Polylogarithm function li
Web14. We know some exact values of the trilogarithm function. Known real analytic values for : where is the Apéry's constant. where is the golden ratio. Using identities for the list above we could also get: or we could write into this alternate form. or there is an alternate form here. We know even less about complex argumented values: WebPolylogarithms. This implements the Polylogarithm and some related functions that were needed (Harmonic numbers, Stieltjes constants, and Bernoulli numbers and polynomials). …
Polylogarithm function li
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WebWe can avoid the need for complex arithmetic in this case by substituting the expression: ∫ 0 x t 3 d t e t − 1 = − 6 Li 4 ( e − x) − 6 x Li 3 ( e − x) − 3 x 2 Li 2 ( e − x) − x 3 Li 1 ( e − x) + π 4 … WebJun 19, 2015 · The Lerch zeta function III. Polylogarithms and special values. Jeffrey C. Lagarias, W.-C. Winnie Li. This paper studies algebraic and analytic structures associated with the Lerch zeta function, extending the complex variables viewpoint taken in part II. The Lerch transcendent is obtained from the Lerch zeta function by the change of variable ...
Web2.2 The Bloch-Wigner-Ramakrishnan-Zagier-Wojtkowiak polylogarithm There are also one-valued variants Pm of each m-logarithm function; their name “Bloch-Wigner … WebFeb 9, 2024 · The dilogarithm function. Li2(x) =: ∞ ∑ n=1 xn n2, Li 2 ( x) =: ∑ n = 1 ∞ x n n 2, (1) studied already by Leibniz, is a special case of the polylogarithm function. Lis(x) =: ∞ …
WebJan 22, 2024 · Description. Compute the polylogarithm function Li_s (z) , initially defined as the power series, Li_ {s+1} (z) = Int [0..z] (Li_s (t) / t) dt. Currently, mainly the case of … WebThe 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 …
WebLi River, rivers in Cheenae an Thailand; Li (surname), a Cheenese surname whiles transliteratit Lee. Li (李) Lí (黎) Lì (利) 51 (nummer), written as "LI" in Roman numerals; Li …
WebCell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["PolyLog", "[", RowBox[List["n_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox ... flushmounted outdoor wall porch lightsWebarxiv:math/0306226v2 [math.pr] 3 apr 2004 limiting distributions for additive functionals on catalan trees james allen fill and nevin kapur flush mounted over sink lightingWebMar 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 … green frog with orange feetWebMar 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; green frog with red feetWebThis 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 … flush mounted overhead bug lightWebPolylogarithm and Geometric Progression. Polylogarithm is connected to the infinite geometric progression sum \operatorname {Li}_0 (x)=\sum_ {n=1}^\infty x^n=\dfrac {x} {1 … green frog with yellow spotsIn 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 … 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): A source of confusion is that some computer algebra systems See more For particular cases, the polylogarithm may be expressed in terms of other functions (see below). Particular values for the … See more Any of the following integral representations furnishes the analytic continuation of the polylogarithm beyond the circle of … See more For z ≫ 1, the polylogarithm can be expanded into asymptotic series in terms of ln(−z): where B2k are the Bernoulli numbers. Both versions hold for all … See more greenfrom coffee grinder