Exactness and convergence properties of some recent numerical quadrature formulas for supersingular integrals of periodic functions

In a recent work, we developed three new compact numerical quadrature formulas for finite-range periodic supersingular integrals \(I[f]={\mathop\int{\!\!\!\!\!\!=}}^{\,\,b}_{\!\!a} f(x)\,dx\), where \(f(x)=g(x)/(x-t)^3,\) assuming that \(g\in C^\infty [a,b]\) and f(x) is T-periodic, \(T=b-a\). With \(h=T/n\), these numerical quadrature formulas read We also showed that these formulas have spectral accuracy; that is, In the present work, we continue our study of these formulas for the special case in which \(f(x)=\frac{\cos \frac{\pi (x-t)}{T}}{\sin ^3\frac{\pi (x-t)}{T}}\,u(x)\), where u(x) is in \(C^\infty ({\mathbb {R}})\) and is T-periodic. Actually, we prove that \({\widehat{T}}{}^{(s)}_n[f]\), \(s=0,1,2,\) are exact for a class of singular integrals involving T-periodic trigonometric polynomials of degree at most \(n-1\); that is, We also prove that, when u(z) is analytic in a strip \(\big |\text {Im}\,z\big |<\sigma \) of the complex z-plane, the errors in all three \({\widehat{T}}{}^{(s)}_n[f]\) are \(O(e^{-2n\pi \sigma /T})\) as \(n\rightarrow \infty \), for all practical purposes.