On the ambiguity of functions represented by divergent power series

TL;DR

This paper investigates the ambiguity in defining functions from divergent power series, especially those represented by Laplace-Borel integrals with complex contours, and explores methods to reduce this ambiguity in quantum chromodynamics applications.

Contribution

It extends the class of integration contours for asymptotic series, including self-intersecting curves, and provides estimates on remainders, advancing understanding of function ambiguity.

Findings

Extended contour class allows more functions to share the same asymptotic expansion.

Provided estimates on remainders for various contour types.

Discussed ambiguity reduction methods in the context of QCD.

Abstract

Assuming the asymptotic character of divergent perturbation series, we address the problem of ambiguity of a function determined by an asymptotic power expansion. We consider functions represented by an integral of the Laplace-Borel type, with a curvilinear integration contour. This paper is a continuation of results recently obtained by us in a previous work. Our new result contained in Lemma 3 of the present paper represents a further extension of the class of contours of integration (and, by this, of the class of functions possessing a given asymptotic expansion), allowing the curves to intersect themselves or return back, closer to the origin. Estimates on the remainders are obtained for different types of contours. Methods of reducing the ambiguity by additional inputs are discussed using the particular case of the Adler function in QCD.