Mutlivariable integral, How to compute it?

Question

Can anybody please tell me, how to evaluate a multivariate integral with a gaussian weight function. $$\int_{-\infty}^{\infty} \mathrm{d}x_{1}\,\mathrm{d}x_{2}\,\mathrm{d}x_{3}\,\mathrm{d}x_{4}\dots\,\mathrm{d}x_{n}\exp\left(\sum_{j}x_{j}^2\right)\times f(x_{1},x_{2},x_{3},x_{4}....x_{n})$$ where $$f(x_{1},x_{2},x_{3},x_{4}....x_{n})=\prod_{j} \frac{1}{\sqrt{(1+i\,k(x_{j}^2-x_{j+1}^2)^2)}}$$

I need a hint to solve this integral.

I have found that, I can relate this to a partition function $\mathcal{Z}$. Any ideas to solve this Integral?

This post imported from StackExchange Physics at 2014-03-06 21:15 (UCT), posted by SE-user Sijo Joseph

Comments

@SijoJoseph: The integral does not converge for any $n$. Are you sure you typed it correctly?

This post imported from StackExchange Physics at 2014-03-06 21:15 (UCT), posted by SE-user DumpsterDoofus

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I am pretty sure that the exponential function should read $\exp\left(-\sum_{j} x_j^2\right)$. (More like a Gaussian distribution.) Then the integral converges because of the upper bound given by a Gaussian distribution. I think you are familiar with the integration trick for the Gaussian distribution (changing from Cartesian to Polar coordinates). I am afraid that this does not work here because of the term under the square root. :-(

This post imported from StackExchange Physics at 2014-03-06 21:15 (UCT), posted by SE-user Tobias

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As it was said by @Tobias, there is a missing sign in the exponential. Is the value of $k$ small ($k\ll1$) ? If that is so, an expansion of $f$ in a series of terms like $k^nx_i^px_j^q$ should allow estimating the integral.