Hi everyone. The problem I have to face is to perform a taylor series expansion of the integral(adsbygoogle = window.adsbygoogle || []).push({});

[tex] \int_{-\infty}^{\infty}\frac{e^{-\sum_{i}\frac{x_{i}^{2}}{2\epsilon}}}{\sqrt{2\pi\epsilon}^{N}}\cdot e^{f(\{x\})}dx_{i}\ldots dx_{N} [/tex]

with respect to variance [tex]\epsilon[/tex]. I find some difficulties because the function in the integral has a singularity in [tex]\epsilon=0[/tex] and this makes calculation a bit strange, but it must be possible to calculate the taylor series coefficients (I need them only at second order). In fact, for zero variance the entire integral reduces to f(0) besause the gaussian function becomes a Dirac's delta distribution centered in 0.

The function f is a polynomial of degree 4 in all x variables and contains only even powers and terms like x_i x_j, so it is not possible to factor it into product of single terms dependending only from i index.

I have tried to expand f in series and use the moment generating function of the gaussian, but neither this trick seems to work. Nor I have had much luck expanding the gaussian. Anyone has some ideas? Thank you.

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# Taylor expansion of gaussian integral with respect to variance

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