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 $$\begin{aligned} {\widehat{T}}^{(0)}_{3,n}[f]&=h\sum ^{n-1}_{j=1}f(t+jh)-\frac{\pi ^2}{3}\,g'(t)\,h^{-1} +\frac{1}{6}\,g'''(t)\,h, \quad h=\frac{T}{n},\\ {\widehat{T}}^{(1)}_{3,n}[f]&=h\sum ^n_{j=1}f(t+jh-h/2)-\pi ^2\,g'(t)\,h^{-1},\quad h=\frac{T}{n},\\ {\widehat{T}}^{(2)}_{3,n}[f]&=2h\sum ^n_{j=1}f(t+jh-h/2)- \frac{h}{2}\sum ^{2n}_{j=1}f(t+jh/2-h/4),\quad h=\frac{T}{n}. \end{aligned}$$ 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, $$\begin{aligned} {\widehat{T}}^{(s)}_{m,n}[f]-I[f]=o(n^{-\mu })\quad \text {as}\, {n\rightarrow \infty }\quad \forall \mu >0. \end{aligned}$$ 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.