Complex Gaussian Quadrature of Oscillatory Integrals Articles uri icon

publication date

  • January 2009

start page

  • 197

end page

  • 219

issue

  • 112

volume

  • 2

International Standard Serial Number (ISSN)

  • 0029-599X

Electronic International Standard Serial Number (EISSN)

  • 0945-3245

abstract

  • We construct and analyze Gauss-type quadrature rules with complex- valued nodes and weights to approximate oscillatory integrals with stationary points of high order. The method is based on substituting the original interval of integration by a set of contours in the complex plane, corresponding to the paths of steepest descent. Each of these line integrals shows an exponentially decaying behaviour, suitable for the application of Gaussian rules with non-standard weight functions. The results differ from those in previous research in the sense that the constructed rules are asymptotically optimal, i.e., among all known methods for oscillatory integrals they deliver the highest possible asymptotic order of convergence, relative to the required number of evaluations of the integrand.