Evaluations of hypersingular integrals using Gaussian quadrature

A Gaussian quadrature formula for hypersingular integrals with second-order singularities is developed based on previous Gaussian quadrature formulae for Cauchy principal value integrals. The formula uses classical orthonormal polynomials, and the formula is then specialized to the case of Legendre and Chebyshev polynomials. Numerical experiments are carried out using the current formula and a previous formula developed by Kutt. It is found that the two methods generally give similar results, and in some cases the current method works better. It has also been shown that the current method allows the choice of an appropriate weight which can increase the convergence rate and the accuracy of the results.