deletions | additions       diff --git a/untitled.tex b/untitled.tex  index 7a54437..144ad4d 100644  --- a/untitled.tex  +++ b/untitled.tex 
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\end{eqnarray}  where  \begin{eqnarray}  G^>_{\beta q}(\omega+\epsilon) q}(\epsilon)  = \sum_\gamma [G_{\beta\gamma}^r(\omega) [G_{\beta\gamma}^r(\epsilon)  V_{\gamma q} g^{>}_{q}(\omega)+ G_{\beta\gamma}^>(\omega) g^{>}_{q}(\epsilon)+ G_{\beta\gamma}^>(\epsilon)  V_{\gamma q} g^{a}_{q}(\omega)] g^{a}_{q}(\epsilon)]  \end{eqnarray}  \begin{eqnarray}  G^{<,h}_{\gamma k}(\omega+\epsilon) = \sum_\beta [G_{\gamma\beta}^r(\omega+\epsilon) V^*_{\beta k} g^{<,h}_{k}(\omega)+ G_{\gamma\beta}^<(\omega+\epsilon) V^*_{\beta k} g^{a,h}_{k}(\omega+\epsilon)] 
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Then, we have  \begin{eqnarray}  &&N^>(\omega)=(e^2/h)\sum_{k\beta,q\gamma, \nu\mu} \int \frac{d\epsilon}{2\pi} V_{\beta k} V^*_{\gamma q}   [G_{\beta\nu}^r(\omega+\epsilon) [G_{\beta\nu}^r(\epsilon)  V_{\nu q} g^{>}_{q}(\omega+\epsilon)+ G_{\beta\nu}^>(\omega+\epsilon) G_{\beta\nu}^>(\epsilon)  V_{\nu q} g^{a}_{q}(\omega+\epsilon)] \\ \nonumber  &&[G_{\gamma\mu}^r(\epsilon) V^*_{\mu k} g^{<,h}_{k}(\epsilon)+ g^{<,h}_{k}(\epsilon+\omega)+  G_{\gamma\mu}^<(\epsilon) V^*_{\mu k} g^{a,h}_{k}(\epsilon)] g^{a,h}_{k}(\epsilon+\omega)]  \end{eqnarray}  \begin{eqnarray}  N^>(\omega)&=&(e^2/h)\sum_{k\beta,q\gamma, \nu\mu} \int \frac{d\epsilon}{2\pi}   [G_{\beta\nu}^r(\omega+\epsilon) [G_{\beta\nu}^r(\epsilon)  V_{\nu q} g^{>}_{q}(\omega+\epsilon) V^*_{\gamma q} G_{\gamma\mu}^r(\epslion) V^*_{\mu k} g^{<,h}_{k}(\epsilon) g^{<,h}_{k}(\omega+\epsilon)  V_{\beta k}] \\ \nonumber  &&[G_{\beta\nu}^r(\omega+\epsilon) &&[G_{\beta\nu}^r(epsilon)  V_{\nu q} g^{>}_{q}(\omega+\epsilon) V^*_{\gamma q} G_{\gamma\mu}^<(\epsilon) V^*_{\mu k} g^{a,h}_{k}(\epsilon) g^{a,h}_{k}(\omega+\epsilon)  V_{\beta k}] \\ \nonumber  &&[G_{\beta\nu}^>(\omega+\epsilon) &&[G_{\beta\nu}^>(\epsilon)  V_{\nu q} g^{a}_{q}(\omega+\epsilon) V^*_{\gamma q} G_{\gamma\mu}^r(\epsilon) V^*_{\mu k} g^{<,h}_{k}(\epsilon) g^{<,h}_{k}(\omega+\epsilon)  V_{\beta k}] \\ \nonumber  &&[G_{\beta\nu}^>(\omega+\epsilon) &&[G_{\beta\nu}^>(\epsilon)  V_{\nu q} g^{a}_{q}(\omega+\epsilon) V^*_{\gamma q} G_{\gamma\mu}^<(\epsilon) V^*_{\mu k} g^{a,h}_{k}(\epsilon) V_{\beta k}] \end{eqnarray}  Inserting the expressions for the self-energies we get  \begin{eqnarray}