Using the finite-temperature Green’s functions given in this section, compute the real self-energy correction for a free electron that corresponds to the diagram shown in Figure 8.18, known as a “tadpole” diagram, for optical phonons with constant energy h̄ω_LO. To have a sensible answer, you will need to multiply your integral by a factor e^iηE, to make the integrand vanish in the limit of large imaginary E in the upper half of the complex plane, and take the limit η→ 0.

Fig. 8.18 k P k Another diagram for the self-energy correction of a free electron due to phonon interaction.