Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Asymptotic expansion of $ \int\sp {\pi/2}\sb 0J\sp 2\sb \nu(\lambda\,{\rm cos}\,\theta)\,d\theta$

Author: R. Wong
Journal: Math. Comp. 50 (1988), 229-234
MSC: Primary 41A60; Secondary 33A40
MathSciNet review: 917830
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: An asymptotic expansion is obtained, as $ \lambda \to + \infty $, for the integral

$\displaystyle I(\lambda ) = \int_0^{\pi /2} {J_v^2(\lambda \cos \theta )\;d\theta ,} $

where $ {J_v}(t)$ is the Bessel function of the first kind and $ v > - \tfrac{1}{2}$. This integral arises in studies of crystallography and diffraction theory. We show in particular that $ I(\lambda ) \sim \ln \lambda /\lambda \pi $.

References [Enhancements On Off] (What's this?)

  • [1] N. Bleistein & R. A. Handelsman, Asymptotic Expansions of Integrals, Holt, Rinehart and Winston, New York, 1975.
  • [2] B. C. Carlson & J. L. Gustafson, "Asymptotic expansion of the first elliptic integral," SIAM J. Math. Anal., v. 16, 1985, pp. 1072-1092. MR 800798 (87d:33002)
  • [3] O. I. Marichev, Handbook of Integral Transforms of Higher Transcendental Functions, Theory and Algorithmic Tables, Ellis Horwood Ltd., West Sussex, England, 1983. MR 689711 (84f:00017)
  • [4] F. W. J. Olver, Asymptotics and Special Functions, Academic Press, New York, 1974. MR 0435697 (55:8655)
  • [5] B. J. Stoyanov &. R. A. Farrell, "On the asymptotic evaluation of $ \smallint _0^{\pi /2}J_0^2(\lambda \sin x)\;dx$," Math. Comp., v. 49, 1987, pp. 275-279. MR 890269 (88e:41067)
  • [6] R. Wong, "Explicit error terms for asymptotic expansions of Mellin convolutions," J. Math. Anal. Appl., v. 72, 1979, pp. 740-756. MR 559402 (81a:41049)
  • [7] R. Wong, "Error bounds for asymptotic expansions of integrals," SIAM Rev., v. 22, 1980, pp. 401-435. MR 593856 (82a:41030)
  • [8] R. Wong, Applications of Some Recent Results in Asymptotic Expansions, Proc. 12th Winnipeg Conf. on Numerical Methods of Computing, Congress. Numer., v. 37, 1983, pp. 145-182. MR 703584 (84f:41031)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 41A60, 33A40

Retrieve articles in all journals with MSC: 41A60, 33A40

Additional Information

Keywords: Asymptotic expansion, Bessel functions, Mellin transforms
Article copyright: © Copyright 1988 American Mathematical Society

American Mathematical Society