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On the calculation of highly oscillatory integrals with an algebraic singularity. (English) Zbl 1209.65030
The authors present a numerical method for computing highly oscillatory integrals containing an algebraic singularity of the form $$ \int_{0}^{1} f(x) \mathrm e^{\mathrm i \omega x^{-r}} \mathrm d x, $$ where $\omega$ and $r$ are positive real numbers and $f$ is an analytic function. By the change of variable $t = x^{-r}$ the singularity is removed and the integral is transformed into an integral on the infinite range $[1,\infty)$. For the evaluation of this improper integral the authors choose the limit of proper integration rules based on the numerical steepest descent method [cf. {\it D. Huybrechs} and {\it S. Vandewalle}, SIAM J. Numer. Anal. 44, No. 3, 1026--1048 (2006; Zbl 1123.65017)], where after analytic continuation Cauchy’s integral theorem applies and a more suitable integration path is selected. Along the new integration path a standard Gauss-Laguerre quadrature is used, which achieves high convergence rates in the frequency $\omega$ and the number of quadrature points. Numerical examples are presented and compared to integration methods for oscillatory integrals with an algebraic singularity based on Gauß rules for orthogonal polynomials with respect to special weight functions [cf. {\it A. I. Hascelik}, J. Comput. Appl. Math. 223, No. 1, 399--408 (2009; Zbl 1155.65023)].

MSC:
65D32Quadrature and cubature formulas (numerical methods)
41A55Approximate quadratures
65T40Trigonometric approximation and interpolation (numerical methods)
42A16Fourier coefficients, special Fourier series, etc.
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References:
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