Implementing the complex integral method with the transformed Clenshaw–Curtis quadrature

Gauss–Laguerre quadrature plays an important role in implementing the numerical steepest decent method for computing highly oscillatory integrals. However, it consumes too much time when the analytic region of the integrand is narrow. In this paper, we analyze the convergence rate of the transformed Clenshaw–Curtis quadrature, and show that this method also shares the property that the higher the oscillation, the better the calculation. Moreover, it is efficient to compute highly oscillatory integral with a nearly singularity. Numerical tests are performed to verify our given results.