Exponential Convergence of Some Recent Numerical Quadrature Methods for Hadamard Finite Parts of Singular Integrals of Periodic Analytic Functions (original) (raw)

Let I[f] = ⨎ b a f(x) dx, f(x) = g(x) (x − t)m , m = 1, 2, . . . , a < t < b, assuming that g ∈ C∞[a, b] such that f(x) is T -periodic, T = b−a, and f(x) ∈ C∞(Rt) with Rt = R\{t+kT}∞k=−∞. Here ⨎a f(x) dx stands for the Hadamard Finite Part (HFP) of the singular integral ∫ b a f(x) dx that does not exist in the regular sense. In a recent work, we unified the treatment of these HFP integrals by using a generalization of the Euler–Maclaurin expansion due to the author and developed a number of numerical quadrature formulas T̂ (s) m,n[f] of trapezoidal type for I[f] for all m. For example, three numerical quadrature formulas of trapezoidal type result from this approach for the case m = 3, and these are T̂ (0) 3,n[f] = h n−1 ∑ j=1 f(t + jh) − π 3 g ′(t)h−1 + 1 6 g ′′′(t)h, h = Tn , T̂ (1) 3,n[f] = h n ∑ j=1 f(t + jh − h/2) − π g′(t)h−1, h = Tn , T̂ (2) 3,n[f] = 2h n ∑ j=1 f(t + jh − h/2) − h 2 2n ∑ j=1 f(t + jh/2 − h/4), h = Tn . For all m and s, we showed that all of the numerica...