From Taylor series of analytic functions to their global analysis

TL;DR

This paper investigates how the Taylor coefficients of analytic functions determine their ability to be extended globally, introducing a new summation method to analyze their singularities, zeros, and asymptotic behavior.

Contribution

It presents a novel summation technique converting sums into integrals, providing new insights into the global properties of analytic functions based on their Taylor coefficients.

Findings

Conditions on Taylor coefficients for global continuation

A new summation method for sums and expansions

Results on singularities, zeros, and asymptotics

Abstract

We analyze the conditions on the Taylor coefficients of an analytic function to admit global analytic continuation, complementing a recent paper of Breuer and Simon on general conditions for natural boundaries to form. 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.