A regular covariant method is proposed for expanding the one-loop effective action in powers of curvatures. The expansion is nonlocal and corresponds to summation of the local Schwmger-DeWitt series. The effective equations are computed for both euclidcan and lorentzian signatures of spacetime. The diagrammatic technique for expectation values is used to prove that the nonlocal effective equations for the mean field in the in-vacuum state can be obtained from euclidean equations by a special analytic continuation.