The reciprocal of a Borel summable function is Borel summable
type: Item
instance of: scholarly article
title: The reciprocal of a Borel summable function is Borel summable
zbMATH Open document ID: 0574.40003
DOI: 10.1007/BF01206138
author: G. Mennessier
author: Guy Auberson
published in: Communications in Mathematical Physics
publication date: 1985-01-01T00:00:00Z
review text: It is shown that the Borel summability of a function f is preserved by inversion, with the only proviso that the first derivative of f does not vanish at the origin. More precisely, if: i) f(z) is holomorphic in \(K(\lambda,R)=\{z| Re(e^{i\theta}/z)>1/R,\quad | \vartheta | \leq \lambda \}\) (\(\lambda\leq \pi /2)\), ii) \(f(z)=\sum^{M- 1}_{n=1}f_ nz^ n+R_ M(z)\) with \(f_ 1\neq 0\), and \(| R_ M(z)\leq AM!| \sigma z|^ M\) \(\forall z\in K(\lambda,R)\) for some \(\sigma >0\), then \(\exists R_ 1\) and \(\sigma_ 1\) such that \(f^{-1}\) \((f^{-1}\cdot =1)\) is well defined in \(K(\lambda,R_ 1)\) and obeys i) and ii). The asymptotic expansion of \(f^{-1}\) is obtained by the formal inversion of ii). This may be useful to justify formal manipulations of perturbation expansions in quantum mechanics and quantum field theory.
Mathematics Subject Classification ID: 40G10
Mathematics Subject Classification ID: 30B10
zbMATH DE Number: 3916892
zbMATH Keywords: divergent expansions
zbMATH Keywords: Borel summability
MaRDI profile type: Publication
cites work: Some properties of Borel summable functions
cites work: An improvement of Watson’s theorem on Borel summability
description: scientific article; zbMATH DE number 3916892