(a) Write down the Matsubara sum for the electron self-energy diagram for phonon interaction shown in Figure 8.17, and resolve the sum using the method of residues given here.

(b) Using the identities of this section, show that you get the same answer as in Section 8.10 when using analytic continuation, in the limit of low electron density.

Fig. 8.17 Diagram for the self-energy correction of a free electron due to phonon interaction.