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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
DOI: https://doi.org/10.1090/S0025-5718-1988-0917830-2
MathSciNet review: 917830
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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 $.

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1988-0917830-2
Keywords: Asymptotic expansion, Bessel functions, Mellin transforms
Article copyright: © Copyright 1988 American Mathematical Society