We consider the numerical computation of finite-range singular integrals I[f] = (sic)(a)(b) f(x)dx, f(x) = g(x)/(x - t)(m), m = 1, 2, . . . , a < t < b, that are defined in the sense of Hadamard Finite Part, assuming that g is an element of C-infinity [a, b] and f (x) is an element of C-infinity (R-t) is T-periodic with f is an element of C-infinity (R-t), R-t = R\{t + kT}(k =) (-infinity) (infinity), T = b - a. Using a generalization of the Euler-Maclaurin expansion developed in [A. Sidi, Euler-Maclaurin expansions for integrals with arbitrary algebraic endpoint singularities. Math. Comp., 81:2159-2173, 2012], we unify the treatment of these integrals. For eachm, we develop a number of numerical quadrature formulas (T) over cap ((s))(m,n)[f] of trapezoidal type for I [f]. For example, three numerical quadrature formulas of trapezoidal type result from this approach for the case m = 3, and these are (T) over cap ((0))(3,n)[f] = h Sigma(n-1)(j=1) f(t + jh) = pi(2)/3 g'(t)h(-1) + 1/6g"'(t)h, h = T/n, (T) over cap ((1))(3,n)[f] = h Sigma(n)(j=1) f(t + jh - h/2) - pi(2) g'(t)h(-1), h = T/n, (T) over cap ((2))(3,n)[f] = 2h Sigma(n)(j=1) f(t + jh - h/2) - h/2 Sigma(2n)(j=1) f(t + jh/2 - h/4), h = T/n. For all m and s, we show that all of the numerical quadrature formulas (T) over cap ((s))(m,n)[f] have spectral accuracy; that is, (T) over cap ((s))(m,n)[f] - I[f] = o(n(-mu)) as n -> infinity for all mu > 0. We provide a numerical example involving a periodic integrand with m = 3 that confirms our convergence theory. We also show how the formulas (T) over cap ((s))(m,n)[f] can be used in an efficient manner for solving supersingular integral equations whose kernels have a (x - t)(-3) singularity. A similar approach can be applied for all m.
