The Integral Representations For A Class Of Generalized Poly-Analytic Functions And The Applications To Boundary Value Problems

The theory of integral representations is an important research branch in the complex analysis and harmornic analysis,which plays an import role in solving differential boundary value problems and investigating singular integral equations.In this paper,several integral representations denfined on the Lyapunov curves for a class of generalizedβanalytic functions with higher order,are systematically investigated including those with or without Carleman shift.On this basis,its applications to the Carleman type outer boundary value problems,Schwarz problems and Hilbert problems are obtained.In the first chapter,the background and current situation of the theory for the integral representations and its closed related boundary value problems are reviewed and summarized in short.In the second chapter,the theory of integral representations forβanalytic function is well improved.By means of induction and construction,the Cauchy-Pompeiu integral formula for the functions spaceC^（m）（D）is established,and therefore leads to the Cauchy-Pompeiu integral representation of generalizedβpolyanalytic functions with higher order denfined on the bounded domain.Furthermore,the integral representation for Schwartz-Poisson-Pompeiu defined on the unit cicle is also established,and its application to the Schwartz problem defined on the unit cicle is obtained.The mani results extend the integral representation formulas established by H Begehr,A Tungatarov and others.In the third chapter,integral representations for polyanlytic functions are extened to the case ofβpolyanalytic functions with higher order,which give rise to the theory of integral representations forβpolyanalytic functions with higher order.By introducingβCauchy kernel with high order and decomposition theorem forβpolyanalytic functions with higher order,combining Fredholm integral equations and differential transformation,integral representations of generalizedβpolyanalytic functions with higher order with Carleman shifts including bounded and unbounded domains are obtained respectively.The main results generalize some classical integral representation formulas developed by the Soviet mathematician G.S.Litivinchuk,and also enrich and extend the results due to Russian scholars Kats and Katz.In the fourth chapterer,using the previously established integral representation theory of higher order analytic functions,by constructing weakly singular kernel functions,and using the Fredholm integral equation method,various integral representations with Carleman shift for the first and higher orderβanalytic functions in an unbounded and simply connected regions are established in detail,and their applications to the Carleman outer boundary value problems are also obtained.The main results enrich and extend the theory of boundary value problems forβanalytic functions studied by scholars such as RA Blaya and J B Reyes,and also extend the theory of boundary value problems for poly-analytic functions on the unit circle.In the last chapterer,the main content of this paper is summarized,and makes some constructive predictions and explanations for the boundary value problems with Carleman shift and the integral representations defined on the non-rectifiable curves later.