I know this is a very naive question but during my quantum optics study, I often come across the following kind of integrals. Sometimes there is an identity to get it solved but other times there is no identity. Could someone guide me how these are solved like I really want to understand how we evaluate integrals with powers like $d^2 \alpha$?
$$a=\int d^2 \alpha \exp[-\alpha \gamma^* + \alpha^* \gamma]$$
$\alpha$ is a complex number here, so the best way to think about this is as $$ d^2\alpha = dx\,dy\,, $$ where $\alpha = x+i y$. In that case, $$ a=\int d^2 \alpha \exp[-\alpha \gamma^* + \alpha^* \gamma] =\int dx\,dy\,\exp\left(-(x+iy)\gamma^* + (x-y)\gamma\right)\,, $$ in which case you can just perform these integrals as real integrals. In this case, the integrals are straight-forward because they are just exponentials, but Gaussian integrals come up a lot in quantum optics as well. Conveniently, Gaussian integrals can be done by hand.
