Asymptotic estimates in perturbation theory and the structure of functional integrals

The asymptotic series of perturbation theory are investigated for the example of the perturbation series for the ground-state energy of an anharmonic oscillator. The functional integrals in the Feynman expression for the ground-state energy are continued to the plane of complex values of the coupling constant $g$. The discontinuity of the functional integral across the cut $g\leqslant 0$ is calculated by the method of stationary phase. Because the extremals of the action are complex at unphysical values of $g$, the question arises of a transition in the functional integrals to integration with respect to a complex functional variable. By a special change of variable in the functional integral, this problem is reduced to the investigation of an ordinary integral.