A note on asymptotic evaluation of some Hankel transforms

Frenzen, C. L.; Wong, R.

doi:10.1090/S0025-5718-1985-0804942-4

A note on asymptotic evaluation of some Hankel transforms
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by C. L. Frenzen and R. Wong PDF

Math. Comp. 45 (1985), 537-548 Request permission

Abstract:

Asymptotic behavior of the integral \[ {I_f}(w) = \int _0^\infty {{e^{ - {x^2}}}{J_0}(wx)f({x^2})\;x\;dx} \] 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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