Irrational series II Summation by packages

TL;DR

This paper investigates the convergence of exponential series with positive exponents in complex neighborhoods and introduces a summation method called 'packages' that handles convergence issues through grouping terms with similar exponents.

Contribution

It introduces and analyzes the notion of summation by packages for exponential series, providing conditions under which these series can be summed despite convergence challenges.

Findings

Summation by packages can handle series bounded in logarithmic neighborhoods.

Massive cancellations occur when summing within packages.

The method extends the understanding of convergence for exponential sums.

Abstract

Discrete sums of exponentials $g(w) = \sum a_{\beta} \mathrm{e}^{\beta w}$ with positive exponents may converge not normally in neighborhoods $H$ of $-\infty$ which do not contain half-planes. We study different notions of convergence for these series and in particular the intuitive notion of summation by packages. Indeed, joining in packages the terms in the sum $g(w)$ whose exponents are close together, and summing first inside each package may result in massive cancellations. We show that discrete sums $g(w)$ which are bounded in what we call logarithmic neighborhoods can always be summated by packages.