Convergence results for piecewise linear quadratures for Cauchy principal value integrals

Rabinowitz, Philip

doi:10.1090/S0025-5718-1988-0958639-3

Convergence results for piecewise linear quadratures for Cauchy principal value integrals
HTML articles powered by AMS MathViewer

by Philip Rabinowitz PDF

Math. Comp. 51 (1988), 741-747 Request permission

Abstract:

Conditions on k and f are given for the pointwise and uniform convergence to the Cauchy principal value integral \[ \int {\frac {{k(x)f(x)}}{{x - \lambda }} dx} ,\quad - 1 < \lambda < 1,\] of a sequence of integrals of piecewise linear approximations to $f(x)$ or ${g_\lambda }(x) = (f(x) = f(\lambda ))/(x - \lambda )$. The important special case, $k(x) = {(1 - x)^\alpha }{(1 + x)^\beta }$, is considered in detail.

References

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, DOI 10.1090/S0025-5718-1987-0878702-4
A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Tables of integral transforms. Vol. II, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1954. Based, in part, on notes left by Harry Bateman. MR 0065685
Apostolos Gerasoulis, Piecewise-polynomial quadratures for Cauchy singular integrals, SIAM J. Numer. Anal. 23 (1986), no. 4, 891–902. MR 849289, DOI 10.1137/0723057
Philip Rabinowitz, Numerical integration in the presence of an interior singularity, J. Comput. Appl. Math. 17 (1987), no. 1-2, 31–41. MR 884259, DOI 10.1016/0377-0427(87)90036-7
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
Charles E. Stewart, On the numerical evaluation of singular integrals of Cauchy type, J. Soc. Indust. Appl. Math. 8 (1960), 342–353. MR 114101

Orthogonal Polynomials