Abstract
In this paper sufficient conditions are derived to ensure the convergence of the Elliott and Hunter types of quadrature rules for the evaluation of weighted Cauchy principal-value integrals of the form:

The simultaneous convergence in the interval (−1, 1) of both quadratures was established for a class of Hölder-continuous functionsf(f∈H μ ). Corrections of some previous statements on the subject of convergence of such quadratures are also included.
Moreover, a simple derivation of the Hunter and Elliott types of quadrature rules for the evaluation of the derivative of thep-th-order of the abovestated integral was given and sufficient conditions for the convergence of the Hunter-type quadrature were obtained. Thus, the convergence of this integral was ensured for functionsf such thatf (p) ∈H μ .
Zusammenfassung
In diesem Artikel sind hinreichende Bedingungen, welche die Konvergenz von Quadratursätzen des Elliott-und Hunter-Typus für die Bestimmung von gewichteten Cauchy Hauptwert-Integralen der Form

sicherstellen, hergeleitet.
Die gleichzeitige Konvergenz beider Quadraturen im Intervall (−1, +1) wurde für eine Klasse von Hölderstetigen Funktionenf(f∈H μ ) nachgewiesen. Im Artikel sind auch Korrekturen von gewissen früheren Darlegungen über die Konvergenz von solchen Quadraturen enthalten.
Ferner wurde eine einfache Herleitung der Elliott-und Hunterschen Quadratursätze für die Bestimmung derp-ten Ableitung des obenstehenden Integrals gegeben und hinreichende Bedingungen für die Konvergenz der Hunterschen Quadratur wurden erhalten. Die Konvergenz dieses Integrals wurde somit für Funktionenf, für welchef (p) ∈H μ gilt, sichergestellt.
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.References
Paget, D. F., Elliott, D.: An algorithm for the numerical evaluation of certain Cauchy principal value integrals. Numer. Math.19, 373–385 (1972).
Hunter, D. B.: Some Gauss-type formulae for the evaluation of Cauchy principal values of integrals. Numer. Math.19, 419–424 (1972).
Chawla, M. M., Ramakrishnan, T. R.: Modified Gauss-Jacobi quadrature formulas for the numerical evaluation of Cauchy type singular integrals. BIT14, 14–21 (1974).
Tsamasphyros, G., Theocaris, P. S.: Sur une méthode générale de quadrature pour des intégrales du type Cauchy. Symposium of Applied Mathematics, Salonica, August 1976, and Rev. Roum. Sci. Techn., Sér. Méc. Appl.25, 839–856 (1980).
Theocaris, P. S., Tsamasphyros, G.: Numerical solution of systems of singular integral equations with variable coefficients. Report, National Technical University, November 1976, also Appl. Analysis9, 37–52 (1979).
Elliott D., Paget, D. F.: On the convergence of a quadrature rule for evaluating certain Cauchy principal value integrals. Numer. Math.23, 311–319 (1975).
Elliott, D., Paget, D. F.: On the convergence of a quadrature rule for evaluating certain Cauchy principal value integrals: An addendum. Numer. Math.25, 287–289 (1976).
Tsamasphyros, G. J., Theocaris, P. S.: On the convergence of a Gauss quadrature rule for evaluation of Cauchy type singular integrals. BIT17, 458–464 (1977).
Elliott, D., Paget, D. F.: Gauss type quadrature rules for Cauchy principal value integrals. Math. Comp.33, 301–309 (1979).
Elliott, D.: On the convergence of Hunter's quadrature rule for Cauchy principal value integrals. BIT19, 457–469 (1979).
Haftmann, R.: Quadraturformeln vom Gausstyp für singuläre Integrale und ihre Anwendung zur Lösung singulärer Integralgleichungen. Wissenschaftliche Informationen17, Karl-Marx-Stadt (1980).
Ioakimidis, N. I.: On the numerical evaluation of derivatives of Cauchy principal value integrals. Computing27, 81–88 (1981).
Szegö, G.: Orthogonal Polynomials. Amer. Math. Soc. Colloq. Publ.23 (1959). Providence: American Math. Soc.
Theocaris, P. S.: Asymmetric branching of cracks. J. Appl. Mech.44, 611–618 (1977).
Chawla, M. M., Kumar, S.: Convergence for Cauchy principal value integrals. Computing23, 67–72 (1979).
Kalandiya, A. I.: Mathematical methods of two-dimensional elasticity. Moscow: Mir 1975.
Gakhov, F. D.: Boundary value problems. Oxford: Pergamon Press 1966.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Tsamasphyros, G., Theocaris, P.S. On the convergence of some quadrature rules for Cauchy principal-value and finite-part integrals. Computing 31, 105–114 (1983). https://doi.org/10.1007/BF02259907
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02259907