Polynomial expansions of analytic functions

Boas, R. P.; Buck, R. C.

doi:10.1007/978-3-662-25170-6

Cited by 126 publications

(54 citation statements)

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“…As a general rule, both have better performance than the direct summation formula (5), although this depends on whether the polylogarithm or the Hurwitz zeta is the desired quantity.…”

Section: Performancementioning

confidence: 95%

“…14] as a Mittag-Leffler type decomposition [5]. A general discussion of the monodromy is given in a later section.…”

Section: Duplication Formulamentioning

confidence: 99%

“…The polylogarithm, sometimes called the fractional polylogarithm to emphasize that s need not be an integer, is given by the sum Li s (z) = ∞ n=1 z n n s (5) This sum is absolutely convergent on the complex half-plane σ > 0 when |z| < 1, and in σ > 1 when |z| ≤ 1. The polylogarithm has a non-trivial analytic structure.…”

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confidence: 99%

“…The other sheets of the polylogarithm also have a branch point at z = 0, which is not naively obvious from the form (5). The set of possible paths that navigate from sheet to sheet form a group, known as the monodromy group of the function; a detailed exposition of the monodromy will be given in this paper.…”

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confidence: 99%

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An efficient algorithm for accelerating the convergence of oscillatory series, useful for computing the polylogarithm and Hurwitz zeta functions

Vepstas

2008

Numer Algor

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This paper sketches a technique for improving the rate of convergence of a general oscillatory sequence, and then applies this series acceleration algorithm to the polylogarithm and the Hurwitz zeta function. As such, it may be taken as an extension of the techniques given by Borwein's "An efficient algorithm for computing the Riemann zeta function" by Borwein for computing the Riemann zeta function, to more general series. The algorithm provides a rapid means of evaluating Li s (z) for general values of complex s and a kidney-shaped region of complex z values given by z 2 /(z − 1) < 4. By using the duplication formula and the inversion formula, the range of convergence for the polylogarithm may be extended to the entire complex z-plane, and so the algorithms described here allow for the evaluation of the polylogarithm for all complex s and z values. Alternatively, the Hurwitz zeta can be very rapidly evaluated by means of an Euler-Maclaurin series. The polylogarithm and the Hurwitz zeta are related, in that two evaluations of the one can be used to obtain a value of the other; thus, either algorithm can be used to evaluate either function. The Euler-Maclaurin series is a clear performance winner for the Hurwitz zeta, while the Borwein algorithm is superior for evaluating the polylogarithm in the kidney-shaped region. Both algorithms are superior to the simple Taylor's series or direct summation. The primary, concrete result of this paper is an algorithm allows the exploration of the Hurwitz zeta in the critical strip, where fast algorithms are otherwise unavailable. A discussion of the monodromy group of the polylogarithm is included. L. Vepštas (B)

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“…As a general rule, both have better performance than the direct summation formula (5), although this depends on whether the polylogarithm or the Hurwitz zeta is the desired quantity.…”

Section: Performancementioning

confidence: 95%

“…14] as a Mittag-Leffler type decomposition [5]. A general discussion of the monodromy is given in a later section.…”

Section: Duplication Formulamentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

See 2 more Smart Citations

An efficient algorithm for accelerating the convergence of oscillatory series, useful for computing the polylogarithm and Hurwitz zeta functions

Vepstas

2008

Numer Algor

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“…There are many authors who have studied polynomials and their properties (see [1][2][3][4][5][6][7][8][9][10]). The polynomials are applied in many areas of mathematics, for instance, continued fractions, operator theory, analytic functions, interpolation, approximation theory, numerical analysis, electrostatics, statistical quantum mechanics, special functions, number theory, combinatorics, stochastic processes, sorting, and data compression.…”

Section: Introductionmentioning

confidence: 99%