Determination of ${\bf f(\infty)}$ from the Asymptotic Series for ${\bf f(x)}$ About ${\bf x=0}$

C. M. Bender; S. Boettcher

doi:10.1063/1.530577

paper

Determination of $f (\infty)$ from the Asymptotic Series for $f (x)$ About $x = 0$

arXiv:hep-lat/9312031 · doi:10.1063/1.530577

Abstract

A difficult and long-standing problem in mathematical physics concerns the determination of the value of $f (\infty)$ from the asymptotic series for $f (x)$ about $x = 0$ . In the past the approach has been to convert the asymptotic series to a sequence of Padé approximants ${P_{n}^{n} (x)}$ and then to evaluate these approximants at $x = \infty$ . Unfortunately, for most physical applications the sequence ${P_{n}^{n} (\infty)}$ is slowly converging and does not usually give very accurate results. In this paper we report the results of extensive numerical studies for a large class of functions $f (x)$ associated with strong-coupling lattice approximations. We conjecture that for large $n$ , $P_{n}^{n} (\infty) \sim f (\infty) + B / ln n$ . A numerical fit to this asymptotic behavior gives an accurate extrapolation to the value of $f (\infty)$ .

plain TEX, 7 pages, 7 figures included, WASH-U-HEP-93

Determination of f(∞) from the Asymptotic Series for f(x) About x=0

Abstract

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Determination of $f (\infty)$ from the Asymptotic Series for $f (x)$ About $x = 0$