Efficient quadrature of highly oscillatory integrals with algebraic singularities (Q455896)

The authors transform the oscillatory integral NEWLINE\[NEWLINE I[f] = \int_0^1x^{\alpha}f(x)e^{i\omega x^{-\beta}}\,dx, \quad \omega, \beta > 0, \;\alpha + \beta > -1, NEWLINE\]NEWLINE where \(f\) is a non-oscillatory, sufficiently smooth function on [0,1], into asymptotic series in inverse powers of the frequency \(\omega\). Then, the Filon-type and the Clenshaw-Curtis-Filon-type methods are available for the computation of \(I[f]\). For large \(\omega\), the efficiency and validity of these methods are shown by both some numerical experiments and theoretical results.