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Computing Integrals Involved Oscillatory Functions and Gaussian Function with a Small Standard Deviation

This talk is concerned with the numerical computation of two kinds of integrals. The integrand of the first kind is a highly oscillatory function, where the amplitude function may have weak singularities and the oscillator has stationary points of a certain order. The integrand of the second kind is a product of a smooth function and the Gaussian function with a small standard deviation. Computing those integrals is of importance in wide application areas ranging from quantum chemistry, electrodynamics, fluid mechanics, statistics, probability theory, image, computerized tomography, and signal processing. The calculation of those integrals is widely perceived as a challenge issue. Two classes of composite numerical quadrature rules are presented for computing these integrals. One class of quadrature rules has a polynomial order of convergence and the other class has an exponential order of convergence.