Composite trapezoidal quadrature for computing hypersingular integrals on interval

In this paper, composite trapezoidal quadrature for numerical evaluation of hypersingular integrals was first introduced. By Taylor expansion at the singular point $y$, error functional was obtained. We know that the divergence rate of $O(h^{-p}),p=1,2$, and there were no roots of the special function for the first part in the error functional. Meanwhile, for the second part of the error functional, the divergence rate was $O(h^{-p+1}),p=1,2$, but there were roots of the special function. We proved that the convergence rate could reach $O(h^2)$ at superconvergence points far from the end of the interval. Two modified trapezoidal quadratures are presented and their convergence rate can reach $O(h^2)$ at certain superconvergence points or any local coordinate point. At last, several examples were presented to test our theorem.