Past exam of the theoretical physics II course of the University of Cambridge / 2014 / Paper 1 / 2 / e

Take $\alpha =I/(i\hslash T)$ in the supplied Poisson summation identity, initially with a convergent regulator. Its left side is the winding sum. Multiplying its right side by the prefactor in part d givesConsequentlyRelabelling $q=-l$ gives exactly the spectral answer. The square-root branch is fixed by continuation from positive imaginary-time damping; the limiting kernel has the required periodic delta-function normalization.