A new method to sum divergent power series: educated match

Abstract

A new method to sum divergent power series: educated match AbstractWe present a method to sum Borel-and Gevrey-summable asymptotic series by matching the series to be summed with a linear combination of asymptotic series of known functions that themselves are scaled versions of a single, appropriate, but otherwise unrestricted, function Φ. Both the scaling and linear coefficients are calculated from Padé approximants of a series transformed from the original series by Φ. We discuss in particular the case that Φ is (essentially) a confluent hypergeometric function, which includes as special cases the standard Borel-Padé and Borel-Leroy-Padé methods. A particular advantage is the mechanism to build knowledge about the summed function into the approximants, extending their accuracy and range even when only a few coefficients are available. Several examples from field theory and Rayleigh-Schrödinger perturbation theory illustrate the method.

Fast summation of divergent series and resurgent transseries from Meijer- G approximants

“…When it comes to sum divergent series, Borel-Padé has become the dominant approach [1,15,41]. There are various reasons for the popularity of this approach but it can be argued that the most important of these is its algorithmic simplicity.…”

“…This partition function is commonly used to benchmark resummation techniques-see Refs. [15,64] for two recent examples. The first few terms of the asymptotic expansion about g ¼ 0 read…”

“…The interested reader can now look to the closely related approach put forward in Ref. [15]-both approaches take very different routes beyond first order.…”

“…On the other hand, extracting physically relevant information from the asymptotic series at larger coupling constants calls, almost invariably, for resummation techniques as exemplified by the LoSurdo-Stark effect [5,6] and field assisted excitonic ionization in layered materials [7], anharmonic oscillators in quantum mechanics [8,9], ϕ 4 theory in QFT [10], quantum chromodynamics [11], string perturbation theory [12,13], and diagrammatic Monte Carlo techniques in condensed matter physics [14]. Given the ubiquity of divergent series in physics, research on summation techniques remains an active research area [15,16].…”

Resurgence of one-point functions in a matrix model for 2D type IIA superstrings

“…In case that the original integration contour is a finite interval not passing through a saddle point, we have to choose an appropriate integration path in order for resurgence to work.1 Resurgence structure has been studied in various models and theories based on several motivations: see e.g. in quantum mechanics [9][10][11][12][13][14][15][16][17][18][19][20][21], string theories [22][23][24] as well as quantum field theories .…”