Comment on “Generalization of Keldysh’s theory”

We reassess some mathematical aspects of the theory by Keldysh [Sov. Phys. JETP 20, 1307 (1965)]. In particular, we discuss in detail what closed contour is associated with the symbol {integral} introduced by Keldysh and we expose how the contour involved is conveniently deformed to calculate the integral by the saddle-point method. The integral cannot be reduced to a contribution of a single pole that could be evaluated via the residue theorem, contrary to the statement in the recent work by Mishima et al. [Phys. Rev. A 66, 033401 (2002)].