I'm trying to calculate the two dimensional Fourier integral
$\iint \mathrm d\vec{R} \; (x^2 + y^2) \; e^{-2 \sqrt{ x^2 + y^2 + z^2}} \; e^{i\vec{K}\vec{R}} \;,$
with $\vec{R}=(x,y)$. Switching to polar coordinates I arrive at
$\int_0^{\infty} \mathrm d{R} \int_0^{2\pi} \mathrm d\Phi \; R^3 \; e^{-2 \sqrt{ R^2 + z^2}} \; e^{i K R \cos(\Phi)} \;.$
The angle integration yields a Bessel function and the remaining integral reads
$2 \pi \int_0^{\infty} \mathrm d{R} \; R^3 \; e^{-2 \sqrt{ R^2 + z^2}} \; J_0(K R) \;.$
Using the transfromations $x^2 = R^2 + z^2$ and $y = \frac{x}{|z|}$ this can also be written as
$2 \pi |z|^4 \int_{1}^{\infty} \mathrm d{y} \; y \; (y^2 - 1) \; e^{-2 |z| y} \; J_0\left(K |z| \sqrt{y^2 - 1}\right) \;.$
And this is where I'm stuck. I have no idea how to carry out the integration. I also checked various integral tables but without success. Can anyone give me pointers?
Edit 1
Following Andrew's advice I employed the expansion
$J_0\left(K |z| \sqrt{y^2 - 1}\right) = \sum_{m=0}^\infty \frac{(-1)^m}{m! \; \Gamma(m+1)} \left( \frac{K |z|}{2} \right)^{2m} (y^2 - 1)^m \;.$
The integral then becomes
$2 \pi |z|^4 \sum_{m=0}^\infty \frac{(-1)^m}{m! \; \Gamma(m+1)} \left( \frac{K |z|}{2} \right)^{2m} \int_{1}^{\infty} \mathrm d{y} \; y \; (y^2 - 1)^{m+1} \; e^{-2 |z| y} \;.$
I found the integral tabulated in Gradshteyn/Ryzhik (3.389 4.) and finally arrived at
$\frac{2 \pi}{\sqrt{\pi}} \sum_{m=0}^\infty \frac{(-1)^m \; (m+1)}{m!} \left( \frac{K}{2} \right)^{2m} |z|^{m+\frac{5}{2}} K_{m+\frac{5}{2}}\left( 2|z| \right) \;,$
where $K_{m+\frac{5}{2}}$ is the irregular modified cylindrical Bessel function of order $m+\frac{5}{2}$. Unfortunately it doesn't seem like this expression can be simplified any further. Moreover, I'd expect the convergence to be rather slow, since for $m\rightarrow\infty$ with $|z| > 1$ the only limiting term is the factorial $m!$ in the denominator.