Numerical Steepest Descent Method for Hankel Type of Hypersingular Oscillatory Integrals in Electromagnetic Scattering Problems

We present a fast and accurate numerical scheme for approximating hypersingular integrals with highly oscillatory Hankel kernels. The main idea is to first change the integration path by Cauchy’s theorem, transform the original integral into an integral on [a,+∞], and then use the generalized Gauss Laguerre integral formula to calculate the corresponding integral. This method has the advantages of high-efficiency, fast convergence speed. Numerical examples show the effect of this method.