A new method of the decision of the theory of elasticity plane boundary problems at the final (big) defomations has been offered. Regional problems of a Signorini's model, which describes the deflected mode of an elastic body at final deformations, are offered to be studied by means of complex structure introduction in the area of coordinates and movings. It allows to construct the approximate solutions, depended on two holomorphic functions, in the form of special holomorphic decomposition. This method allows to allocate the solution, providing the approximate come up to the boundary requirements, from the received solutions manifold and thus to estimate the sharpness of the equations satisfaction.