The Boundary Value Problems With Square Roots And The Inverse Riemann Boundary Value Problems For Bianalytic Functions

Boundary value problem for analytic functions is one of the most important branches in the theory of complex functions. They are not only important in theory but also extensive in application. Nonlinear Riemann boundary value problems and nonlinear Hilbert boundary value problems will be studied in this paper, because they have been researched seldom. The research of bianalytic functions, polyanalytic functions and inverse Riemann boundary value problems is an important question for discussion in recently years. This paper will also discuss a kind of inverse Riemann boundary value problems for bianalytic functions.Firstly, the solution of a kind of Riemann boundary value problem with square roots on an open smoothly arc is considered. By analyzing the structure of the unknown function, the problem is transformed into a typical Riemann boundary value problem on an open arc, and its solutions as well as the conditions of its solvability are given according to the known theory.Secondly, Hilbert boundary value problem with square roots in the upper half-plane is discussed. As above, by analyzing the structure of the unknown function, the problem is transformed into a typical Hilbert boundary value problem in the upper half-plane. Then the problem is further equivalent to a kind of Riemann boundary value problem on the real axis by extension of symmetry with respect to the real axis. As a result, we establish the theorems of its solvability according to the known theory.Finally, A class of inverse Riemann boundary value problem for bianalytic functions on the real axis is proposed. By eliminating parametic unknown functions and using the notation of matrix form which will be generalized easily, the problem is trans-...