正实轴上的Riemann边值问题

本文研究正实轴上的Riemann边值问题.首先,引入沿正实轴剖开的复平面上的全纯函数在无穷远点和原点处主部及阶的概念,相比于经典意义下,这个概念更为广泛.其次,讨论了正实轴上Cauchy型积分和Cauchy主值积分在无穷远点和原点处的性质.基于此,以正实轴为跳跃曲线的分区全纯函数的Riemann边值问题得以详细解决.这个过程有别于经典意义下有限曲线上的Riemann边值问题,且比整个实轴上的Riemann边值问题更为复杂.最后,作为例子讨论了一类矩阵值函数的边值问题,该问题对于正实轴上正交多项式的渐近分析有重要意义.     

关于实轴上多解析函数Riemann边值问题的可解性

本文研究了实轴上具有不同因子的多解析函数的Riemann边值问题的可解性．利用所谓的转化法．建立了Riemann问题的可解性与其相联问题的解之间的关系。该结果推广了解析函数的相应理论。     

N—解析函数的Riemamm—Hilbert问题

在N-解析函数类中,对于无穷直线上的Riemann-Hilbert边值问题,通过轴的对称扩张法将其转化为在附加条件下相应的Riemann边值问题,从而建立了其齐次和非齐次问题的可解性理论。     

Koch曲线上的齐次Riemann边值问题

当L为典型的分形曲线一Koch曲线时,提出了Riemann边值问题,但在一般情况下,在Koch曲线上所做的Cauchy型积分无意义.当对已知函数G(z),g(z)增加一定的解析条件,同时利用一列Cauchy型积分的极限函数,对定义在Koch曲线上的齐次Riemann边值问题进行了讨论,并得到与经典解析函数边值问题相类似的结果.     

边界过原点的任意半平面中的RH边值问题

给出边界过原点的任意半平面中RH边值问题的提法,借助于解析函数的对称扩张将此问题转化为无穷直线上的Riemann边值问题,讨论了该问题的求解并得到该问题的一般解及可解性定理.     

一般周期Riemann边值问题关于跳跃曲线的稳定性

本文研究了一般周期Riemann边值问题关于跳跃曲线的稳定性.利用解析函数边值理论和不等式分析理论,获得了一般周期Riemann边值问题的解及其关于跳跃曲线的误差估计.     

实轴上非正则型Riemann边值问题

本文研究了实轴上一类特殊非正则型Riemann边值问题.利用Peano导数构造出一种广义Hermite插值多项式,获得了该问题的可解条件和解的封闭形式.     

K-解析函数的Hilbertert边值问题

定义和讨论了K-解析函数在典型域S~+={z:|z(k)|1}外的K-对称扩张函数,利用它把K-解析函数的Hilbert边值问题转化为Riemann边值问题,得到了K-解析函数类F(D(k))中Hilbert边值问题与Dirichlet边值问题的可解条件及其解的表达式.而解析函数和共轭解析函数都是K-解析函数的特例,所得结果,包含了解析函数和共轭解析函数中的相应结论.     

超解析函数在不可求长闭曲线上的Riemann边值问题

本文研究超解析函数在不可求长Jordan闭曲线上的Riemann边值问题,用完全不同于可求长曲线上的Riemann问题的处理方法,证明了问题的可解性结论,得到了问题的一般解的表示式及可解的充分必要条件。     

双解析函数的一般复合边值问题关于边界曲线的稳定性

开口弧段Γ上的双解析函数的Riemann边值问题与单位圆周L上双解析函数的Hilbert边值问题复合而成的一般复合边值问题,当L与Γ发生微小的光滑摄动后,借助于推广的拉甫伦捷夫近似于圆的共形映射,将星形域映为单位圆域,从而得出摄动后的问题的解的表达式,同时讨论了解的稳定性情况,并给出误差估计.     

ON RIEMANN BOUNDARY VALUE PROBLEM FOR POLYANALYTIC FUNCTIONS ON THE REAL AXIS

In this article, Riemann boundary value problem with different factors for polyanalytic functions on the real axis is studied. The expression of solution and sufficient and necessary condition for solvability of the non-homogeneous Riemann boundary value problem are obtained.     

On Boundary Value Problems of Polyanalytic Functions on the Real Axis

In this article, we discuss the Riemann boundary value problems and the Hilbert boundary value problems of polyanalytic functions on the real axis, and both the explicit expressions of solutions and the conditions of solvability are obtained.     

On Riemann problems for single‐periodic polyanalytic functions

In this article, Riemann‐type boundary‐value problem of single‐periodic polyanalytic functions has been investigated. By the decomposition of single‐periodic polyanalytic functions, the problem is transformed into n equivalent and independent Riemann boundary‐value problems of single‐periodic analytic functions, which has been discussed in details according to two growth orders of functions. Finally, we obtain the explicit expression of the solution and the conditions of solvability for Riemann problem of the single‐periodic polyanalytic functions.     

The Schwarz problem for analytic functions in torus-related domains

The Cauchy kernel is one of the two significant tools for solving the Riemann boundary value problem for analytic functions. For poly-domains, the Cauchy kernel is modified in such a way that it corresponds to a certain symmetry of the boundary values of holomorphic functions in poly-domains. This symmetry is lost if the classical counterpart of the one-dimensional form of the Cauchy kernel is applied. It is also decisive for the establishment of connection between the Riemann–Hilbert problem and the Riemann problem. Thus, not only the Schwarz problem for holomorphic functions in poly-domains is solved, but also the basis is established for solving some other problems. The boundary values of functions, holomorphic in poly-domains, are classified in the Wiener algebra. The general integral representation formulas for these functions, the solvability conditions and the solutions of the corresponding Schwarz problems are given explicitly. A necessary and sufficient condition for the boundary values of a holomorphic function for arbitrary poly-domains is given. At the end, well-posed formulations of the torus-related problems are considered.     

Riemann boundary value problems for some K-regular functions in clifford analysis

In this paper, we study the R_m (m > 0) Riemann boundary value problems for regular functions, harmonic functions and bi-harmonic functions with values in a universal clifford algebra C(V_n,n). By using Plemelj formula, we get the solutions of R_m (m > 0) Riemann boundary value problems for regular functions. Then transforming the Riemann boundary value problems for harmonic functions and bi-harmonic functions into the Riemann boundary value problems for regular functions, we obtain the solutions of R_m (m > 0) Riemann boundary value problems for harmonic functions and bi-harmonic functions.     

The RH boundary value problem of the k-monogenic functions

In this paper we study the Riemann and Hilbert problems of k-monogenic functions. By using Euler operator, we transform the boundary value problem of k-monogenic functions into the boundary value problems of monogenic functions. Then by the Almansi-type theorem of k-monogenic functions, we get the solutions of these problems.