The reciprocal of a Borel summable function is Borel summable
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.
scientific article; zbMATH DE number 3916892