Boundary Value Problems For Polyanalytic Functions

In this dissertation, firstly, we systematically study several basic boundary value problems (BVPs) for polyanalytic (p.a.) functions and reduced p.a. functions, which include Riemann BVPs, Hilbert BVPs, Hasemann BVPs and compound BVPs. Boundary conditions may be defined on the real axis or a simple closed smooth contour. In order to give out reasonable growth conditions for BVPs, the order of p.a. function at infinity is defined, which generalizes the conception of order for analytic function. We use two methods, the canonical method and the conversion method, to obtain the condition of solvability and the solutions for these BVPs. In simple words, the canonical method is connected with the so-called poly-Cauchy type integral and the conversion method is to change these BVPs into equivalent BVPs for analytic functions or systems of analytic functions by the decomposition theorem of p.a. functions. In order to make use of the canonical method to solve Riemann BVPs of p.a. functions with the same factors on the real axis, we introduce the poly-Cauchy integral defined on the real axis and investigate its basic boundary properties. And, in order to find the solutions of all kinds of BVPs by the conversion method, we give out three simple decomposition theorems of p.a. function and explain them in algebra language. For Riemann BVPs with the same factors, we use those two methods to obtain solutions of two different types and prove they may be conversed each other. The main method applied in this dissertation is the conversion method because it is effective for all kinds of BVPs. Secondly, we study Riemann BVPs and Hasemann BVPs for metaanalytic (m.a.) functions. By the appropriate transformation, those problems may be changed into the equivalent BVPs of p.a. functions.The dissertation consists of seven chapters and is arranged as follows.In Chapter 1, we mainly introduce some background and actuality for BVPs of analytic and polyanalytic function and related fields, and briefly introduce our work and open problems.In Chapter 2, we firstly introduce three simple decomposition theorems of p.a. functions and explain them from the view of algebra module. And those decomposition theorems are basis of the conversion method. And then the Liouville type theorem of polyentire function and its proof are introduced. Finally, we give out the definition of order of p.a. function at infinity and the poly-Cauchy type integral defined on the real axis. The poly-Cauchy integral defined on the simple closed smooth curve is firstly studied by H.Begehr, and it is difficult to study the poly-Cauchy integral defined on the real axis. The context of this chapter is the work basis of this dissertation.IVIn Chapter 3, we firstly seek for the solutions of Riemann BVPs of p.a. function with the same factors on the real axis by the poly-Cauchy integral, and the expression of solutions and the condition of solvability are obtained. And then we use the conversion method to solve Riemann BVPs of p.a. function with the different factors on the real axis and obtain the expression of solutions and the condition of solvability. For Riemann BVPs with the same factors, we obtain solutions of different type by the two methods and prove they are equivalent. And smoothness condition of known function in boundary condition is decreased when we use the conversion method to solve BVPs. Therefore, the conversion method is more good. Finally, we give out the relationship between the solvability of Riemann BVPs of p.a. function and the solution of its adjoint problems.In Chapter 4, we mainly study all kinds of Riemann BVPs of p.a. function on the closed curve, and the expression of solution and the condition of solvability are obtained. Firstly, Riemann BVPs of bianalytic (b.i.) functions are discussed in details, and its boundary condition is defined on the unit circumference or a simple smooth closed curve. In order to make use of symmetric principle of the circle, we use the first decomposition theorem to solve Riemann BVPs of b.i. function with the...