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].
Original language | English |
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Pages (from-to) | 192-213 |
Number of pages | 22 |
Journal | Journal of Computational and Applied Mathematics |
Volume | 207 |
Issue number | 2 |
Early online date | 8 Dec 2006 |
DOIs | |
Publication status | Published - 15 Oct 2007 |
Externally published | Yes |
Keywords
- Airy–Hardy integrals
- Bessel functions of fractional order
- Bifurcation set
- Caustic
- Complex whisker
- CUSPINT
- Cuspoid integrals
- Evenoid integrals
- Oddoid integrals
- Oddenoid integrals
- Oscillating integrals
- Singularity theory
- Z2
- -symmetry