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On the evaluation of Cauchy principal value integrals of oscillatory functions. (English) Zbl 1190.65043

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.

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
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References:

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