Linear-source multivariate Gaussian integral (${\int }_{{\mathbb{R}}^n}{e}^{-u^{T}Au/2+b^{T}u}du=(2\pi {)}^{n/2}(\det A{)}^{-1/2}{e}^{b^T{A}^{-1}b/2}$)

For real symmetric positive-definite $A$, complete the square with $v=u-A^{-1}b$. Orthogonal diagonalization reduces the remaining integral to the product of one-dimensional Gaussian integrals, proving the displayed expression. Positivity ensures convergence and the positive determinant square root. Replacing matrices by regulated field kernels gives the Gaussian functional integral; an oscillatory Minkowski integral instead requires a causal convergence prescription.