The superconvergence of composite Newton–Cotes rules for Hadamard finite-part integral on a circle

We study the general (composite) Newton–Cotes rules for the computation of Hadamard finite-part integral on a circle with the hypersingular kernel $${\sin^{-2}\frac{x-s}2 }$$ and focus on their pointwise superconvergence phenomenon, i.e., when the singular point coincides with some a priori known point, the convergence rate is higher than what is globally possible. We show that the superconvergence rate of the (composite) Newton–Cotes rules occurs at the zeros of a special function $${\Phi_k(\tau)}$$ and prove the existence of the superconvergence points. The relation between $${\Phi_k(\tau)}$$ and $${\mathcal{S}_k(\tau)}$$ defined in Wu and Sun (Numer Math 109:143–165, 2008) is established, and the efficient calculation of Cotes coefficients is also discussed. Several numerical examples are provided to validate the theoretical analysis.