Recently, we derived some new numerical quadrature formulas of trapezoidal rule type for the singular integrals I(1)[u] = integral(b)(a)(cot pi(x-t)/T)u(x) dx and I-(2)[u] = integral(b)(a)(csc(2) pi(x-t)/T)u(x) dx with b - a = T and u(x) a T-periodic continuous function on R. These integrals are not defined in the regular sense, but are defined in the sense of Cauchy Principal Value and Hadamard Finite Part, respectively. With h = (b - a)/n, n = 1, 2,..., the numerical quadrature formulas Qn((1))[u] for I((1)[)u] and Qn((2))[u] for I-(2)[u] are Qn((1))[u] = h (n)Sigma(j=1) f(t+jh-h/2), f(x) = (cot pi(x-t)/T)u(x), and Qn((2))[u] = h (n)Sigma(j=1) f(t+jh-h/2) - T(2)u(t)h(-1), f(x) = (csc(2) pi(x-t)/T)u(x) We provided a complete analysis of the errors in these formulas under the assumption that u is an element of C-infinity (R) and is T-periodic. We actually showed that, I-(1)[u] - Q(n)((1))[u] = O(n(-mu)) and I-(2)[u] - Q(n)((2))[u] = O(n(-mu)) as n ->infinity, for all mu > 0. In this note, we analyze the errors in these formulas under the weaker assumption that u is an element of C-s (R) for some finite integer s. By first regularizing these integrals, we prove that, if u((s+1)) is piecewise continuous, then I-(1)[u] - Q(n)((1))[u] = O(n(-s-1/2)) as n ->infinity, if s >= 1, and I-(2)[u] - Q(n)((2))[u] = O(n(-s-1/2)) as n ->infinity, if s >= 2 We also extend these results by imposing different smoothness conditions on u((s+1)). Finally, we append suitable numerical examples. (C) 2014 IMACS. Published by Elsevier B.V. All rights reserved.
