Suppose I have some probability distribution in $\mathbb{R}^n$ and I want to calculate the expectation value of $$\left\langle e^{-i\vec{q}\cdot\vec{r}}\right\rangle^n.$$ My professor says that the answer is $$e^{n\left(-i\vec{q}\cdot\left\langle\vec{r}\right\rangle-\frac{1}{2}q_\mu q_\nu C^{\mu\nu}_2+O\left(q^3\right)\right)},\qquad C_2^{\mu\nu}:=\left\langle\left(r^\mu-\left\langle r^\mu\right\rangle\right)\left(r^\nu-\left\langle r^\nu\right\rangle\right)\right\rangle$$ and one gets it simply by expanding the exponential, distributing the averages and then regrouping. I am however unable to do this. Can somebody help me with this?


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