Two-Dimensional Gaussian integral in complex coordinates. 
(4) A two-dimensional Gaussian integral in complex coordinates:
$$
\begin{aligned}
&\frac{1}{\pi} \int \exp \left(-\mu \alpha^{2}-\nu \alpha^{* 2}-z^{*} \alpha+z \alpha^{*}-|\alpha|^{2}\right) \mathrm{d}^{2} \alpha \\
&\quad=\frac{1}{\tau} \exp \left[-\left(\mu z^{2}+\nu z^{* 2}+z z^{*}\right) / \tau^{2}\right], \quad \tau=\sqrt{1-4 \mu \nu}, \quad \mu+\nu<1 .
\end{aligned}
$$

(Transcribed from picture)
Can anyone suggest some text reference where I can get to learn about the result and about Two-dimensional Gaussian integral in complex coordinates in general? Any Mathematical Physics text reference is preferred. I am having problems dealing with them in my study of Quantum Optics.
 A: *

*It follows from my Phys.SE answer here that OP's Gaussian integral becomes
$$ \begin{align} I~:=~&\int_{\mathbb{R}^2} \!  \frac{\mathrm{d}{\rm Re}\alpha \wedge \mathrm{d}{\rm Im}\alpha}{\pi}~ 
\exp\left\{-\mu \alpha^{2}-\nu \alpha^{* 2}-z^{*} \alpha+z \alpha^{*}-|\alpha|^{2}\right\}\cr
~=~&\int_{\mathbb{C}} \!  \frac{\mathrm{d}\alpha^{\ast} \wedge \mathrm{d}\alpha}{2\pi i}~ 
\exp\left\{ -\frac{1}{2}A^TSA +Z^TA\right\}\cr
~=~&\sqrt{\frac{-1}{\det(S)}}\exp\left\{\frac{1}{2}Z^TS^{-1}Z \right\}\cr
~=~&\frac{1}{\tau} \exp \left\{-\frac{\mu z^2+\nu z^{* 2}+z z^{*}}{\tau^2}\right\}.
\end{align}$$


*Here we have defined
$$ \begin{align} 
A~:=~&\begin{pmatrix} \alpha   \cr \alpha^{\ast} \end{pmatrix}, \cr
Z~:=~&\begin{pmatrix} -z^{\ast}   \cr z \end{pmatrix}, \cr
J~:=~&\begin{pmatrix} 1 & i \cr 1 & -i \end{pmatrix}, \cr 
S~:=~&\begin{pmatrix} 2\mu & 1 \cr 1 & 2\nu \end{pmatrix}, \cr
\tau^2~:=~&-\det(S), \cr
S^{-1}~=~&\frac{1}{\tau^2}\begin{pmatrix} -2\nu & 1 \cr 1 & -2\mu \end{pmatrix}, 
\end{align}$$


*Moreover, the Gaussian integral $I$ is convergent if ${\rm Re}(J^TSJ)>0$ is positive definite.
