Abstract
The methods of complex analysis constitute the classical direction in the study of elliptic equations and mixed-type equations on the plane and fundamental results have now been obtained. In the early 60s of the last century, a new theoretical-functional approach was developed for elliptic equations and systems based on the use of functions analytic by Douglis. In the works of A.P. Soldatov and Yeh, it turned out that in the theory of elliptic equations and systems, Douglis analytic functions play an important role. These functions are solutions of a first-order elliptic system generalizing the classical Cauchy-Riemann system. In this paper, the Fredholm solvability of the generalized Neumann problem for a high-order elliptic equation on a plane is investigated. The equivalence of the solvability condition of the generalized Neumann problem with the complementarity condition (Shapiro-Lopatinsky condition) is proved. The formula for the index of the specified problem in the class of functions under study is calculated.