Terminating Poincare asymptotic expansion of the Hankel transform of entire exponential type functions

Abstract

We perform an asymptotic evaluation of the Hankel transform, $\int_0^{\infty}J_{\nu}(\lambda x) f(x)\mathrm{d}x$, 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 Poincar{\'e} expansion without any trailing subdominant exponential terms. That is the Hankel transform evaluates exactly into a polynomial in inverse $\lambda$ as $\lambda$ approaches infinity.