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Polylogarithm

WebThe 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. These are sufficient to evaluate it numerically, with reasonable efficiency, in all cases. 1. Definition The polylogarithm may be defined as the function Li p ... 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.

Zeta functions, L-series and polylogarithms - mpmath

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 … WebMar 3, 1997 · We prove a special representation of the polylogarithm function in terms of series with such numbers. Using … Expand. 1. PDF. Save. Alert. Identities Involving Generalized Harmonic Numbers and Other Special Combinatorial Sequences. Huyile Liang; Mathematics. 2012; how bad stage kidney failure https://thepowerof3enterprises.com

What does polylogarithm mean? - Definitions.net

WebThe polylogarithm function is an important function for integration, and finding seemingly complicated sum. Polylogarithm is connected to the infinite geometric progression sum ... 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 … 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 how many months pregnant until the gender

Zeta Functions & Polylogarithms—Wolfram Language …

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Polylogarithm

How to use polylogarithm function in c++?

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 … Webgives the Nielsen generalized polylogarithm function . Details. Mathematical function, suitable for both symbolic and numerical manipulation.. . . PolyLog [n, z] has a branch cut …

Polylogarithm

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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 WebThe Polylogarithm package provides C, C++ and Fortran implementations of various polylogarithms, including the real and complex dilogarithm, trilogarithm, and (Standard and Glaisher) Clausen functions. The implementations have been fully tested against the literature and many other implementations and are highly optimized for fast numerical ...

WebDifferentiation (12 formulas) PolyLog. Zeta Functions and Polylogarithms PolyLog[nu,z] WebOct 8, 1999 · Historically, the polylogarithm has attracted specialists and non-specialists alike with its lovely evaluations. Much the same can be said for Euler sums (or multiple harmonic sums), which, within the past decade, have arisen in combinatorics, knot theory and high-energy physics. More recently, we have been forced to consider …

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-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 ... WebWe associate to a multiple polylogarithm a holomorphic 1-form on the universal abelian cover of its domain. We relate the 1-forms to the symbol and variation matrix and show that the 1-forms naturally define a lift of …

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 …

WebThe polylogarithm , also known as the Jonquière's function, is the function. (1) defined in the complex plane over the open unit disk. Its definition on the whole complex plane then … how bad translate to spanishIn 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): how bad was dahmer beatenWeba 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) … how bad was hurricane sandyWebDec 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 ... how bad was inflation in 2022WebThe 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 ... how bad was life in the trenchesWebThere'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. … how bad was jeremy renner injuredWebApr 12, 2024 · In this paper, we introduce and study a new subclass S n β,λ,δ,b (α), involving polylogarithm functions which are associated with differential operator. we also obtain coefficient estimates ... how bad was sin before the flood