Conformal and Uniformizing Maps in Borel Analysis

TL;DR

This paper explores how conformal and uniformizing maps, combined with Padé approximants, enhance the extraction of physical information from divergent perturbative series in Borel analysis.

Contribution

It introduces a method using conformal and uniformizing maps with Padé approximants to improve the analysis of finite perturbative data in physical applications.

Findings

Enhanced precision in extrapolating perturbative series

Connection between Padé approximants and electrostatic potential theory

Method applicable to divergent series in physics

Abstract

Perturbative expansions in physical applications are generically divergent, and their physical content can be studied using Borel analysis. Given just a finite number of terms of such an expansion, this input data can be analyzed in different ways, leading to vastly different precision for the extrapolation of the expansion parameter away from its original asymptotic regime. Here we describe how conformal maps and uniformizing maps can be used, in conjunction with Pad'e approximants, to increase the precision of the information that can be extracted from a finite amount of perturbative input data. We also summarize results from the physical interpretation of Pad'e approximations in terms of electrostatic potential theory.