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A Note on Asymptotic Evaluation of Some Hankel Transforms

C. L. Frenzen and R. Wong
Mathematics of Computation
Vol. 45, No. 172 (Oct., 1985), pp. 537-548
DOI: 10.2307/2008143
Stable URL: http://www.jstor.org/stable/2008143
Page Count: 12
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A Note on Asymptotic Evaluation of Some Hankel Transforms
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Abstract

Asymptotic behavior of the integral $$I_f(w) = \int^\infty_0 e^{-x^{2}}J_0(wx)f(x^2)x dx$$ 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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