# Generalized complex Gaussian integrals with linear terms

1. Aug 13, 2012

### jraynor

Hi there!

It's fairly simple to show the following identity of a multivariate Gaussian integral with linear terms, if $H$ is a Hermitean matrix and ${}^*$ denotes complex conjugation:

$$\int \frac{dx_1^* \, dx_1 ... dx_n^* \, dx_n}{(2\pi i)^n} e^{-\sum_{ij} x_i^* H_{ij} x_j + \sum_i (J^*_i x_i + J_i x_i^*)} = (\det H)^{-1} e^{\sum_{ij} J_i^* (H^{-1})_{ij} J_j}$$
(One just introduces the new variable $y_i = x_i - \sum_j (H^{-1})_{ij} J_j$, uses that $H$ is Hermitean and solves the remaining purely Gaussian integral.)

In the book "Quantum Many-Particle Systems" by Negele and Orland (p. 34), they say that this identity also holds if $H$ just has a positive Hermitean part but they don't show how to prove it...can someone help me? I've searched on Google and PF for a quite a while but found nothing.

Thanks a lot!