On high precision methods for computing integrals involving Bessel functions
Author:
Bruno Gabutti
Journal:
Math. Comp. 33 (1979), 10491057
MSC:
Primary 65D30
MathSciNet review:
528057
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Abstract: The technique of Bakhvalov and Vasil'eva for evaluating Fourier integrals is generalized to integrals involving exponential and Bessel functions.
 [1]
N. S. BAKHVALOV & L. G. VASIL'EVA, "Evaluation of the integrals of oscillating functions by interpolation at nodes of Gaussian quadratures," USSR Comp. Math. and Math. Phys., v. 8, 1968, pp. 241249.
 [2]
J.
Balázs and P.
Turán, Notes on interpolation. IX (Approximate
representation of Fouriertransform), Acta Math. Acad. Sci. Hungar.
16 (1965), 215–220. MR 0174907
(30 #5098)
 [3]
W.
Barrett, Convergence properties of Gaussian quadrature
formulae, Comput. J. 3 (1960/1961), 272–277. MR 0128073
(23 #B1117)
 [4]
C.
W. Clenshaw and A.
R. Curtis, A method for numerical integration on an automatic
computer, Numer. Math. 2 (1960), 197–205. MR 0117885
(22 #8659)
 [5]
R.
J. Glauber, Highenergy collision theory, Lectures in
theoretical physics, Vol. I. Lectures delivered at the Summer Institute for
Theoretical Physics, University of Colorado, Boulder, 1958 (edited by W. E.
Brittin and L. G. Dunham), Interscience Publishers, New YorkLondon, 1959,
pp. 315–414. MR 0107488
(21 #6213)
 [6]
I. S. GRADSHTEYN & I. M. RYZHIK, Table of Integral Series and Products, Academic Press, New York, 1965.
 [7]
Peter
Linz, A method for computing Bessel function
integrals, Math. Comp. 26 (1972), 509–513. MR 0303687
(46 #2823), http://dx.doi.org/10.1090/S00255718197203036878
 [8]
I.
M. Longman, Note on a method for computing infinite integrals of
oscillatory functions, Proc. Cambridge Philos. Soc.
52 (1956), 764–768. MR 0082193
(18,515f)
 [9]
J.
N. Lyness, Adjusted forms of the Fourier
coefficient asymptotic expansion and applications in numerical
quadrature, Math. Comp. 25 (1971), 87–104. MR 0290020
(44 #7205), http://dx.doi.org/10.1090/S00255718197102900202
 [10]
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 0433932
(55 #6902)
 [11]
R. PIESSENS & A. HAEGEMANS,"Numerical calculation of Fourier transform integrals," Electron. Lett., v. 9, 1973, pp. 108109.
 [12]
N.
M. Steen, G.
D. Byrne, and E.
M. Gelbard, Gaussian quadratures for the integrals
₀^{∞}𝑒𝑥𝑝(𝑥²)𝑓(𝑥)𝑑𝑥
and
₀^{𝑏}𝑒𝑥𝑝(𝑥²)𝑓(𝑥)𝑑𝑥,
Math. Comp. 23 (1969), 661–671. MR 0247744
(40 #1005), http://dx.doi.org/10.1090/S00255718196902477443
 [1]
 N. S. BAKHVALOV & L. G. VASIL'EVA, "Evaluation of the integrals of oscillating functions by interpolation at nodes of Gaussian quadratures," USSR Comp. Math. and Math. Phys., v. 8, 1968, pp. 241249.
 [2]
 J. BALÀZS & P. TURAN, "Notes on interpolation. IX," Acta Math. Acad. Sci. Hungar., v. 16, 1965, pp. 215220. MR 0174907 (30:5098)
 [3]
 W. BARRETT, "Convergence properties of Gaussian quadrature formulas," Comput. J., v. 3, 1960, pp. 272277. MR 0128073 (23:B1117)
 [4]
 C. W. CLENSHAW & A. R. CURTIS, "A method for numerical integration on an automatic computer," Numer. Math., v. 2, 1960, pp. 197205. MR 0117885 (22:8659)
 [5]
 R. J. GLAUBER, Lectures in Theoretical Physics, Vol. 1, Interscience, New York, 1959. MR 0107488 (21:6213)
 [6]
 I. S. GRADSHTEYN & I. M. RYZHIK, Table of Integral Series and Products, Academic Press, New York, 1965.
 [7]
 P. LINZ, "A method for computing Bessel function integrals," Math. Comp., v. 26, 1972, pp. 509513. MR 0303687 (46:2823)
 [8]
 I. M. LONGMANN, "Note on a method for computing infinite integrals of oscillatory functions," Proc. Cambridge Philos. Soc., v. 52, 1956, pp. 764768. MR 0082193 (18:515f)
 [9]
 J. N. LYNESS, "Adjusted forms of the Fourier coefficient asymptotic expansion and applications in numerical quadrature," Math. Comp., v. 25, 1971, pp. 86104. MR 0290020 (44:7205)
 [10]
 T. N. L. PATTERSON, "On high precision methods for the evaluation of Fourier integrals with finite and infinite limits," Numer. Math., v. 27, 1976, pp. 4152. MR 0433932 (55:6902)
 [11]
 R. PIESSENS & A. HAEGEMANS,"Numerical calculation of Fourier transform integrals," Electron. Lett., v. 9, 1973, pp. 108109.
 [12]
 N. M. STEEN, G. D. BYRNE & E. M. GELBARD, "Gaussian quadratures for the integrals and ," Math. Comp., v. 23, 1969, pp. 661671. MR 0247744 (40:1005)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718197905280575
PII:
S 00255718(1979)05280575
Article copyright:
© Copyright 1979 American Mathematical Society
