Resurgence Analysis of Meromorphic Transforms

TL;DR

This paper develops a comprehensive asymptotic analysis framework for meromorphic transforms, establishing their resummability and applying the theory to classical special functions.

Contribution

It introduces a novel asymptotic expansion method for meromorphic transforms and provides explicit formulas for resurgent functions and Stokes coefficients.

Findings

Asymptotic expansions are established under decay assumptions.

The transforms are shown to be Borel resummable.

Classical functions like the gamma and zeta functions are derived as examples.

Abstract

We consider meromorphic transforms given by meromorphic kernels and study their asymptotic expansions under a certain rescaling. Under decay assumptions we establish the full asymptotic expansion in the rescaling parameter of these transforms and provide global estimates for error terms. We show that the resulting asymptotic series is Borel resummable and we provide formulae for the resulting resurgent function, which allows us to give formulae for the Stokes coefficients. A number of classical functions are obtained by applying such meromorphic transforms to elementary functions, of which, the Faddeev quantum dilogarithm, the Euler gamma function, the Riemann zeta function, the Gauss hypergeometric function and the Airy function are excellent examples of our general theory.