Resurgence of Gevrey-1 Functions

A classical strategy for making sense of a divergent asymptotic series is to divide itscoefficients by n!, forming its Borel transform, and hope this tamer series can be summedand reconstructed into a genuine function. Écalle’s theory of resurgent functions studiesthis reconstruction for divergent formal series in depth, with a rich singularity structurein the Borel plane. We ask a much more modest question: if one starts from an honestfunction whose derivatives already satisfy the growth bound that would make such a Boreltransform converge (a Gevrey-1 bound), what can be said? We show the answer is completeand elementary: for a smooth Gevrey-1 function, the Borel series converges to the functionitself on a disk of uniform radius at every basepoint, so the resurgence structure collapses— there is no singularity to resum. We also show, with an explicit example, that “smooth”cannot be dropped from the hypothesis: a bare derivative-growth bound, without genuinedifferentiability, is not enough.