Conformal and uniformizing maps in Borel analysis

@article{Costin2021ConformalAU,
  title={Conformal and uniformizing maps in Borel analysis},
  author={O. Costin and Gerald V. Dunne},
  journal={The European Physical Journal Special Topics},
  year={2021}
}
  • O. Costin, G. Dunne
  • Published 2021
  • Physics, Mathematics
  • The European Physical Journal Special Topics
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 Padé approximants, to increase… Expand
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References

SHOWING 1-10 OF 58 REFERENCES
Perturbative Expansions in QCD Improved by Conformal Mappings of the Borel Plane
Perturbation expansions appear to be divergent series in many physically interesting situations, including in quantum field theories like quantum electrodynamics (QED) and quantum chromodynamicsExpand
Resummation of Diagrammatic Series with Zero Convergence Radius for Strongly Correlated Fermions.
TLDR
It is demonstrated that a summing up series of Feynman diagrams can yield unbiased accurate results for strongly correlated fermions even when the convergence radius vanishes, and the theoretically conjectured fourth virial coefficient is reconciled. Expand
Borel summability: Application to the anharmonic oscillator
We prove that the energy levels of an arbitrary anharmonic oscillator ( x 2 m and in any finite number of dimensions) are determined uniquely by their Rayleigh-Schrodinger series via a (generalized)Expand
Representation of conformal maps by rational functions
TLDR
This work proves a sequence of four theorems establishing that in any conformal map of the unit circle onto a region with a long and slender part, there must be a singularity or loss of univalence exponentially close to the boundary, and polynomial approximations cannot be accurate unless of exponentially high degree. Expand
Logarithmic Potential Theory with Applications to Approximation Theory
We provide an introduction to logarithmic potential theory in the complex plane that particularly emphasizes its usefulness in the theory of polynomial and rational approximation. The reader isExpand
Resurgent extrapolation: rebuilding a function from asymptotic data. Painlevé I
  • O. Costin, G. Dunne
  • Mathematics, Physics
  • Journal of Physics A: Mathematical and Theoretical
  • 2019
Extrapolation is a generic problem in physics and mathematics: how to use asymptotic data in one parametric regime to learn about the behavior of a function in another parametric regime. For example:Expand
Uniformization and Constructive Analytic Continuation of Taylor Series
We analyze the general mathematical problem of global reconstruction of a function with least possible errors, based on partial information such as n terms of a Taylor series at a point, possiblyExpand
From Useful Algorithms for Slowly Convergent Series to Physical Predictions Based on Divergent Perturbative Expansions
This review is focused on the borderline region of theoretical physics and mathematics. First, we describe numerical methods for the acceleration of the convergence of series. These provide a usefulExpand
Reconstructing Nonequilibrium Regimes of Quantum Many-Body Systems from the Analytical Structure of Perturbative Expansions
We propose a systematic approach to the non-equilibrium dynamics of strongly interacting many-body quantum systems, building upon the standard perturbative expansion in the Coulomb interaction. HighExpand
Asymptotics and Borel Summability
Introduction Expansions and approximations Formal and actual solutions Review of Some Basic Tools The Phragmen-Lindelof theorem Laplace and inverse Laplace transforms Classical AsymptoticsExpand
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