Fourier transform of a Bessel function multiplied by a Gaussian

Abstract

An analytical result is given for the exact evaluation of an integral which arises in the analysis of acoustic radiation from wave packet sources: $I_{mn}(\beta ,q)={\int }_{-\infty }^{\infty }{e}^{-{\beta }^2{x}^2-iqx}{x}^{m+1/2}{J}_{n+1/2}(x)dx,$ where $m$ and $n$ are non-negative integers, and $J_{n+1/2}(\cdot )$ is a Bessel function of order $n+1/2$.

Submitted to Journal of Physics A