I am trying to solve the following integral, $$\int dz_1d\bar{z}_1\cdots dz_nd\bar{z}_n\:\exp(-\sum_{i,j}\bar{z}_i A_{ij}z_j),$$ where $A_{ij}$ is an $n\times n$ hermitian matrix. I know how to do this for the usual Gaussian integral with real coordinates, namely, let $A = O^{-1}\cdot D\cdot O$ where $D$ is a diagonal matrix; then let $y_i = O_{ij}x_j$, and simplify until you find $D_{ii}y^2_i$ and use a previous Gaussian integral which gives an answer of $\pi^{n/2}(\det{A})^{-1/2}$. I would like to do something similar for the complex integral, where I would perform the same change of variables, but then check for poles and use Cauchy's Residue theorem.
This was an identity given to me to use for solving path integrals, but I would like to know where it comes from (the above explantaion originally came from Zee's textbook, but for this more 'complex' integral (literally), I would like some guidance).
I would attempt what I said above, but I am not confident in using Residue Theorem and performing contour integrals (hence I am here). Any help/hints are appreciated, thanks!
