TITLE: Integration of a Gaussian multiplied by a Bessel function
QUESTION [5 upvotes]: I currently have a hard time figuring out the following integral:

Integrate[r*Exp[-r^2/h^2]*BesselJ[0,i*k*r/z],{r,0,a}]

I've written it down in the Mathematica typeset and hope you can help me!
Thanks 

$$\int_0^a r e^{-\frac{r^2}{h^2}} J_0\left(\dfrac{i k r}{z}\right) \, dr$$

I have tried integration by parts, but I cannot get rid of the resulting integral either. I can keep doing integration by parts, but I will also have an integral I cannot solve

REPLY [3 votes]: A somewhat related integral (over infinite interval)
$$
  \int_0^\infty J_0(\alpha t)\exp(-\gamma^2 t^2)t\,dt=\frac{1}{2\gamma^2}\exp\bigg(-\frac{\alpha^2}{4\gamma^2}\bigg)
$$
was used by F. W. Crawford and J. A Tataronis, Journal of Applied Physics, Volume 36, pp. 2930-2934 (1965), in their equation (8) (correcting for a small typo). They cite A. Erdélyi, "Higher Transcendental Functions" (McGraw-Hill Book Company, Inc., New York, 1953).