Application of quadrature rules for Cauchy-type integrals to the generalized Poincaré-Bertrand formula

Author:
N. I. Ioakimidis

Journal:
Math. Comp. **44** (1985), 199-206

MSC:
Primary 65D32; Secondary 65R20

DOI:
https://doi.org/10.1090/S0025-5718-1985-0771041-X

MathSciNet review:
771041

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Abstract | References | Similar Articles | Additional Information

Abstract: The classical Poincaré-Bertrand transposition formula for the inversion of the order of integration in repeated Cauchy-type integrals is generalized in accordance with a new interpretation of Cauchy-type integrals. Next, the Gauss-Jacobi quadrature rule is applied, in a particular case of the generalized Poincaré-Bertrand formula, to both members of this formula and it is proved that this formula still remains valid (after the approximation of the integrals by quadrature sums). Two simple applications of this result, one concerning the convergence of a quadrature rule for repeated Cauchy-type integrals, and the other the numerical solution of singular integral equations, are made. Further generalizations and applications of the present results follow easily.

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DOI:
https://doi.org/10.1090/S0025-5718-1985-0771041-X

Article copyright:
© Copyright 1985
American Mathematical Society