Unified compact numerical quadrature formulas for Hadamard finite parts of singular integrals of periodic functions

We consider the numerical computation of finite-range singular integrals that are defined in the sense of Hadamard Finite Part, assuming that \(g\in C^\infty [a,b]\) and \(f(x)\in C^\infty ({\mathbb {R}}_t)\)  is T-periodic with \(f \in C^\infty ({\mathbb {R}}_t),\)   \({\mathbb {R}}_t={\mathbb {R}}{\setminus }\{t+ kT\}^\infty _{k=-\infty }\),  \(T=b-a\). 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 \({\widehat{T}}^{(s)}_{m,n}[f]\) 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 For all m and s, we show that all of the numerical quadrature formulas \({\widehat{T}}^{(s)}_{m,n}[f]\) have spectral accuracy; that is, We provide a numerical example involving a periodic integrand with \(m=3\) that confirms our convergence theory. We also show how the formulas \({\widehat{T}}{}^{(s)}_{3,n}[f]\) 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.