# multivariate gaussian integral with linear constraint

I am interested in the calculation of the multivariate gaussian integral $$\begin{equation*} \int_{\mathbb R^N} \mathrm d x_1\cdots \mathrm d x_N \; \prod_{j=1}^Ne^{-\frac{x_j^2}{2\sigma_j^2}}\delta\left(x-\sum_{l=1}^N c_l x_l\right) \end{equation*}$$ for a set of $$c_l\in\mathbb R$$. Is there a way to solve the integral with this linear constraint? If so, how? As a simple separation of variables does not apply anymore, I would guess that a smart change of variables could be a way to tackle the calculation.

Typically one uses the representation $$\delta(f(x))={1\over 2\pi}\int dk e^{i k f(x)}$$
• I see, that's convenient. Then I guess it's a matter of closing the square for the $N$ integrals in $x_j$; there's only an additional integration over a simple exponential in $k$ – Graz Nov 14 '19 at 9:32
• Actually, you will find that the integration will be an exponential in $k^2$ ( a Gaussian) – user619894 Nov 14 '19 at 10:00