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I know, lots of threads and online information about Gaussian integrals. But still, I couldn't find what I am looking for: Is there a general formula for the integral

[tex] \int_{\mathbb{R}^d} d^d y \left|\vec{y}\right| \exp(-\alpha \vec{y}^2)[/tex]

where y is a vector of arbitrary dimension d and alpha is a complex number? It could be done in hyperspherical coordinates but its cumbersome. So I wanted to look up a general expression but couldn't find one. Can anyone help?

Thanks a lot

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# Multidim. Gaussian integral with linear term

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