Consider the integral

$$ \int_{\mathbb{R}^n}dx\,e^{-\frac12 x^TAx}=\frac{(2\pi)^{n/2}}{\sqrt{\det A}} $$

where $A=A^T$ is a symmetric $n\times n$ complex matrix with positive definite real part.

**Question:** can we explicitly calculate this integral (for complex $A$) without using analytic continuation?

**Motivation:** the standard proof of the above result starts off with a real $A$ and uses Cholesky decomposition to decouple the integral into $n$ one-dimensional Gaussian. (Diagonalizing $A$ with an orthogonal matrix with Jacobian $J=1$ essentially does the same.) Then one argues that, as long as the real part of $A$ remains positive definite, both sides are holomorphic and by analytic continuation the integral must have the value of the right hand side even for complex $A$, (see a good discussion on this here).

My question is motivated by the observation that for $n=1$ everything is scalar, $\det A=A$, and one can prove the above result for $\Re A>0$ using Cauchy theorem and contour integration with complex Jacobian $J=\sqrt{A}$, $\arg\sqrt{A}\in(-\pi/4,\pi/4)$. There is no need for analytic continuation (unless of course you want to go to $\Re A<0$), see the proof here.

So I wonder if there exists a direct proof for $n>1$ using some variant of Cauchy theorem in $\mathbb{C}^n$? Or some other way of integration with substitution using complex Jacobians, without having to rely on analytic continuation?