I'd like to verify the result of this integral, or find if I've made a mistake. In the following, $\mathbf x, \mathbf a, \mathbf b$ are all real vectors in $\mathrm R^3$. I do the following:
- group the dot products $\mathbf x \cdot \mathbf a$, $\mathbf x \cdot i \mathbf b$
- complete the square
- change variable and compute the (3D) gaussian integral
$\int d^3 \mathbf x \ \mathbf x e^{-(\mathbf x^2 + 2 \mathbf a \cdot \mathbf x + i 2 \mathbf b \cdot \mathbf x)}$
$=\int d^3 \mathbf x \ \mathbf x \ e^{-(\mathbf x + \tilde {\mathbf x}_0)^2}e^{\ \tilde {\mathbf x}_0^2}$
$=e^{\ \tilde {\mathbf x}_0^2}\int d^3 \mathbf x' \ (\mathbf x' - \tilde{\mathbf x}_0) \ e^{-\mathbf x'^2}$
$= e^{\ \tilde {\mathbf x}_0^2}\left[0 - \pi^{3/2}\tilde{\mathbf x}_0\right] $
where $\tilde{\mathbf x}_0 \equiv \mathbf a + i \mathbf b$ and $\mathbf x' \equiv \mathbf x + \tilde{\mathbf x}_0$.
I think this should be as trivial as the real case, but I'm a little uneasy about making assumption in a complex vector space. In particular, wikipedia suggests that the dot product of complex vectors is given as: $\mathbf{a} \cdot \mathbf{b} = \sum{a_i \overline{b_i}}$. I think this should not apply, as all of my vectors live in the same real space (despite the imaginary unit making one term imaginary). Your help/verification is much appreciated.
