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

Authors:
C. A. Hobbs;J. N. L. Connor;N. P. Kirk
Affiliations:
Department of Mathematical Sciences, School of Technology, Oxford Brookes University, Wheatley Campus, Oxford OX33 1HX, UK;School of Chemistry, The University of Manchester, Manchester M13 9PL, UK;Department of Mathematical Sciences, University of Liverpool, Liverpool L69 3BX, UK
Venue:
Journal of Computational and Applied Mathematics
Year:
2007

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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].