  In this paper, we present a Clenshaw-Curtis-Filon-type method for the weakly
singular oscillatory integral with Fourier and Hankel kernels. By interpolating
the non-oscillatory and nonsingular part of the integrand at $(N+1)$
Clenshaw-Curtis points, the method can be implemented in $O(N\log N)$
operations. The method requires the accurate computation of modified moments.
We first give a method for the derivation of the recurrence relation for the
modified moments, which can be applied to the derivation of the recurrence
relation for the modified moments corresponding to other type oscillatory
integrals. By using recurrence relation, special functions and classic
quadrature methods, the modified moments can be computed accurately and
efficiently. Then, we present the corresponding error bound in inverse powers
of frequencies $k$ and $\omega$ for the proposed method. Numerical examples are
provided to support the theoretical results and show the efficiency and
accuracy of the method.
