For a problem in magnetism, I have to calculate the expectation value (average) for the variable $S_x=\cos \theta$ such that
$$\langle \cos\theta\rangle = \mathfrak{R}(\langle e^{i\theta} \rangle)= \mathfrak{R}(e^{\langle i \theta\rangle}\cdot e^{-\frac{1}{2}\langle \theta^2\rangle})=e^{-\frac{1}{2}\langle \theta^2 \rangle} $$
which of course only works if one knows, that:
For any set of Gaussian distributed variables $\{\theta_i\}$ we have the indentity $$ \langle \exp\sum_ia_i\theta_i\rangle=\exp\bigg(\sum_{i,j}\frac{a_i a_j}{2}\langle \theta_i \theta_j \rangle \bigg) $$
which in our case is just $$\langle e^{\theta}\rangle=e^{\langle \theta\rangle+\frac{1}{2}\langle\theta^2\rangle}$$
Now, how do I derive that, or alternativey, does this identity have a name?