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

Catherine Hobbs, J. N.L. Connor, N. P. Kirk

Research output: Contribution to journalArticlepeer-review

9 Citations (Scopus)

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 languageEnglish
Pages (from-to)192-213
Number of pages22
JournalJournal of Computational and Applied Mathematics
Volume207
Issue number2
Early online date8 Dec 2006
DOIs
Publication statusPublished - 15 Oct 2007
Externally publishedYes

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

Fingerprint

Dive into the research topics of 'Theory and numerical evaluation of oddoids and evenoids: Oscillatory cuspoid integrals with odd and even polynomial phase functions'. Together they form a unique fingerprint.

Cite this