Summary: Asymptotic behavior of the integral
is investigated, where is the Bessel function of the first kind and w is a large positive parameter. It is shown that decays exponentially like , , 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.