A note on asymptotic evaluation of some Hankel transforms

Abstract

Asymptotic behavior of the integral \[ I f ( w ) = ∫ 0 ∞ e − x 2 J 0 ( w x ) f ( x 2 ) x d x {I_f}(w) = \int _0^\infty {{e^{ - {x^2}}}{J_0}(wx)f({x^2})\;x\;dx} \] is investigated, where J 0 ( x ) {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 ) {I_f}(w) decays exponentially like e − γ w 2 {e^{ - \gamma {w^2}}} , γ > 0 \gamma > 0 , when f ( z ) f(z) is an entire function subject to a suitable growth condition. A complete asymptotic expansion is obtained when f ( z ) f(z) is a meromorphic function satisfying the same growth condition. Similar results are given when f ( z ) f(z) has some specific branch point singularities.