quantum field theory - Questions about the Dyson equation

I'm studying finite temperature many-body perturbation theory, and am trying to understand The Dyson equation. In particular, I'm using Mattuck - A guide to Feynman diagrams in the many body problem.

There appear to be two types of Dyson equation: one which is valid in the special case of no external potential and with diagrams calculated in $(k,w)$-space,

\begin{equation} G(k,w) = \frac{1}{G_0 (k,w)^{-1} - \Sigma (k,w)}. \end{equation}

And one valid at other times which is an integral equation for $G$ which can be different depending on what space you're in and whether you have an external potential or not, but is something like

\begin{equation} G(k,\tau-\tau') = G_0(k,\tau-\tau') + \int_0^\beta d\tau_1 d\tau_2 G_0(k,\tau- \tau_2) \Sigma(k,\tau_2-\tau_1) G(k,\tau_1-\tau'). \end{equation}

So, my questions:

1. 
   

   What is an external potential? I can't find an actual definition in the book. At first I assumed it meant the part of the system you want to treat perturbatively, I.e. the Hamiltonian is $H=H_0+H_1$ where $H_1$ is small and can there for be considered a perturbation. But obviously this can't be the case otherwise there's no expansion to use in a Dyson equation.

   
   


2. 
   

   Why is the second Dyson equation useful? - How does it allow partial summations?

   
   



3. 
   

   Every book I've looked at seems to give slightly different rules for calculating the Feynman diagrams. Any thoughts on which is correct? (for example, Negele - Quantum many-particle systems is different to Mattuck).