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
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