Unified Compact Numerical Quadrature Formulas for Hadamard Finite Parts of Singular Integrals of Periodic Functions Avram Sidi Computer Science Department Technion - Israel Institute of Technology Haifa 32000, Israel E-mail:
[email protected] URL: http://www.cs.technion.ac.il/~asidi February 15, 2021 arXiv:2102.06461v1 [math.NA] 12 Feb 2021 Abstract We consider the numerical computation of finite-range singular integrals b g x I f = f x dx, f x = , m = , ,..., a < t < b, −( ) m 1 2 [ ] Ua ( ) ( ) x t ( ) ∞ that are defined in the sense of Hadamard Finite Part, assuming that g ∈ C a,b and ∞ R R R ∞ [ ] f x ∈ C t is T -periodic with t = t + kT k=−∞, T = b − a. Using a gen- eralization( ) 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 de- s velop a number of numerical quadrature formulas T̂m,n( ) f of trapezoidal type for I f . For example, three numerical quadrature formulas of trapezoida[ ] l type result from[ this] approach for the case m = 3, and these are n−1 2 0 π ′ −1 1 ′′′ T T f = h f t + jh − g t h + g t h, h = , ̂3(,n) = 3 6 n [ ] j=1 ( ) ( ) ( ) n 1 2 ′ −1 T T f = h f t + jh − h 2 − π g t h , h = , ̂3(,n) = n [ ] j=1 ( / ) ( ) n 2n 2 h T T3( ) f = 2h f t + jh − h 2 − f t + jh 2 − h 4 , h = .