# Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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## Noninterpolatory integration rules for Cauchy principal value integralsHTML articles powered by AMS MathViewer

by P. Rabinowitz and D. S. Lubinsky
Math. Comp. 53 (1989), 279-295 Request permission

## Abstract:

Let $w(x)$ be an admissible weight on $[ - 1,1]$ and let $\{ {p_n}(x)\} _0^\infty$ be its associated sequence of orthonormal polynomials. We study the convergence of noninterpolatory integration rules for approximating Cauchy principal value integrals $I(f;\lambda ):=\oint w(x)\frac {{f(x)}}{{x - \lambda }} dx,\quad \lambda \in ( - 1,1).$ This requires investigation of the convergence of the expansion $I(f;\lambda ) \sim \sum \limits _{k = 0}^\infty {(f,{p_k}){q_k}(\lambda ),\quad \lambda \in ( - 1,1),}$ in terms of the functions of the second kind $\{ {q_k}(\lambda )\} _0^\infty$ associated with w, where $(f,{p_k}):=\int _{ - 1}^1 {w(x)f(x){p_k}(x) dx\quad {\text {and}}\quad {q_k}(\lambda } ):=\oint w(x)\frac {{{p_k}(x)}}{{x - \lambda }} dx,$ $k = 0,1,2, \ldots ,\lambda \in ( - 1,1)$.
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