# Technical Report CS-2021-02

 TR#: CS-2021-02 Class: CS Title: Exponential Convergence of Some Recent Numerical Quadrature Methods for Hadamard Finite Parts of Singular Integrals of Periodic Analytic Functions Authors: Avram Sidi PDF Currently accessibly only within the Technion network Abstract: Let $$I[f]=\intBar^b_a f(x)\,dx,\quad f(x)=\frac{g(x)}{(x-t)^m},\quad m=1,2,\ldots,\quad a0.$$ 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 $|\Im z|<\sigma$ for some $\sigma>0$, excluding the poles of order $m$ at the points $t+kT$, $k=0,\pm1,\pm2,\ldots.$ For $m=1,2,3,4$ and relevant $s$, we prove that $$\widehat{T}^{(s)}_{m,n}[f]-I[f]=O\big(\exp(-2\pi n\rho/T)\big)\quad\text{as n\to\infty}\quad \forall \rho<\sigma.$$ Copyright The above paper is copyright by the Technion, Author(s), or others. Please contact the author(s) for more information

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