Dear Class,
For your homework, please numerically (brute force) evaluate the first 
self energy diagram in the Migdal Eliashberg approach. 

    (1)                        g^2           2w(k-k')
Sigma (k,iwn)  = T SUM   ------------- ------------------------
                   k iwn  iwn'-eps(k') (iwn-iwn')^2 - w(k-k')^2
                   

             k-k'  iwn-iwn'
             ---------
            /         \
          /             \
        /                 \
      /       k' iwn'       \
     o-----------<-----------o k iwn
     g                       g
      
choose some realistic parameters (or something close but which
will make the calculation simpler), and show that it weakly depends
on k, but strongly upon iwn.  What does strong or weak mean in this
context (i.e. compared to what)?

Then evaluate the first crossing graph discussed in class (you
may approximate the contribution to the self energy as k independent
if you wish).  

    (2)       
Sigma (k,iwn) =


             ---------
            /         \
          /             \
        /                 \
      /                     \
     o------<----o-----<-----o-----<-----o 
                  \                     /
                    \                 /
                      \             /
                        \         /
                         ---------         
                                                          (1)      
Show that this contribution is negligible compared to Sigma (k,iwn)
(you could also evaluate simplest the vertex correction and compare 
it to g instead).           

Please work on this together, and prepare one presentation of the 
results.