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Computation of integrals with oscillatory and singular integrands using Chebyshev expansions. (English) Zbl 1255.65073
Summary: We present a general method for computing oscillatory integrals of the form $\int_{-1}^1f(x)G(x)e^{i{\omega }x}dx$, where $f$ is sufficiently smooth on $[-1,1]$, $\omega $ is a positive parameter and $G$ is a product of singular factors of algebraic or logarithmic type. Based on a Chebyshev expansion of $f$ and the properties of Chebyshev polynomials, the proposed method for such integrals is constructed with the help of the expansion of the oscillatory factor $e^{i{\omega }x}$. Furthermore, due to numerically stable recurrence relations for the modified moments, the devised scheme can be employed to compute oscillatory integrals with algebraic or logarithmic singularities at the end or interior points of the interval of integration. Numerical examples are provided to confirm our analysis.

65D32Quadrature and cubature formulas (numerical methods)
65D30Numerical integration
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