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

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

1993-12-06

arxiv.org/abs/hep-lat/9312031v1

J.Math.Phys. 35 (1994) 1914-1921

Physics

High Energy Physics

High Energy Physics - Lattice

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

Scientific paper

10.1063/1.530577

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 Pad\'e 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 $f(\infty )$.

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