Generalized noninterpolatory rules for Cauchy principal value integrals

Author:
Philip Rabinowitz

Journal:
Math. Comp. **54** (1990), 271-279

MSC:
Primary 65D30

DOI:
https://doi.org/10.1090/S0025-5718-1990-0990601-6

MathSciNet review:
990601

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Consider the Cauchy principal value integral

*w*and , then an approximation to is given by . If, in turn, we approximate by , then we get a double sequence of approximations to . We study the convergence of this sequence by relating it to the sequence of approximations associated with which has been investigated previously.

**[1]**Giuliana Criscuolo and Giuseppe Mastroianni,*On the convergence of an interpolatory product rule for evaluating Cauchy principal value integrals*, Math. Comp.**48**(1987), no. 178, 725–735. MR**878702**, https://doi.org/10.1090/S0025-5718-1987-0878702-4**[2]**Luigi Gatteschi,*On some orthogonal polynomial integrals*, Math. Comp.**35**(1980), no. 152, 1291–1298. MR**583506**, https://doi.org/10.1090/S0025-5718-1980-0583506-X**[3]**Peter Henrici,*Applied and computational complex analysis. Vol. 3*, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1986. Discrete Fourier analysis—Cauchy integrals—construction of conformal maps—univalent functions; A Wiley-Interscience Publication. MR**822470****[4]**I. P. Natanson,*Constructive function theory*, Vol. II (transl. by J. R. Schulenberger), Ungar, New York, 1955.**[5]**D. F. Paget,*Generalized product integration*, Ph.D. Thesis, Univ. of Tasmania, Hobart, 1976.**[6]**R. Piessens,*Modified Clenshaw-Curtis integration and applications to numerical computation of integral transforms*, Numerical integration (Halifax, N.S., 1986) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 203, Reidel, Dordrecht, 1987, pp. 35–51. MR**907110****[7]**P. Rabinowitz,*Some practical aspects in the numerical evaluation of Cauchy principal value integrals*, Internat. J. Comput. Math.**20**(1986), 283-298.**[8]**Philip Rabinowitz,*The convergence of noninterpolatory product integration rules*, Numerical integration (Halifax, N.S., 1986) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 203, Reidel, Dordrecht, 1987, pp. 1–16. MR**907108****[9]**P. Rabinowitz and D. S. Lubinsky,*Noninterpolatory integration rules for Cauchy principal value integrals*, Math. Comp.**53**(1989), no. 187, 279–295. MR**972372**, https://doi.org/10.1090/S0025-5718-1989-0972372-4**[10]**Philip Rabinowitz and William E. Smith,*Interpolatory product integration for Riemann-integrable functions*, J. Austral. Math. Soc. Ser. B**29**(1987), no. 2, 195–202. MR**905804**, https://doi.org/10.1017/S0334270000005713

Retrieve articles in *Mathematics of Computation*
with MSC:
65D30

Retrieve articles in all journals with MSC: 65D30

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1990-0990601-6

Keywords:
Cauchy principal value integrals,
numerical integration,
noninterpolatory integration rules,
orthogonal polynomials

Article copyright:
© Copyright 1990
American Mathematical Society