Polylogarithm

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August undefined, 2024

Weba reﬁnement involving a “lifting” from R to C/(2πi)mQ of the mth polylogarithm function. The natural setting for all of this is algebraic K-theory and the conjectures about polylogarithms lead to a purely algebraic (conjectural) … WebDec 11, 2024 · Abstract. Gamma and Polylogarithm identities completely deduced, producing others related identities and applied in solving some definite integrals.The analysis involves Riemann Zeta, Dirichlet ...

Complete Fermi–Dirac integral - Wikipedia

WebThis function is defined in analogy with the Riemann zeta function as providing the sum of the alternating series. η ( s) = ∑ k = 0 ∞ ( − 1) k k s = 1 − 1 2 s + 1 3 s − 1 4 s + …. The eta … Webs(z) resembles the Dirichlet series for the polylogarithm function Li s(z). Nice reviews of the theory of such functions are given by Lewin [2,19] and Berndt [10]. Cvijović published integral representations of the Legendre chi functio [20], which are thus likely to provide, via χ 2(z), expressions for Li 2(z)−Li 2(−z). 4 Conclusion shark vector art

Polylogarithm - MATLAB polylog - MathWorks France

WebThe polylogarithm function (or Jonquière's 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 Riemann's zeta function [1]. The polylogarithm function appears in the Fermi–Dirac and Bose–Einstein … WebJun 26, 2015 · 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) Share. Improve this … population of carinthia

PolyLog Function - Wolfram Demonstrations Project

Spence

WebThere's a GPL'd C library, ANANT - Algorithms in Analytic Number Theory by Linas Vepstas, which includes multiprecision implementation of the polylogarithm, building on GMP. … 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 … See more In the case where the order $${\displaystyle s}$$ is an integer, it will be represented by $${\displaystyle s=n}$$ (or $${\displaystyle s=-n}$$ when negative). It is often convenient to define Depending on the … See more • For z = 1, the polylogarithm reduces to the Riemann zeta function Li s ⁡ ( 1 ) = ζ ( s ) ( Re ⁡ ( s ) > 1 ) . {\displaystyle \operatorname {Li} … See more 1. As noted under integral representations above, the Bose–Einstein integral representation of the polylogarithm may be extended to negative orders s by means of See more The dilogarithm is the polylogarithm of order s = 2. An alternate integral expression of the dilogarithm for arbitrary complex argument z is (Abramowitz & Stegun 1972, § 27.7): See more For particular cases, the polylogarithm may be expressed in terms of other functions (see below). Particular values for the polylogarithm may thus also be found as particular … See more Any of the following integral representations furnishes the analytic continuation of the polylogarithm beyond the circle of convergence z = 1 of the defining power series. 1. The polylogarithm can be expressed in terms of the integral … See more For z ≫ 1, the polylogarithm can be expanded into asymptotic series in terms of ln(−z): where B2k are the See more shark vectorWebApr 30, 2024 · In mathematics, the polylogarithm (also known as Jonquière ʹ s function, for Alfred Jonquière) is a special function Li s (z) of order s and argument z.Only for special values of s does the ... shark vector black

"Webpolylog(2,x) is equivalent to dilog(1 - x). The logarithmic integral function (the integral logarithm) uses the same notation, li(x), but without an index.The toolbox provides the logint function to compute the logarithmic integral function.. Floating-point evaluation of the polylogarithm function can be slow for complex arguments or high-precision numbers. " - Polylogarithm

Polylogarithm

WebDifferentiation (12 formulas) PolyLog. Zeta Functions and Polylogarithms PolyLog[nu,z] WebIt appears that the only known representations for the Riemann zeta function ((z) in terms of continued fractions are those for z = 2 and 3. Here we give a rapidly converging continued-fraction expansion of ((n) for any integer n > 2. This is a special case of a more general expansion which we have derived for the polylogarithms of order n, n > 1, by using the …

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Web, when s 1, … , s k are positive integers and z a complex number in the unit disk. For k = 1, this is the classical polylogarithm Li s (z).These multiple polylogarithms can be defined also in terms of iterated Chen integrals and satisfy shuffle relations.Multiple polylogarithms in several variables are defined for s i ≥ 1 and z i < 1(1 ≤ i ≤ k) by In mathematics, Spence's function, or dilogarithm, denoted as Li2(z), is a particular case of the polylogarithm. Two related special functions are referred to as Spence's function, the dilogarithm itself: and its reflection. For z < 1, an infinite series also applies (the integral definition constitutes its analytical extension to the complex plane):

WebMar 24, 2024 · The trilogarithm Li_3(z), sometimes also denoted L_3, is special case of the polylogarithm Li_n(z) for n=3. Note that the notation Li_3(x) for the trilogarithm is unfortunately similar to that for the logarithmic integral Li(x). The trilogarithm is implemented in the Wolfram Language as PolyLog[3, z]. Plots of Li_3(z) in the complex … WebThe polylogarithm function is an important function for integration, and finding seemingly complicated sum. Polylogarithm is connected to the infinite geometric progression sum ...

WebZeta Functions and Polylogarithms PolyLog [ nu, z] Identities. Recurrence identities. General cases. Involving two polyilogarithms. Involving several polylogarithms. WebMay 18, 2009 · The nth order polylogarithm Li n (z) is defined for z ≦ 1 by ([4, p. 169], cf. [2, §1. 11 (14) and § 1. 11. 1]). The definition can be extended to all values of z in the z …

WebSome other important sources of information on polylogarithm functions are the works of References and . In References [ 5 ] and [ 6 ], the authors explore the algorithmic and analytic properties of generalized harmonic Euler sums systematically, in order to compute the massive Feynman integrals which arise in quantum field theories and in certain …

WebInformally, a cluster polylogarithm is a homotopy-invariant iterated integral ż γ ÿ i “ dlogpai 1q ... dlogpai nq ‰ on XsmpCq where for each ithere exists a cluster containing cluster variables ai 1,...,a i n. We call the latter condition cluster adjacency, it was inspired by [DFG18]. Consider the following simplest example. shark vector black and whiteWebPolylogarithms of Numeric and Symbolic Arguments. polylog returns floating-point numbers or exact symbolic results depending on the arguments you use. Compute the … shark version of crosswaveWebThe Wolfram Language supports zeta and polylogarithm functions of a complex variable in full generality, performing efficient arbitrary-precision evaluation and implementing extensive symbolic transformations. Zeta — Riemann and generalized Riemann zeta function. RiemannSiegelZ RiemannSiegelTheta StieltjesGamma RiemannXi. shark vector imageWebnthe weight (or transcendentality) of the polylogarithm. Multiple polylogarithms de ned as power series Li n 1;:::;n k(x1;:::;x k) = X 1 p 1<::: shark velcro replacementWebMay 18, 2009 · The nth order polylogarithm Li n (z) is defined for z ≦ 1 by ([4, p. 169], cf. [2, §1. 11 (14) and § 1. 11. 1]). The definition can be extended to all values of z in the z-plane cut along the real axis from 1 to ∝ by the formula [2, §1. 11(3)]. Then Li n (z) is regular in the cut plane, and there is a differential recurrence relation ... shark vehicleWebIn mathematics, the complete Fermi–Dirac integral, named after Enrico Fermi and Paul Dirac, for an index j is defined by = (+) +, (>)This equals + ⁡ (), where ⁡ is the polylogarithm.. Its … shark velcro watch bandWebThe Polylogarithm is also known as Jonquiere's function. It is defined as ∑ k = 1 ∞ z k / k n = z + z 2 / 2 n +... The polylogarithm function arises, e.g., in Feynman diagram integrals. It also arises in the closed form of the integral of the Fermi-Dirac and the Bose-Einstein distributions. The special cases n=2 and n=3 are called the ... shark valley airboat tours