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&=& \frac{-4e^2}{\hbar^2}\int_{-\infty}^\infty \frac{d\epsilon}{2\pi} \sum_{\beta\gamma\alpha\delta} [G^r_{\beta \alpha}(\epsilon) \Gamma_{\alpha\delta} G^a_{\delta \gamma}(\epsilon) \Gamma_{\alpha\delta} [(1-f_{e}(\epsilon))f_{h}(\epsilon+\omega)+(1-f_{h}(\epsilon)) f_{h}(\epsilon+\omega)]  \end{eqnarray}  \begin{eqnarray}  S^{>,2}(\omega) = \frac{-4e^2}{\hbar^2} \int_{-\infty}^\infty \frac{d\epsilon}{2\pi} \sum_{\beta\gamma\alpha\delta} G^r_{\beta \alpha}(\epsilon) \Gamma_{\alpha\delta}(\epsilon) G^a_{\delta \gamma}(\epsilon) [-i\Gamma_{\gamma\gamma}] G^{r}_{\gamma\beta}(\omega+\epsilon) \Gamma_{\beta\beta}[(1-f_{e}(\epsilon))f_{h}(\epsilon)+(1-f_{h}(\epsilon)) f_{h}(\epsilon)] \Gamma_{\beta\beta}[(1-f_{e}(\epsilon))f_{h}(\epsilon+\omega)+(1-f_{h}(\epsilon)) f_{h}(\epsilon+\omega)]  \end{eqnarray}  \begin{eqnarray}  S^{>,3}(\omega)= \frac{-4e^2}{\hbar^2} \frac{2i e^2}{\hbar^2}  \int_{-\infty}^\infty \frac{d\epsilon}{2\pi} \sum_{\beta\gamma\alpha\delta} G^r_{\beta \alpha}(\epsilon) \Gamma_{\alpha\delta} G^a_{\delta \gamma}(\epsilon) [-i\Gamma_{\gamma\gamma}] G^{<}_{\gamma\beta}(\omega+\epsilon) [i\Gamma_{\beta\beta}][(1-f_{e}(\epsilon))+(1-f_{h}(\epsilon))] \end{eqnarray}  We replace $G^{<}_{\gamma\beta}(\omega+\epsilon) = 2i G^{r}_{\gamma\beta}(\omega+\epsilon)(f_{e}(\omega+\epsilon)+f_{h}(\omega+\epsilon))G^{a}_{\gamma\beta}(\omega+\epsilon)$, then  \begin{eqnarray}  S^{>,3}(\omega)= \frac{-4 e^2}{\hbar^2} \int_{-\infty}^\infty \frac{d\epsilon}{2\pi} \sum_{\beta\gamma\alpha\delta} G^r_{\beta \alpha}(\epsilon) \Gamma_{\alpha\delta} G^a_{\delta \gamma}(\epsilon) [-i\Gamma_{\gamma\gamma}] G^{r}_{\gamma\beta}(\omega+\epsilon) G^{a}_{\gamma\beta}(\omega+\epsilon) [i\Gamma_{\beta\beta}][(1-f_{e}(\epsilon))+(1-f_{h}(\epsilon))] [f_{e}(\epsilon+\omega)+f_{h}(\epsilon+\omega)]  \end{eqnarray}  We replace $G^{<}_{\gamma\beta}(\omega+\epsilon) = 2i (G^{r}_{\gamma\beta}(\omega+\epsilon)(f_{e}(\omega+\epsilon)+f_{h}(\omega+\epsilon))G^{a}_{\gamma\beta}(\omega+\epsilon)$