Richardson Extrapolation on Some Recent Numerical Quadrature Formulas for Singular and Hypersingular Integrals and Its Study of Stability

Recently, we derived some new numerical quadrature formulas of trapezoidal rule type for the integrals I (1)[g] = ∫ b a g(x) x−t dx and I (2)[g] = ∫ b a g(x) (x−t)2 dx . These integrals are not defined in the regular sense; I (1)[g] is defined in the sense ofCauchy PrincipalValuewhile I (2)[g] is defined in the sense of Hadamard Finite Part. With h = (b − a)/n, n = 1, 2, . . ., and t = a+kh for some k ∈ {1, . . . , n−1}, t being fixed, the numerical quadrature formulas Q n [g] for I (1)[g] and Q n [g] for I (2)[g] are Q(1) n [g] = h n ∑ j=1 f (a + jh − h/2), f (x) = g(x) x − t ,