I want to calculate the Fourier transform, but I don't understand the equation transformation.
Let $A$ be a real symmetric non degenerate matrix with eigenvalues $\lambda_1, \dots \lambda_n$.
\begin{align*}\int e^{-\xi x } e^{ i x^t A x - \epsilon x^2} dx &= \int e^{ - \xi Oy}e^{y^t O^t A O y - \epsilon (Oy)^2} dy \\ &= \int e^{ - i \xi Oy} e^{i \Sigma j \lambda_j y_j ^2 - \epsilon y^2} dy \\&= \prod_j \int_{\mathbb R} \dots\int_{\mathbb R} e^{ -i\xi O y}e^{i(\lambda_j - \epsilon)y_j^2} dy_1 \dots dy_n \\&= \prod_j \sqrt{\frac{\pi}{\epsilon-i \lambda_j}}e^{- \frac{(O^t \xi)_j^2}{4(\epsilon - i \lambda_j)}} \\&\rightarrow \prod_j \sqrt{\frac{\pi}{-i \lambda_j}}e^{- \frac{(O^t \xi)_j^2}{\lambda_j}} \quad (\epsilon \rightarrow 0) \\&=\pi^{\frac{n}{2}} e^{\textrm{sgn}A i \frac{\pi}{4}} \frac{1}{\sqrt{\det A}} e^{-i \frac{\xi^t A^{-1} \xi}{4}} \end{align*}
Where, $O$ is the matrix that diagonalizes $A$.
I referred to Lemma 1.6 in https://bpb-us-e1.wpmucdn.com/blogs.rice.edu/dist/8/4754/files/2016/09/dHHU_Chap1-14tliki.pdf
I don't understand the final equation transformation.
\begin{align*}\prod_j e^{- \frac{(O^t \xi)_j^2}{\lambda_j}} = e^{-i \frac{\xi^t A^{-1} \xi}{4}} \end{align*}
So, the problem is:
\begin{align*} \prod_j \frac{1}{\sqrt{- i \lambda_j}}= e^{\textrm{sgn} A i \frac{\pi}{4}} \frac{1}{\sqrt{\det A}} \end{align*}
$\textrm{sgn} A$ may be the number of positive eigenvalues minus the number of negative eigenvalues of $A$.