To estimate a Feynman path integral for a nonrelativistic particle with one degree of freedom in an arbitrary potentialV(x), it is proposed to use a functional method of steepest descent, the analog of the method for finite-dimensional integrals, without going over to the Euclidean form of the theory. The concepts of functional Cauchy—Riemann conditions and Cauchy theorem in a complex function space are introduced and used essentially. After the choice in this space of a “contour of steepest descent,” the original Feynman integral is reduced to a functional integral of a decreasing exponential. In principle, the obtained result can serve as a basis for constructing the measure of Feynman path integrals.