CONSTRUCTION OF AREAS USING THE LINE OF LEVEL OF HARMONIC FUNCTIONS IN COMPLEX AREAS

Abstract

The study of the asymptotic behavior of solutions of singularly perturbed equations with analytic functions in complex domains is reduced to the study of some special functions with a small parameter (like integrals containing a large parameter). The study of such functions is hampered by the fact that it is necessary to select a certain part from a given region and choose integration paths that ensure that the functions under consideration are bounded with respect to a small parameter. In this paper, we have developed a method based on the lines of levels of harmonic functions, constructing an area. Then, sufficient conditions for boundedness of special functions with respect to a small parameter are proved. All constructions are accompanied by corresponding figures. In the future, the results of this work can be used for the theory of singularly perturbed equations in the complex domain.