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How to compute the following integral?

$$\int_{\|x\|^2\leq R} \exp(-x^\ast G x+2\mathcal{Re}(x^\ast a)) \,dx,$$ where $x$ is an $M \times 1$ vector ($M\gg 1$), $G$ is a positive definite matrix, and $a$ is an arbitrary vector.

This does not seem to be an instance of the Itzykson Zuber integral.

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I propose to diagonalize the matrix $G$, making a change of variables by an orthogonal matrix, and next separe the variables; I am reduced so to calculate: $$ \int_{|x| \leq r} \exp ( a x^2 + bx) dx $$ but I have no real calculus for such integrals except if $r=+\infty$ because: $$ \int_{\mathbb R} \exp(-\pi x^2) dx =1 $$

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    $\begingroup$ How can you "separate the variables" when they are coupled by the condition $\sum_{n}x_n^2<R$ ? $\endgroup$
    – Carlo Beenakker
    Commented Jan 19, 2017 at 22:08
  • $\begingroup$ I propose to make a recurrence with $\sum_{k=1}^{n-1} x_k^2 < 1 - x_n^2$. $\endgroup$
    – Antoine Balan
    Commented Jan 19, 2017 at 22:13

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