Theory and numerical evaluation of oddoids and evenoids: Oscillatory cuspoid integrals with odd and even polynomial phase functions

Kirk, N. P.; Connor, Jonathan; Hobbs, Catherine

Abstract

The properties of oscillating cuspoid integrals whose phase functions are odd and even polynomials are investigated. These integrals are called oddoids and evenoids, respectively (and collectively, oddenoids). We have studied in detail oddenoids whose phase functions contain up to three real parameters. For each oddenoid, we have obtained its Maclaurin series representation and investigated its relation to Airy-Hardy integrals and Bessel functions of fractional orders. We have used techniques from singularity theory to characterise the caustic (or bifurcation set) associated with each oddenoid, including the occurrence of complex whiskers. Plots and short tables of numerical values for the oddenoids are presented. The numerical calculations used the software package CUSPINT [N.P. Kirk, J.N.L. Connor, C.A. Hobbs, An adaptive contour code for the numerical evaluation of the oscillatory cuspoid canonical integrals and their derivatives, Comput. Phys. Commun. 132 (2000) 142-165]. © 2006 Elsevier B.V. All rights reserved.

• Journal Article Type: Article
• Publication Date: Oct 15, 2007
• Publicly Available Date: Jun 8, 2019
• Journal: Journal of Computational and Applied Mathematics
• Print ISSN: 0377-0427
• Publisher: Elsevier
• Peer Reviewed: Peer Reviewed
• Volume: 207
• Issue: 2
• Pages: 192-213
• DOI: https://doi.org/10.1016/j.cam.2006.10.079
• Keywords: Airy-Hardy integrals, Bessel functions of fractional order, bifurcation set, caustic, cuspoid integral, oddoid integral, evenoid integral, oddenoid integral, oscillating integrals, singularity theory, Z2-symmetry
• Public URL: https://uwe-repository.worktribe.com/output/1024312
• Publisher URL: http://dx.doi.org/10.1016/j.cam.2006.10.079