doi:10.1111/sapm.12539engFerreira, CheloLópez, José L.Pérez Sinusía, EsterAsymptotic approximation of a highly oscillatory integral with application to the canonical catastrophe integralsART-2023-131696We consider the highly oscillatory integral () ∶= ∫ ∞ −∞ (+2+) () for large positive values of , − < ≤ , and positive integers with 1 ≤ ≤ , and () an entire function. The standard saddle point method is complicated and we use here a simplified version of this method introduced by López et al. We derive an asymptotic approximation of this integral when → +∞ for general values of and in terms of elementary functions, and determine the Stokes lines. For ≠ 1, the asymptotic behavior of this integral may be classified in four different regions according to the even/odd character of the couple of parameters and ; the special case =1 requires a separate analysis. As an important application, we consider the family of canonical catastrophe integrals Ψ(1, 2,…,) for large values of one of its variables, say , and bounded values of the remaining ones. This family of integrals may be written in the form () for appropriate values of the parameters , and the function (). Then, we derive an asymptotic approximation of the family of canonical catastrophe integrals for large ||. The approximations are accompanied by several numerical experiments. The asymptotic formulas presented here fill up a gap in the NIST Handbook of Mathematical Functions by Olver et al.2023http://zaguan.unizar.es/record/12118410.1111/sapm.12539http://zaguan.unizar.es/record/121184oai:zaguan.unizar.es:121184STUDIES IN APPLIED MATHEMATICS 150, 1 (2023), 254-275by-nc-ndhttp://creativecommons.org/licenses/by-nc-nd/3.0/es/info:eu-repo/semantics/openAccess