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

[tex]\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} [/tex]

(One just introduces the new variable [itex]y_i = x_i - \sum_j (H^{-1})_{ij} J_j[/itex], uses that [itex]H[/itex] 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 [itex]H[/itex] 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!

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# Generalized complex Gaussian integrals with linear terms

Can you offer guidance or do you also need help?

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