I am having some difficulty doing integrals like
$$ \int_{-\infty}^\infty e^{2|\alpha - \gamma|^2} e^{-|\gamma|^2/\mu} d^2\gamma $$
where $\alpha$ and $\gamma$ are complex numbers, $\mu$ is real, and $d^2\gamma = d\text{Re}(\alpha)\,d\text{Im}(\alpha)$ up to a factor of $\pi$. I see that this is some kind of Gaussian integral and I expect (and hope) to get a Gaussian out.
My question is: should I be doing this integral by writing $\alpha$ and $\gamma$ in terms of real and imaginary parts, and working from there, or is there an easier way to do it?
If somebody could point me to a resource that discusses integrals like the above, or show an easier way to do this integral, that would be appreciated.