these other formulas for π can all be written as formula (1.2) plus a rational multiple of the identity 0= ∞ i=0 1 16i −8 8i+1 8 8i+2 4 8i+3 8 8i+4 2 8i+5 2 8i+6 − 1 8i+7 The proof of Zeta Functions and Polylogarithms. > Strictly speaking x*sin(x) is not invertible near x Integral #5 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. It can be plotted for complex values ; for example, along the celebrated critical line for Riemann's zeta function. Clausen Functions. While all the series converge in the interior of the unit disk, the behavior on the boundary depends on the value of . Implementation of Polylogarithm function need to be similar to that of Mathematica or Python (can return complex values) and defined for non-integer value (here we have 3/2). Vermaseren, Int. Mathematica Moravica, Vol. Documenta Mathematica 15 (2010) 1–34 Relations of Multiple Polylogarithm Values 5 tion shows that the regularized distribution relations (RDT) do contribute to Modern Phys. Although Rubi’s Int commands are similar in form and function to Mathematica’s Integrate commands, the rules Rubi uses to integrate expressions do not in any way depend on Mathematica’s built-in integrator. done in Maple or Mathematica. Zeta Functions and Polylogarithms. In this paper, we present an implementation of the harmonic polylogarithm of Remiddi and Vermaseren [E. Remiddi, J.A.M. The library should be written in Fortran or C/C++. ( n θ) n s = 1 2 i [ L s ( e i θ) − L s ( e − i θ)] ∑ n = 1 ∞ cos. ⁡. L s ( z) = ∑ n = 1 ∞ z n n s. for the polylogarithm is valid only in the open disk | z | < 1. In mathematics, the Lerch zeta function, sometimes called the Hurwitz–Lerch zeta function, is a special function that generalizes the Hurwitz zeta function and the polylogarithm. In this paper, we present an implementation of the harmonic polylogarithm of Remiddi and Vermaseren [E. Remiddi, J.A.M. Documenta Mathematica 15 (2010) 1–34 Relations of Multiple Polylogarithm Values 5 tion shows that the regularized distribution relations (RDT) do contribute to We define the functions (1) f 1 ( x) = 1 1 − x, f 0 ( x) = 1 x, f −1 ( x) = 1 1 + x. So, Newton method should also work for complex numbers. Shortly after the authors originally In my code i want to solve the Fermi-Dirac-Integral numerically. Classical motivic polylogarithm according to Beilinson and Deligne. The following functions should be able to do the job. 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. Documenta Mathematica 3: 27-133, 1998. Wolfram gives you a selection, of which the polylogarithm page looks most promising. As it turns out, these other formulas for π can all be written as formula (1.2) plus a rational multiple of the identity 0= ∞ i=0 1 16i −8 8i+1 8 8i+2 4 8i+3 8 8i+4 2 8i+5 2 8i+6 − 1 8i+7 The proof of We found the identity (1.2) by a combination of inspired guessing and extensive searching using the PSLQ integer relation algorithm [3],[12]. Evaluation. Zeta Functions & Polylogarithms The Wolfram Language supports zeta and polylogarithm functions of a complex variable in full generality, performing efficient arbitrary-precision evaluation and implementing extensive symbolic transformations. By systematically applying its extensive, coherent collection of symbolic integration rules, Rubi is able to find the optimal antiderivative of large classes of mathematical expressions. Definition. There are also two different commonly encountered normalizations for the function, both denoted, and one of which is … Vermaseren, Int. Opuscula Mathematica Rocznik 2017 Tom Vol. … PolyLog[n, z] gives the polylogarithm function Lin (z). ⁡. Let Li k: $ C$ p → $ C$ p be the p-adic polylogarithms defined by Coleman. Documenta Mathematica Band 15, 2010 Jianqiang Zhao Standard Relations of Multiple Polylogarithm Values at Roots of Unity 1–34 Indranil Biswas and Norbert Hoffmann The Line Bundles on Moduli Stacks of Principal Bundles on a Curve 35–72 Jos´e I. Burgos Gil For the series never converges on the boundary, for it converges everywhere except at , and for it converges everywhere. where a is the integration limit, li_k the Polylogarithm function of order k (see mpmath.polylog) and ζ is the Riemann Zeta function (see scipy.special.zetac). Then Truesdell [35 HPL, a Mathematica implementation of the harmonic polylogarithms. J. You will discover the beauty, patterns, and unexpected … It is named after the Czech mathematician Mathias Lerch [1] . J. The analytic continuation has been treated carefully, allowing the user to keep the control over the definition of the sign of the imaginary parts. Modern Phys. Spread the WikiLove by smiling at someone else, whether it be someone you have had disagreements with in the past or a good friend. Welcome to Rubi, A Rule-based Integrator. 2 (2019), 17–28. the Polylogarithm function to Fermi-Dirac, Bose-Einstein, and classical Maxwell-Boltzmann statistics, which include a convenient description of physical quantities like the density, energy, and chemical potential in terms of this function. Wolfram gives you a selection, of which the polylogarithm page looks most promising. Abstract In this paper, we present an implementation of the harmonic polylogarithm of Remiddi and Vermaseren for Mathematica. Classical motivic polylogarithm according to Beilinson and Deligne. of the polylogarithm functions, once the chemical potential is known. In this article we study the properties of these functions. The hard part is to make sure that all the usual plotting options work correctly, and that the separate parts are registered (aligned) properly in the final superposition. This is part 3 of our series on very nasty logarithmic integrals. Zeta — Riemann and generalized Riemann zeta function Vermaseren, Int. (Berndt 1994). The generalized polylogarithm G(a1,…,an;x) diverges whenever x=a1. From these it is then straightforward to evaluate properties of Bose and Fermi gases. 2.1 How to use it from within C++ The GiNaC open framework for symbolic computation within the C++ programming language does not try to define a language of its own as conventional CAS do. Functional identities. Introduction The properties of ideal Fermi and Bose gases are the starting points for the understanding of the low-temperature behaviour of a broad range of physical systems, including electrons in metals [ 1 , 2 ], the helium liquids [ 3 , 4 ] and systems of trapped gases [ 5 ]. 3: 27-133, 1998. 136 Asifa Tassaddiq, Rana Alabdan Therefore, Polylogarithm functions were first known to C. Truesdell when Mr. H. Jacobson informed him that these function play an important role in his researches on the theory of structure of polymers [7]. Modern Phys. is called Clausen's integral. Here Lin is the polylogarithm of index n. Other “functions” are not even given in closed form, but only as a set of discrete values (for example as measurements in an experiments, or as output from another computer simulation). Functional equations for higher logarithms Functional equations for higher logarithms Gangl, Herbert 2003-09-01 00:00:00 Following earlier work by Abel and others, Kummer gave in 1840 functional equations for the polylogarithm function Li m (z) up to m = 5, but no example for larger m was known until recently. Sqrt[2], Pi, 27 4 It can also do Arbitrary-precision Arithmetic, e.g. Mathematica density and contour Plots with rasterized image representation. Evaluation. It contains an implementation of the product algebra, the derivative properties, series expansion and numerical evaluation. c++ fortran c libraries. Polylogarithm ladders provide the basis for the rapid computations of various mathematical constants by means of the BBP algorithm (Bailey, Borwein & Plouffe 1997)), monodromy group for the polylogarithm (Heisenberg group) The divergence of the … The polylogarithm arises in Feynman diagram integrals (and, in particular, in the computation of quantum electrodynamics corrections to the electrons gyromagnetic ratio), and the special cases and are called the dilogarithm and trilogarithm , respectively. The polylogarithm is implemented in the Wolfram Language as PolyLog [ n , z ].
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