A Note on Asymptotic Evaluation of Some Hankel Transforms
C. L. Frenzen, R. Wong · Mathematics of Computation · 1985
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.