A note on asymptotic evaluation of some Hankel transforms
Authors:
C. L. Frenzen and R. Wong
Journal:
Math. Comp. 45 (1985), 537548
MSC:
Primary 41A60; Secondary 44A15, 65R10
MathSciNet review:
804942
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Asymptotic behavior of the integral is investigated, where is the Bessel function of the first kind and w is a large positive parameter. It is shown that decays exponentially like , , when is an entire function subject to a suitable growth condition. A complete asymptotic expansion is obtained when is a meromorphic function satisfying the same growth condition. Similar results are given when has some specific branch point singularities.
 [1]
Bruno
Gabutti, On high precision methods for
computing integrals involving Bessel functions, Math. Comp. 33 (1979), no. 147, 1049–1057. MR 528057
(80c:65048), http://dx.doi.org/10.1090/S00255718197905280575
 [2]
B. Gabutti & B. Minetti, "A new application of the discrete Laguerre polynomials in the numerical evaluation of the Hankel transform of a strongly decreasing even function," J. Comput. Phys., v. 42, 1981, pp. 277287.
 [3]
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)
 [4]
Richard
A. Handelsman and John
S. Lew, Asymptotic expansion of a class of integral transforms with
algebraically dominated kernels, J. Math. Anal. Appl.
35 (1971), 405–433. MR 0278012
(43 #3744)
 [5]
Fritz
Oberhettinger, Tables of Bessel transforms, SpringerVerlag,
New YorkHeidelberg, 1972. MR 0352888
(50 #5374)
 [6]
F.
W. J. Olver, Asymptotics and special functions, Academic Press
[A subsidiary of Harcourt Brace Jovanovich, Publishers], New YorkLondon,
1974. Computer Science and Applied Mathematics. MR 0435697
(55 #8655)
 [7]
K.
Soni, Asymptotic expansion of the Hankel transform with explicit
remainder terms, Quart. Appl. Math. 40 (1982/83),
no. 1, 1–14. MR 652045
(83d:41040)
 [8]
G.
N. Watson, A Treatise on the Theory of Bessel Functions,
Cambridge University Press, Cambridge, England; The Macmillan Company, New
York, 1944. MR
0010746 (6,64a)
 [9]
R.
Wong, Error bounds for asymptotic expansions of Hankel
transforms, SIAM J. Math. Anal. 7 (1976), no. 6,
799–808. MR 0415224
(54 #3315)
 [10]
R.
Wong, Error bounds for asymptotic expansions of integrals,
SIAM Rev. 22 (1980), no. 4, 401–435. MR 593856
(82a:41030), http://dx.doi.org/10.1137/1022086
 [11]
Ahmed
I. Zayed, Asymptotic expansions of some integral
transforms by using generalized functions, Trans. Amer. Math. Soc. 272 (1982), no. 2, 785–802. MR 662067
(83h:41033), http://dx.doi.org/10.1090/S00029947198206620679
 [1]
 B. Gabutti, "On high precision methods for computing integrals involving Bessel functions," Math. Comp., v. 33, 1979, pp. 10491057. MR 528057 (80c:65048)
 [2]
 B. Gabutti & B. Minetti, "A new application of the discrete Laguerre polynomials in the numerical evaluation of the Hankel transform of a strongly decreasing even function," J. Comput. Phys., v. 42, 1981, pp. 277287.
 [3]
 R. J. Glauber, Lectures in Theoretical Physics, vol. 1, Interscience, New York, 1959. MR 0107488 (21:6213)
 [4]
 R. A. Handelsman & J. S. Lew, "Asymptotic expansion of a class of integral transforms with algebraically dominated kernels," J. Math. Anal. Appl., v. 35, 1971, pp. 405433. MR 0278012 (43:3744)
 [5]
 F. Oberhettinger, Tables of Bessel Transforms, Springer, Berlin, 1972. MR 0352888 (50:5374)
 [6]
 F. W. J. Olver, Asymptotics and Special Functions, Academic Press, New York, 1974. MR 0435697 (55:8655)
 [7]
 K. Soni, "Asymptotic expansion of the Hankel transform with explicit remainder terms," Quart. Appl. Math., v. 40, 1982, pp. 114. MR 652045 (83d:41040)
 [8]
 G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed., Cambridge Univ. Press., Cambridge, 1944. MR 0010746 (6:64a)
 [9]
 R. Wong, "Error bounds for asymptotic expansions of Hankel transforms," SIAM J. Math. Anal., v. 7, 1976, pp. 799808. MR 0415224 (54:3315)
 [10]
 R. Wong, "Error bounds for asymptotic expansions of integrals," SIAM Rev., v. 22, 1980, pp. 401435. MR 593856 (82a:41030)
 [11]
 A. I. Zayed, "Asymptotic expansions of integral transforms by using generalized functions," Trans. Amer. Math. Soc., v. 272, 1982, pp. 785802. MR 662067 (83h:41033)
Similar Articles
Retrieve articles in Mathematics of Computation
with MSC:
41A60,
44A15,
65R10
Retrieve articles in all journals
with MSC:
41A60,
44A15,
65R10
Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198508049424
PII:
S 00255718(1985)08049424
Keywords:
Asymptotic expansion,
Hankel transform,
Bessel functions,
Laplace's method
Article copyright:
© Copyright 1985
American Mathematical Society
