Quadrature formulae for Cauchy principal value integrals of oscillatory kind
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- Math. Comp. 49 (1987), 259-268 Request permission
Abstract:
The problem considered is that of evaluating numerically an integral of the form $f_{ - 1}^1\;{e^{i\omega x}}f(x) dx$, where f has one simple pole in the interval $[ - 1,1]$. Modified forms of the Lagrangian interpolation formula, taking account of the simple pole are obtained, and form the bases for the numerical quadrature rules obtained. Further modification to deal with the case when an abscissa in the interpolation formula is coincident with the pole is also considered. An error bound is provided and some numerical examples are given to illustrate the formulae developed.References
- Milton Abramowitz and Irene A. Stegun (eds.), Handbook of mathematical functions with formulas, graphs, and mathematical tables, Dover Publications, Inc., New York, 1992. Reprint of the 1972 edition. MR 1225604
- N. S. Bahvalov and L. G. Vasil′eva, The calculation of integrals of oscillatory functions by interpolation at the Gaussian quadrature nodes, Ž. Vyčisl. Mat i Mat. Fiz. 8 (1968), 175–181 (Russian). MR 226851
- M. M. Chawla and N. Jayarajan, Quadrature formulas for Cauchy principal value integrals, Computing 15 (1975), no. 4, 347–355 (English, with German summary). MR 415991, DOI 10.1007/BF02260318
- Philip J. Davis and Philip Rabinowitz, Methods of numerical integration, Computer Science and Applied Mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0448814
- David Elliott and D. F. Paget, Gauss type quadrature rules for Cauchy principal value integrals, Math. Comp. 33 (1979), no. 145, 301–309. MR 514825, DOI 10.1090/S0025-5718-1979-0514825-2 L. N. G. Filon, "On a quadrature formula for trigonometric integrals," Proc. Roy. Soc. Edinburgh, v. 49, 1929, pp. 38-47.
- Walter Gautschi, A survey of Gauss-Christoffel quadrature formulae, E. B. Christoffel (Aachen/Monschau, 1979) Birkhäuser, Basel-Boston, Mass., 1981, pp. 72–147. MR 661060 I. S. Gradshteyn & I. M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, New York, 1980.
- D. B. Hunter, Some Gauss-type formulae for the evaluation of Cauchy principal values of integrals, Numer. Math. 19 (1972), 419–424. MR 319355, DOI 10.1007/BF01404924
- N. S. Kambo, Error of the Newton-Cotes and Gauss-Legendre quadrature formulas, Math. Comp. 24 (1970), 261–269. MR 275671, DOI 10.1090/S0025-5718-1970-0275671-2
- S. Kumar, A note on quadrature formulae for Cauchy principal value integrals, J. Inst. Math. Appl. 26 (1980), no. 4, 447–451. MR 605402
- R. K. Littlewood and V. Zakian, Numerical evaluation of Fourier integrals, J. Inst. Math. Appl. 18 (1976), no. 3, 331–339. MR 448822
- I. M. Longman, On the numerical evaluation of Cauchy principal values of integrals, Math. Tables Aids Comput. 12 (1958), 205–207. MR 100356, DOI 10.1090/S0025-5718-1958-0100356-7
- Yudell L. Luke, On the computation of oscillatory integrals, Proc. Cambridge Philos. Soc. 50 (1954), 269–277. MR 62518
- W. G. Bickley, L. J. Comrie, J. C. P. Miller, D. H. Sadler, and A. J. Thompson, Bessel functions. Part II. Functions of positive integer order, British Association for the Advancement of Science, Mathematical Tables, vol. 10, University Press, Cambridge, 1952. MR 0050973
- D. F. Paget and David Elliott, An algorithm for the numerical evaluation of certain Cauchy principal value integrals, Numer. Math. 19 (1972), 373–385. MR 366004, DOI 10.1007/BF01404920
- T. N. L. Patterson, On high precision methods for the evaluation of Fourier integrals with finite and infinite limits, Numer. Math. 27 (1976/77), no. 1, 41–52. MR 433932, DOI 10.1007/BF01399083
- R. Piessens and F. Poleunis, A numerical method for the integration of oscillatory functions, Nordisk Tidskr. Informationsbehandling (BIT) 11 (1971), 317–327. MR 288959, DOI 10.1007/bf01931813
- Robert Piessens and Maria Branders, The evaluation and application of some modified moments, Nordisk Tidskr. Informationsbehandling (BIT) 13 (1973), 443–450. MR 331737, DOI 10.1007/bf01933408
- Anthony Ralston and Philip Rabinowitz, A first course in numerical analysis, 2nd ed., International Series in Pure and Applied Mathematics, McGraw-Hill Book Co., New York-Auckland-Bogotá, 1978. MR 0494814 G. Szegö, Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, R. I., 1975.
- Bing Yuan Ting and Yudell L. Luke, Computation of integrals with oscillatory and singular integrands, Math. Comp. 37 (1981), no. 155, 169–183. MR 616369, DOI 10.1090/S0025-5718-1981-0616369-5
Additional Information
- © Copyright 1987 American Mathematical Society
- Journal: Math. Comp. 49 (1987), 259-268
- MSC: Primary 65D32; Secondary 41A55
- DOI: https://doi.org/10.1090/S0025-5718-1987-0890267-X
- MathSciNet review: 890267