In order to study the validity of the functional integral formula of the Green's function, the large time limit of the time evolution kernel exp (-iHt) of any quantum system is discussed. With the Feynman prescription, giving an infinitesimally small negative imaginary part to quadratic term of the potential in the Hamiltonian, the correct formula for this limit is obtained. It differs from the naive formula and depends on the detailed form of the potential. The usual formula holds for the perturbative expansion in terms of the potential, the proof of which is given by the path integral method. The new expression for the ground state wave function is also obtained in the course of the proof.