Hi I'm Reading a notebook on "light atom interactions" There is this calculation I don't understand of average power - $$P=\overline{-\boldsymbol{E} \cdot \frac{\mathrm{d} D}{\mathrm{~d} t}}=\ldots=-\frac{\omega}{2} \operatorname{Im}\left(\tilde{D} \tilde{E}^*\right)$$ where $\boldsymbol{D}(t)=\operatorname{Re}\left[\tilde{D}(t) \hat{\epsilon} \mathrm{e}^{-\mathrm{i} \omega t}\right]$ and $\boldsymbol{E}(t)=\operatorname{Re}\left[\tilde{E}(t) \hat{\epsilon} \mathrm{e}^{-\mathrm{i} \omega t}\right]$
I want to understand the in between calculation.
And I found in another place something similar, an assertion that if a(t), b(t) has the same single frequency than - $$\begin{aligned} & a(t)=\Re\left[A e^{i \omega t}\right] \\ & b(t)=\Re\left[B e^{i \omega t}\right] \end{aligned} \Rightarrow\langle a(t) b(t)\rangle_{\frac{2 \pi}{\omega}}=\frac{1}{2} \Re\left[\left[A B^*\right]\right.$$
