deletions | additions       diff --git a/untitled.tex b/untitled.tex  index 3b5705b..0e6f464 100644  --- a/untitled.tex  +++ b/untitled.tex 
...
Inserting the expressions for the self-energies we get  \begin{align*}  &N^>(\omega)=(e^2/h)\sum_{k\beta,q\gamma, \nu\mu} \int \frac{d\epsilon}{2\pi}   [G_{\beta\nu}^r(\epsilon) \Sigma_{0,\nu\gamma}^>(\epsilon) G_{\gamma\mu}^r(\omega+\epsilon+) \Sigma_{0,\mu\beta}^{<,h}(\omega+\epsilon+)]+[G_{\beta\nu}^r(\epsilon) \Sigma_{0,\mu\beta}^{<,h}(\omega+\epsilon)]+[G_{\beta\nu}^r(\epsilon)  \Sigma_{0,\nu\gamma}^>(\epsilon) G_{\gamma\mu}^<(\omega+\epsilon) \Sigma_{0,\mu\beta}^{a,h}(\omega+\epsilon)] \\   &+[G_{\beta\nu}^>(\epsilon) \Sigma_{0,\nu\gamma}^a(\epsilon) G_{\gamma\mu}^r(\omega+\epsilon)\Sigma_{0,\mu\beta}^{<,h}(\omega+\epsilon)]+[G_{\beta\nu}^>(\epsilon) \Sigma_{0,\nu\gamma}^a(\epsilon) G_{\gamma\mu}^<(\omega+\epsilon+) G_{\gamma\mu}^<(\omega+\epsilon)  \Sigma_{0,\mu\beta}^{a,h}(\omega+\epsilon)] \end{align*}  \begin{align*}  &N^>(\omega)=(4e^2/h)\sum_{k\beta,q\gamma, \nu\mu} \int \frac{d\epsilon}{2\pi} \times \Biggr\{  [G_{\beta\nu}^r(\epsilon) \Gamma_{\nu\gamma} G_{\gamma\mu}^r(\epsilon+\omega) G_{\gamma\mu}^r(\omega+\epsilon)  \Gamma_{\mu\beta} (1-f_e(\epsilon) f_{h}(\epsilon+\omega)] f_{h}(\omega+\epsilon)]  \\ &+\sum_{\lambda\delta}[G_{\beta\nu}^r(\epsilon) \Gamma_{\nu\gamma} G_{\gamma\lambda}^r(\omega+\epsilon) \Gamma_{\lambda\delta}G^a_{\delta\mu}(\epsilon+\omega)[i\Gamma_{\mu\beta}](1-f_e(\epsilon)(f_{h}(\epsilon+\omega)+f_e(\epsilon+\omega))] \Gamma_{\lambda\delta}G^a_{\delta\mu}(\omega+\epsilon)[i\Gamma_{\mu\beta}](1-f_e(\epsilon)(f_{h}(\epsilon+\omega)+f_e(\omega+\epsilon))]  \\ &+\sum_{\lambda\delta} G_{\beta\lambda}^r(\epsilon)\Gamma_{\lambda\delta} G_{\delta\nu}^a(\epsilon)[i\Gamma_{\nu\gamma}] G_{\gamma\mu}^r(\omega+\epsilon)(1-f_{e}(\omega+\epsilon)+1-f_{h}(\omega+\epsilon))f_h(\epsilon)\\   &+\sum_{\lambda\delta\theta\tau}[G_{\beta\lambda}^r(\epsilon)\Gamma_{\lambda\delta} G_{\delta\nu}^a(\epsilon)[i\Gamma_{\nu\gamma}] G_{\gamma\theta}^r(\omega+\epsilon)\Gamma_{\theta\tau} G_{\tau\mu}^a(\omega+\epsilon)[i\Gamma_{\mu\beta}](1-f_{e}(\epsilon)+1-f_{h}(\epsilon))(f_{e}(\epsilon+\omega)+f_{h}(\epsilon+\omega))] G_{\tau\mu}^a(\omega+\epsilon)[i\Gamma_{\mu\beta}](1-f_{e}(\epsilon)+1-f_{h}(\epsilon))(f_{e}(\omega+\epsilon+)+f_{h}(\epsilon+\omega))]  \Biggr\} \end{align*}  Finally, to obtain $N<(t,t)$ we just change $(1-f)\rightarrow f$, and $f\rightarrow (1-f)$.   \begin{align*}