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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

Authors

N. P. Kirk

Jonathan Connor



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.

Citation

Connor, J. N., Hobbs, C. A., Kirk, N. P., Connor, J., & Hobbs, C. (2007). Theory and numerical evaluation of oddoids and evenoids: Oscillatory cuspoid integrals with odd and even polynomial phase functions. Journal of Computational and Applied Mathematics, 207(2), 192-213. https://doi.org/10.1016/j.cam.2006.10.079

Journal Article Type Article
Publication Date Oct 15, 2007
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

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