Joint use of the Weniger transformation and hyperasymptotics for accurate asymptotic evaluations of a class of saddle-point integrals. II. Higher-order transformations

TL;DR

This paper advances the joint application of hyperasymptotics and Weniger transformation for more accurate evaluation of saddle-point integrals, extending the method to second order and demonstrating improved performance in numerical tests.

Contribution

It develops the second-order hyperasymptotic-Weniger transformation (H-WT) and validates its effectiveness on complex saddle-point integrals beyond the asymptotic limit.

Findings

Second-order H-WT improves accuracy of integral evaluations.

Numerical tests confirm effectiveness on complex saddle integrals.

Method outperforms first-order and traditional asymptotics.

Abstract

The use of hyperasymptotics and the Weniger transformation has been proposed, in a joint fashion, for decoding the divergent asymptotic series generated by the steepest descent on a wide class of saddle-point integrals {evaluated across Stokes sets} [R. Borghi, Phys. Rev. E {\bf 78,} 026703 (2008)]. In the present sequel, the full development of the H-WT up to the second order in H is derived. Numerical experiments, carried out on several classes of saddle-point integrals, including the swallowtail diffraction catastrophe, show the effectiveness of the 2nd-level H-WT, in particular when the integrals are evaluated beyond the asymptotic realm.