Polylogarithm
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): WebThe dilogarithm function (sometimes called Euler’s dilogarithm function) is a special case of the polylogarithm that can be traced back to the works of Leonhard Euler. The function re …
Polylogarithm
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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 ... 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) …
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 … 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 …
Webnthe 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<::: 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 ...
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
WebFeb 5, 2016 · The functions dilogarithm, trilogarithm, and more generally polylogarithm are meant to be generalizations of the logarithm. I first came across the dilogarithm in college when I was evaluating some integral with Mathematica, and they've paid a visit occasionally ever since. Unfortunately polylogarithms are defined in several slightly different and … cliff house folsom californiaWebMay 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 … cliff house floor planWebIn 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 … cliff house fort braggWebSome 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 … boardingates montpellierWebDifferentiation (12 formulas) PolyLog. Zeta Functions and Polylogarithms PolyLog[nu,z] boarding association school jobsWebDec 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 ... cliff house fort bragg californiaIn 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 cliff house galiano