Global reconstruction of analytic functions from local expansions

TL;DR

This paper introduces a novel summation method that transforms local expansions into integral representations, enabling the analysis of global properties of analytic functions such as singularities, asymptotics, and zeros.

Contribution

It presents a new summation technique for converting local series into integrals, revealing global analytic features and Borel summability of divergent series.

Findings

Enables determination of singularities and asymptotics from local expansions

Establishes a duality between global structure and coefficient properties

Proves Borel summability for certain divergent series

Abstract

A new summation method is introduced to convert a relatively wide family of infinite sums and local expansions into integrals. The integral representations yield global information such as analytic continuability, position of singularities, asymptotics for large values of the variable and asymptotic location of zeros. There is a duality between the global analytic structure of the reconstructed function and the properties of the coefficients as a function of their index. Borel summability of a class of divergent series follow as a byproduct.