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Convergence of quadratures for Cauchy principal value integrals

Konvergenz von Quadraturen für Cauchysche Hauptwerte

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Abstract

Quadrature formulas based on the “practical” abscissasx k=cos(k π/n),k=0(1)n, are obtained for the numerical evaluation of the weighted Cauchy principal value integrals

$$\mathop {\rlap{--} \smallint }\limits_{ - 1}^1 (1 - x)^\alpha (1 + x)^\beta (f(x))/(x - a)){\rm E}dx,$$

where α,β>−1 andaε(−1, 1). An interesting problem concerning these quadrature formulas is their convergence for a suitable class of functions. We establish convergence of these quadrature formulas for the class of functions which are Hölder-continuous on [−1, 1].

Zusammenfassung

Ermittelt werden die auf den “praktischen” Abszissenx k=cos(k π/n),k=0(1)n, basierten Quadraturformeln für die numerische Berechnung von Cauchyschen gewichteten Hauptwerten

$$\mathop {\rlap{--} \smallint }\limits_{ - 1}^1 (1 - x)^\alpha (1 + x)^\beta (f(x))/(x - a)){\rm E}dx,$$

wobei α,β>−1 undaε(−1, 1). Ein interessantes Problem bezüglich dieser Quadraturformeln ist ihre Konvergenz für die Klasse von Funktionen, die auf [−1, 1] Hölder-kontinuierlich sind.

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Chawla, M.M., Sheo Kumar Convergence of quadratures for Cauchy principal value integrals. Computing 23, 67–72 (1979). https://doi.org/10.1007/BF02252614

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  • DOI: https://doi.org/10.1007/BF02252614

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