Efficient computation of highly oscillatory integrals with weak singularities by Gauss-type method

Abstract

In this paper, we consider the numerical computation of Hankel transform with weak singularities at the endpoints. By using the analytic continuation, we transform the integral into two line integrals in complex plane, which can be efficiently evaluated by some proper Gauss quadrature rules. The asymptotic orders of the errors on k and ω are derived. These asymptotic orders are demonstrated by several numerical experiments either for fixed k or for fixed ω, which show that the efficiency and accuracy of this method significantly improve as the frequencies increase.