Wang, Haiyong; Xiang, Shuhuang On the evaluation of Cauchy principal value integrals of oscillatory functions. (English) Zbl 1190.65043 J. Comput. Appl. Math. 234, No. 1, 95-100 (2010). The authors are concerned with the numerical evaluation of the Cauchy principal value integrals of oscillatory functions \[ \begin{aligned} I_{\omega}(f;\tau) :=&\displaystyle{\int_{-1}^1e^{i\omega x}\frac{f(x)}{x-\tau}dx}\\ =&\displaystyle{\lim_{\epsilon\to 0^{+}}\int_{|x-\tau|\geq\epsilon}e^{i\omega x}\frac{f(x)}{x-\tau}dx,\,\,-1<\tau<1},\end{aligned} \]where \(f\) is analytic in a sufficiently large region of the complex plane containing \([-1,1]\).Based on analytic continuation, the integrals can be transformed into the problems of integrating two integrals on \([0,+\infty)\) with the integrand that does not oscillate, and that decays exponentially fast, which can be efficiently computed by using the Gauss-Laguerre quadrature rule. The validity of the method is demonstrated in the provision of two numerical experiments and their results. Reviewer: Ana-Maria Acu (Sibiu) Cited in 30 Documents MSC: 65D32 Numerical quadrature and cubature formulas 41A55 Approximate quadratures 42A16 Fourier coefficients, Fourier series of functions with special properties, special Fourier series 65T40 Numerical methods for trigonometric approximation and interpolation Keywords:complex integration method; steepest descent method; Cauchy principal value integrals; oscillatory functions; Gauss-Laguerre quadrature rule; numerical experiments PDFBibTeX XMLCite \textit{H. Wang} and \textit{S. Xiang}, J. Comput. Appl. Math. 234, No. 1, 95--100 (2010; Zbl 1190.65043) Full Text: DOI References: [1] Davis, P. J.; Rabinowitz, P., Methods of Numerical Integration (1984), Academic Press · Zbl 0154.17802 [2] Okecha, G. E., Quadrature formulae for Cauchy principal value integrals of oscillatory kind, Math. Comp., 49, 259-268 (1987) · Zbl 0634.65013 [3] Capobianco, M. R.; Criscuolo, G., On quadrature for Cauchy principal value integrals of oscillatory functions, J. Comput. Appl. Math., 156, 471-486 (2003) · Zbl 1043.65048 [4] Okecha, G. E., Hermite interpolation and a method for evaluating Cauchy principal value integrals of oscillatory kind, Kragujevac J. Math., 29, 91-98 (2006) · Zbl 1224.65060 [5] Milovanović, G. V., Numerical calculation of integrals involving oscillatory and singular kernels and some applications of quadratures, Comput. Math. Appl., 36, 19-39 (1998) · Zbl 0932.65023 [6] Huybrechs, D.; Vandewalle, S., On the evaluation of highly oscillatory integrals by analytic continuation, SIAM J. Numer. Anal., 44, 1026-1048 (2006) · Zbl 1123.65017 [7] Abramowitz, M.; Stegun, I. A., Handbook of Mathematical Functions (1964), National Bureau of Standards: National Bureau of Standards Washington, DC · Zbl 0515.33001 [8] Gradshteyn, I. S.; Ryzhik, I. M., Tables of Integrals, Series, and Products (2000), Academic Press: Academic Press San Diego, CA · Zbl 0981.65001 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.