Unified Compact Numerical Quadrature Formulas for Hadamard Finite Parts of Singular Integrals of Periodic Functions
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We consider the numerical computation of finite-range singular integrals I[f]=^b_a 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∈ C^∞[a,b] and f(x)∈ C^∞(ℝ_t) is T-periodic with ℝ_t=ℝ∖{t+ kT}^∞_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^(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^(0)_3,n[f] =h∑^n-1_j=1f(t+jh)-π^2/3 g'(t) h^-1 +1/6 g”'(t) h, h=T/n, T^(1)_3,n[f] =h∑^n_j=1f(t+jh-h/2)-π^2 g'(t) h^-1, h=T/n, T^(2)_3,n[f] =2h∑^n_j=1f(t+jh-h/2)- h/2∑^2n_j=1f(t+jh/2-h/4), h=T/n.
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