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Mathematics of Computation

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A note on asymptotic evaluation of some Hankel transforms

Authors: C. L. Frenzen and R. Wong
Journal: Math. Comp. 45 (1985), 537-548
MSC: Primary 41A60; Secondary 44A15, 65R10
MathSciNet review: 804942
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Abstract: Asymptotic behavior of the integral

$\displaystyle {I_f}(w) = \int_0^\infty {{e^{ - {x^2}}}{J_0}(wx)f({x^2})\;x\;dx} $

is investigated, where $ {J_0}(x)$ is the Bessel function of the first kind and w is a large positive parameter. It is shown that $ {I_f}(w)$ decays exponentially like $ {e^{ - \gamma {w^2}}}$, $ \gamma > 0$, when $ f(z)$ is an entire function subject to a suitable growth condition. A complete asymptotic expansion is obtained when $ f(z)$ is a meromorphic function satisfying the same growth condition. Similar results are given when $ f(z)$ has some specific branch point singularities.

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Keywords: Asymptotic expansion, Hankel transform, Bessel functions, Laplace's method
Article copyright: © Copyright 1985 American Mathematical Society

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