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

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 $f(\infty )$.

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