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Let assuming that g ∈ C ∞ [ a , b ] such that f(x) is T-periodic, T = b - a , and f (x) ∈ C ∞ (R t) with R t = R \ { t + k T } k = - ∞ ∞ . Here stands for the Hadamard Finite Part (HFP) of the singular integral ∫ a b f (x) d x 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 T ^ m , n (s) [ 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 T ^ 3 , n (0) [ f ] = h ∑ j = 1 n - 1 f (t + j h) - π 2 3 g ′ (t) h - 1 + 1 6 g ′ ′ ′ (t) h , h = T n , T ^ 3 , n (1) [ f ] = h ∑ j = 1 n f (t + j h - h / 2) - π 2 g ′ (t) h - 1 , h = T n , T ^ 3 , n (2) [ f ] = 2 h ∑ j = 1 n f (t + j h - h / 2) - h 2 ∑ j = 1 2 n f (t + j h / 2 - h / 4) , h = T n. For all m and s, we showed that all of the numerical quadrature formulas T ^ m , n (s) [ f ] have spectral accuracy; that is, T ^ m , n (s) [ f ] - I [ f ] = o (n - μ) as n → ∞ ∀ μ > 0. In this work, we continue our study of convergence and extend it to functions f(x) that possess certain analyticity properties. Specifically, we assume that f(z), as a function of the complex variable z, is also analytic in the infinite strip | Im z | < σ for some σ > 0 , excluding the poles of order m at the points t + k T , k = 0 , ± 1 , ± 2 , …. For m = 1 , 2 , 3 , 4 and relevant s, we prove that T ^ m , n (s) [ f ] - I [ f ] = O (exp (- 2 π n ρ / T)) as n → ∞ ∀ ρ < σ. [ABSTRACT FROM AUTHOR]

We consider the numerical computation of I [ f ] = ∫ = a b f (x) d x , the Hadamard finite part of the finite-range singular integral ∫ a b f (x) d x , f (x) = g (x) / (x - t) m with a < t < b and m ∈ { 1 , 2 , ... } , assuming that (i) g ∈ C ∞ (a , b) and (ii) g(x) is allowed to have arbitrary integrable singularities at the endpoints x = a and x = b . We first prove that ∫ = a b f (x) d x is invariant under any legitimate variable transformation x = ψ (ξ) , ψ : [ α , β ] → [ a , b ] , hence there holds ∫ = α β F (ξ) d ξ = ∫ = a b f (x) d x , where F (ξ) = f (ψ (ξ)) ψ ′ (ξ) . Based on this result, we next choose ψ (ξ) such that F (ξ) , the T -periodic extension of F (ξ) , T = β - α , is sufficiently smooth, and prove, with the help of some recent extension/generalization of the Euler–Maclaurin expansion, that we can apply to ∫ = α β F (ξ) d ξ the quadrature formulas derived for periodic singular integrals developed in an earlier work of the author: [A. Sidi, "Unified compact numerical quadrature formulas for Hadamard finite parts of singular integrals of periodic functions." Calcolo, 58, 2021. Article number 22]. We give a whole family of numerical quadrature formulas for ∫ = α β F (ξ) d ξ for each m, which we denote T ^ m , n (s) [ F ] . Letting G (ξ) = (ξ - τ) m F (ξ) , with τ ∈ (α , β) determined from t = ψ (τ) , and letting h = T / n , for m = 3 , for example, we have the three formulas T ^ 3 , n (0) [ F ] = h ∑ j = 1 n - 1 F (τ + j h) - π 2 3 G ′ (τ) h - 1 + 1 6 G ′ ′ ′ (τ) h , T ^ 3 , n (1) [ F ] = h ∑ j = 1 n F (τ + j h - h / 2) - π 2 G ′ (τ) h - 1 , T ^ 3 , n (2) [ F ] = 2 h ∑ j = 1 n F (τ + j h - h / 2) - h 2 ∑ j = 1 2 n F (τ + j h / 2 - h / 4). We show that all of the formulas T ^ m , n (s) [ F ] converge to I[f] as n → ∞ ; indeed, if ψ (ξ) is chosen such that F (i) (α) = F (i) (β) = 0 , i = 0 , 1 , ... , q - 1 , and F (q) (ξ) is absolutely integrable in every closed interval not containing ξ = τ , then T ^ m , n (s) [ F ] - I [ f ] = O (n - q) as n → ∞ , where q is a positive integer determined by the behavior of g(x) at x = a and x = b and also by ψ (ξ) . As such, q can be increased arbitrarily (even to q = ∞ , thus inducing spectral convergence) by choosing ψ (ξ) suitably. We provide several numerical examples involving nonperiodic integrands and confirm our theoretical results. [ABSTRACT FROM AUTHOR]

In a recent work, we developed three new compact numerical quadrature formulas for finite-range periodic supersingular integrals I [ f ] = ∫ = a b f (x) d x , where f (x) = g (x) / (x - t) 3 , assuming that g ∈ C ∞ [ a , b ] and f(x) is T-periodic, T = b - a . With h = T / n , these numerical quadrature formulas read T ^ n (0) [ f ] = h ∑ j = 1 n - 1 f (t + j h) - π 2 3 g ′ (t) h - 1 + 1 6 g ′ ′ ′ (t) h , T ^ n (1) [ f ] = h ∑ j = 1 n f (t + j h - h / 2) - π 2 g ′ (t) h - 1 , T ^ n (2) [ f ] = 2 h ∑ j = 1 n f (t + j h - h / 2) - h 2 ∑ j = 1 2 n f (t + j h / 2 - h / 4). We also showed that these formulas have spectral accuracy; that is, T ^ n (s) [ f ] - I [ f ] = o (n - μ) as n → ∞ ∀ μ > 0. In the present work, we continue our study of these formulas for the special case in which f (x) = cos π (x - t) T sin 3 π (x - t) T u (x) , where u(x) is in C ∞ (R) and is T-periodic. Actually, we prove that T ^ n (s) [ f ] , s = 0 , 1 , 2 , are exact for a class of singular integrals involving T-periodic trigonometric polynomials of degree at most n - 1 ; that is, T ^ n (s) [ f ] = I [ f ] when f (x) = cos π (x - t) T sin 3 π (x - t) T ∑ m = - (n - 1) n - 1 c m exp (i 2 m π x / T). We also prove that, when u(z) is analytic in a strip | Im z | < σ of the complex z-plane, the errors in all three T ^ n (s) [ f ] are O (e - 2 n π σ / T) as n → ∞ , for all practical purposes. [ABSTRACT FROM AUTHOR]

We consider the numerical computation of finite-range singular integrals that are defined in the sense of Hadamard Finite Part, assuming that g ∈ C ∞ [ a , b ] and f (x) ∈ C ∞ (R t) is T-periodic with f ∈ C ∞ (R t) , R t = R \ { t + k T } k = - ∞ ∞ , 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 each m, we develop a number of numerical quadrature formulas T ^ m , n (s) [ 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 ^ 3 , n (0) [ f ] = h ∑ j = 1 n - 1 f (t + j h) - π 2 3 g ′ (t) h - 1 + 1 6 g ′ ′ ′ (t) h , h = T n , T ^ 3 , n (1) [ f ] = h ∑ j = 1 n f (t + j h - h / 2) - π 2 g ′ (t) h - 1 , h = T n , T ^ 3 , n (2) [ f ] = 2 h ∑ j = 1 n f (t + j h - h / 2) - h 2 ∑ j = 1 2 n f (t + j h / 2 - h / 4) , h = T n. For all m and s, we show that all of the numerical quadrature formulas T ^ m , n (s) [ f ] have spectral accuracy; that is, T ^ m , n (s) [ f ] - I [ f ] = o (n - μ) as n → ∞ ∀ μ > 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 ^ 3 , n (s) [ 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. [ABSTRACT FROM AUTHOR]