Resumen
Based on the argument function method and the complex variable function method, we have derived the generalizing solutions to a flat problem on the theory of elasticity using the invariant differential ratios capable of closing the result for the set system of equations. The paper reports the approaches whose application defines not the permitting functions themselves but the conditions for their existence. This makes it possible to expand the range of harmonious functions of varying complexity, which satisfy any boundary conditions in the applied problems that are constantly renewed. Two basic functions have been taken into consideration: trigonometric and fundamental, whose arguments are the unknown coordinate dependences. Introducing the argument functions into consideration changes the approaches to determining permitting dependences because the problem is considerably simplified when establishing a differential relationship among them in the form of the Cauchy-Riemann and Laplace ratios. Several analytical solutions of varying complexity have been presented, which are matched with different boundary conditions. Comparison with the results reported by other authors, at the same initial data, leads to the same result; and when considering the test problem on the interaction between a metal and an elastic half space, it leads to the convergence between defining schemes of force influence on the elastic medium.Thus, a new approach has been proposed to solving a flat problem from the theory of elasticity, which is associated with the use of argument functions, which makes it possible to close the problem via the differential Cauchy-Riemann and Laplace ratios. These generalizations expand the range of harmonious functions that correspond to different boundary conditions for the applied problems.