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

TL;DR

This paper investigates a method to accurately determine the limit of a function at infinity from its asymptotic series near zero, using Padé approximants and a novel asymptotic extrapolation technique.

Contribution

It introduces a new asymptotic behavior conjecture for Padé approximants at infinity, improving the accuracy of extrapolating $f( fty)$ from asymptotic series.

Findings

Padé approximants at infinity converge slowly for physical functions.

A conjectured asymptotic form $P^n_n( ext{} o ext{} ) o f( ext{} o ext{} )+B/ ext{ln} n$.

Numerical fits based on this form yield accurate estimates of $f( ext{} o ext{} )$.

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 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 $f(\infty )$.