Preprint Article Version 1 This version is not peer-reviewed

# On the Approximate Evaluation of Some Oscillatory Integrals

Version 1 : Received: 4 March 2019 / Approved: 5 March 2019 / Online: 5 March 2019 (12:34:07 CET)

A peer-reviewed article of this Preprint also exists.

Beuc, R.; Movre, M.; Horvatić, B. On the Approximate Evaluation of Some Oscillatory Integrals. Atoms 2019, 7, 47. Beuc, R.; Movre, M.; Horvatić, B. On the Approximate Evaluation of Some Oscillatory Integrals. Atoms 2019, 7, 47.

Journal reference: Atoms 2019, 7, 47
DOI: 10.3390/atoms7020047

## Abstract

To determine the photon emission or absorption probability for a diatomic system in the context of the semiclassical approximation it is necessary to calculate the characteristic canonical oscillatory integral which has one or more saddle points. Integrals like that appear in a whole range of physical problems, e.g. the atom-atom and atom-surface scattering and various optical phenomena. A uniform approximation of the integral, based on the stationary phase method is proposed, where the integral with several saddle points is replaced by a sum of integrals each having only one or at most two real saddle points and is easily soluble. In this way we formally reduce the codimension in canonical integrals of "elementary catastrophes" with codimensions greater than 1. The validity of the proposed method was tested on examples of integrals with three saddle points ("cusp" catastrophe) and four saddle points ("swallow-tail" catastrophe).

## Subject Areas

oscillatory integrals; stationary point approximation; semi-classical theory, uniform Airy approximation