Exponential convergence of some recent numerical quadrature methods for Hadamard finite parts of singular integrals of periodic analytic functions

Let assuming that \(g\in C^\infty [a,b]\) such that f(x) is T-periodic, \(T=b-a\), and \(f(x)\in C^\infty (\mathbb {R}_t)\) with \(\mathbb {R}_t=\mathbb {R}{\setminus }\{t+ kT\}^\infty _{k=-\infty }\). Here stands for the Hadamard Finite Part (HFP) of the singular integral \(\int ^b_af(x)\,dx\) that does not exist in the regular sense. In a recent work, we unified the treatment of these HFP integrals by using a generalization of the Euler–Maclaurin expansion due to the author and developed a number of numerical quadrature formulas \(\widehat{T}^{(s)}_{m,n}[f]\) of trapezoidal type for I[f] for all m. 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 showed that all of the numerical quadrature formulas \(\widehat{T}^{(s)}_{m,n}[f]\) have spectral accuracy; that is, In this work, we continue our study of convergence and extend it to functions f(x) that possess certain analyticity properties. Specifically, we assume that f(z), as a function of the complex variable z, is also analytic in the infinite strip \(|{\text {Im}}\,z|<\sigma \) for some \(\sigma >0\), excluding the poles of order m at the points \(t+kT\), \(k=0,\pm 1,\pm 2,\ldots .\) For \(m=1,2,3,4\) and relevant s, we prove that