Polylogarithm
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 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 …
Polylogarithm
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WebDifferentiation (12 formulas) PolyLog. Zeta Functions and Polylogarithms PolyLog[nu,z] WebApr 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 ...
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 … WebPolylogarithms of Numeric and Symbolic Arguments. polylog returns floating-point numbers or exact symbolic results depending on the arguments you use. Compute the …
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 … Webgives the Nielsen generalized polylogarithm function . Details. Mathematical function, suitable for both symbolic and numerical manipulation.. . . PolyLog [n, z] has a branch cut …
WebApr 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 ...
WebDefinition of polylogarithm in the Definitions.net dictionary. Meaning of polylogarithm. What does polylogarithm mean? Information and translations of polylogarithm in the most comprehensive dictionary definitions resource on the web. grandview nursing facilityWebThis 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 … chinese takeaway in latchfordWebMar 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 … grandview nursery tahlequah okWebPolylogarithms of Numeric and Symbolic Arguments. polylog returns floating-point numbers or exact symbolic results depending on the arguments you use. Compute the polylogarithms of numeric input arguments. The polylog function returns floating-point numbers. Li = [polylog (3,-1/2), polylog (4,1/3), polylog (5,3/4)] chinese takeaway in larbertWebThe 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. grandview new york real estateWebMay 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 … grandview nursing and rehabilitation centerWebThe 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 ... chinese takeaway in larne