Some Boundary Value Problems And Correlative Problems For High-order And High-dimensional Equations In Complex Analysis

In this paper, we mainly consider a boundary value problem for a high-order equation in the complex plane and a boundary problem for a hyperbolic equation in four-dimensional space by complex analytic methods. At the same time, we study the hyperbolic numbers and the bi-complex numbers by the method of algebra which are important for solving hyperbolic equations,and this presents more theory arguments for solving more hyperbolic equations by the tool of hyperbolic numbers and bi-complex numbers. At the end of this paper, we give out two properties for operator T in quaternionic analysis which is often used as a important tool for solving elliptic equations in four-dimensional space.In paper [5], the author proved a k-regular function (i.e. the solutions of (?)=0)in G could be uniquely expressed as analytic functions. Furthermore, he worked out some functional properties for the k-regular functions, such as Cauchy integral formula, Cauchy type integral, etc. In the second chapter we discuss a boundary value problem with conjugate value for k-regular functions. By the contract mapping theorem, we prove the existence and the uniqueness of the solution for this problem, generalize some results in recent indexes.In the third chapter, we discuss the hyperbolic numbers and the bi-complex numbers. With the given matrix expressions of the hyperbolic numbers and the bi-complex numbers, they become much easier for us to study.In the forth chapter of this paper, the Riemann-Hilbert boundary value problem for...