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Computation of integrals with oscillatory and singular integrands

Authors: Bing Yuan Ting and Yudell L. Luke
Journal: Math. Comp. 37 (1981), 169-183
MSC: Primary 65D30; Secondary 41A60
MathSciNet review: 616369
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Abstract: This paper is concerned with evaluation of integrals whose integrands are oscillatory and contain singularities at the endpoints of the interval of integration. A typical form is $ G(\theta ) = \smallint _a^bw(x){e^{i\theta x}}f(x)\,dx$, where a and b can be finite or infinite, $ \theta $ is a parameter which is usually large, $ f(x)$ is analytic in the range of integration, and the singularities are encompassed in the weight function $ w(x)$. We suppose that $ f(x)$ can be expanded in series of polynomials which are orthogonal over the interval of integration with respect to $ w(x)$. There are two such expansions for $ f(x)$. One is an infinite series which follows from the usual orthogonality property. The other is a polynomial approximation plus a remainder. The relations between the coefficients in these representations are detailed and methods for the evaluation of these are analyzed. Error analyses are provided. A numerical example is given to illustrate the effectiveness of the schemes developed.

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  • [1] L. N. G. Filon, "On a quadrature formula for trigonometric integrals," Proc. Roy. Soc. Edinburgh, v. 49, 1929, pp. 38-47.
  • [2] Y. L. Luke, "On the computation of oscillatory integrals," Part 2, Proc. Cambridge Philos. Soc., v. 50, 1954, pp. 269-277. MR 0062518 (15:992b)
  • [3] J. N. Lyness, "The calculation of Fourier coefficients by Möbius inversion of the Poisson summation formula. Part I, Functions whose early derivatives are continuous," Math. Comp., v. 24, 1970, pp. 101-135. MR 0260230 (41:4858)
  • [4] J. N. Lyness, "The calculation of Fourier coefficients by the Möbius inversion of the Poisson summation formula. Part II, Piecewise continuous functions and functions with poles near [0, 1]," Math. Comp., v. 25, 1971, pp. 59-78. MR 0293846 (45:2922)
  • [5] J. N. Lyness, "The calculation of Fourier coefficients by the Möbius inversion of the Poisson summation formula. Part III, Functions having algebraic singularities," Math. Comp., v. 25, 1971, pp. 483-493. MR 0297162 (45:6220)
  • [6] Y. L. Luke, "On the error in a certain interpolation formula and in the Gaussian integration formula," J. Austral. Math. Soc. Ser. A (2), v. 19, 1975, pp. 196-209. MR 0386232 (52:7090)
  • [7] Y. L. Luke, B. Y. Ting & M. J. Kemp, "On generalized Gaussian quadrature," Math. Comp., v. 29, 1975, pp. 1083-1093. MR 0388740 (52:9574)
  • [8] N. S. Bakhvalov & L. G. Vasiléva, "The calculation of integrals of oscillatory functions by interpolation of the Gaussian quadrature nodes," Ž. Vyčisl. Mat. i Mat. Fiz., v. 8, 1968, pp. 175-181. MR 0226851 (37:2437)
  • [9] R. Piessens & F. Poleunis, "A numerical method for the integration of oscillatory functions," BIT, v. 11, 1971, pp. 317-327. MR 0288959 (44:6154)
  • [10] T. N. L. Patterson, "On higher precision methods for the evaluation of Fourier integrals with finite and infinite limits," Numer. Math., v. 27, 1976, pp. 41-52. MR 0433932 (55:6902)
  • [11] R. K. Littlewood & V. Zakian, "Numerical evaluation of Fourier integrals," J. Inst. Math. Appl., v. 18, 1976, pp. 331-339. MR 0448822 (56:7127)
  • [12] P. J. Davis & P. Rabinowitz, Numerical Integration, Blaisdell, Boston, Mass., 1967. MR 0211604 (35:2482)
  • [13] J. D. Donaldson & D. Elliott, "A uniform approach to quadrature rules with asymptotic estimates of their remainders," SIAM J. Numer. Anal., v. 9, 1972, pp. 573-602. MR 0317522 (47:6069)
  • [14] C. W. Clenshaw & A. R. Curtis, "A method for numerical integration on an automatic computer," Numer. Math., v. 2, 1960, pp. 197-205. MR 0117885 (22:8659)
  • [15] G. Freud, Orthogonal Polynomials, Pergamon Press, Oxford and New York, 1971.
  • [16] A. C. Aitken, "On Bernoulli's numerical solution of algebraic equations," Proc. Roy. Soc. Edinburgh, v. 46, 1926, pp. 289-305.
  • [17] I. P. Natanson, Constructive Function Theory, Vols. 1, 2 and 3, Ungar, New York, 1964.
  • [18] J. Prasad, "On an approximation of a function and its derivatives," Publ. Inst. Math., v. 14 (28), 1972, pp. 129-132. MR 0407524 (53:11297)
  • [19] A. Erdélyi et al., Higher Transcendental Functions, Vol. 2, McGraw-Hill, New York, 1954.
  • [20] Y. L. Luke, The Special Functions and Their Approximations, Vols. 1 and 2, Academic Press, New York and London, 1969.
  • [21] W. Gautschi, "Computational aspects of three-term recurrence relations," SIAM Rev., v. 9, 1967, pp. 24-82. MR 0213062 (35:3927)
  • [22] J. Wimp, On recursive computation, Report ARL 69-0186, Aerospace Res. Labs., Wright-Patterson Air Force Base, Ohio, 1969. MR 0253554 (40:6768)
  • [23] J. Wimp, "Recent development in recursive computation," SIAM Studies in Appl. Math., Vol. VI, Philadelphia, Pa., 1970, pp. 101-123.
  • [24] Y. L. Luke, "On the error in the Padé approximants for a form of the incomplete gamma function including the exponential function," SIAM J. Math. Anal., v. 6, 1975, pp. 829-839. MR 0385413 (52:6275)
  • [25] B. Y. Ting, Evaluation of Integrals Whose Integrands Are Oscillatory and Singular, Ph.D. Thesis, University of Missouri, Kansas City, Mo., 1979.

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Keywords: Fourier coefficients, numerical quadrature, integration of oscillatory and singular integrands, interpolation
Article copyright: © Copyright 1981 American Mathematical Society

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