We perform an asymptotic evaluation of the Hankel transform, \(\int _0^{\infty }J_{\nu }(\lambda x) f(x)\textrm{d}x\) , for arbitrarily large \(\lambda \) of an entire exponential type function, f(x), of type \(\tau \) by shifting the contour of integration in the complex plane. Under the situation that \(J_{\nu }(\lambda x)f(x)\) has an odd parity with respect to x and the condition that the asymptotic parameter \(\lambda \) is greater than the type \(\tau \) , we obtain an exactly terminating Poincaré expansion without any trailing subdominant exponential terms. That is, the Hankel transform evaluates exactly into a polynomial in inverse \(\lambda \) as \(\lambda \) approaches infinity.
