Condensed matter problem with Feynman integral

[aaaa202]

Im working on deriving the final expression on the attached picture. The problem is that of a particle in a potential coupled to a "bath" of harmonic oscillators but I'm not sure how you arrive at the final expression. First of all, why are you allowed to assume that q is periodic on the interval [0,T^-1]? And secondly how should I use this? Should I plug in the Fourier series expansion of q? And in that case what should I do with the integral in dt' (which I suppose has been transformed to an integral in τ by t'->-iτ)

 

[DrDu]

Did you have a look on how the Matsubara formalism works, e.g. in Fetter/ Valecka?