Integration of a Gaussian multiplied by a Bessel function 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
 A: 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).
A: $\int_0^are^{-\frac{r^2}{h^2}}J_0\left(\dfrac{ikr}{z}\right)dr$
$=\int_0^are^{-\frac{r^2}{h^2}}I_0\left(\dfrac{kr}{z}\right)dr$
$=\int_0^are^{-\frac{r^2}{h^2}}\sum\limits_{m=0}^\infty\dfrac{k^{2m}r^{2m}}{4^mz^{2m}(m!)^2}dr$
$=\int_0^a\sum\limits_{m=0}^\infty\dfrac{k^{2m}r^{2m+1}e^{-\frac{r^2}{h^2}}}{4^mz^{2m}(m!)^2}dr$
$=\int_0^a\sum\limits_{m=0}^\infty\dfrac{k^{2m}r^{2m}e^{-\frac{r^2}{h^2}}}{2^{2m+1}z^{2m}(m!)^2}d(r^2)$
$=\int_0^{a^2}\sum\limits_{m=0}^\infty\dfrac{k^{2m}r^me^{-\frac{r}{h^2}}}{2^{2m+1}z^{2m}(m!)^2}dr$
$=\left[\sum\limits_{m=0}^\infty\sum\limits_{n=0}^m\dfrac{k^{2m}h^{2m-2n+2}r^ne^{-\frac{r}{h^2}}}{2^{2m+1}z^{2m}m!n!}\right]_0^{a^2}$ (can be obtained from http://en.wikipedia.org/wiki/List_of_integrals_of_exponential_functions)
$=\sum\limits_{m=0}^\infty\sum\limits_{n=0}^m\dfrac{k^{2m}h^{2m-2n+2}a^{2n}e^{-\frac{a^2}{h^2}}}{2^{2m+1}z^{2m}m!n!}-\sum\limits_{m=0}^\infty\dfrac{k^{2m}h^{2m+2}}{2^{2m+1}z^{2m}m!}$
$=\sum\limits_{m=0}^\infty\sum\limits_{n=0}^m\dfrac{k^{2m}h^{2m-2n+2}a^{2n}e^{-\frac{a^2}{h^2}}}{2^{2m+1}z^{2m}m!n!}-\dfrac{h^2e^\frac{k^2h^2}{4z^2}}{2}$
