无穷直线上K-解析函数的 Riemann 边值问题与 Hilbert 边值问题

首先引入了无穷直线上(分片)K-解析函数的Cauchy型K-积分的概念,利用K-对称变换的方法研究了Cauchy型K-积分的某些性质,然后借助函数在无穷直线上的指标与这些Cauchy型K-积分的性质,得到了在无穷直线上K-解析函数类中的Riemann边值问题的可解条件和解的表达式以及它们与指标之间的关系;进一步利用半平面内的K-对称扩张函数,把Hilbert边值问题转化为无穷直线X上的Riemann边值问题,又得到了Hilbert边值问题的可解条件和解的表达式.而解析函数和共轭解析函数都是K-解析函数的特例,所得结果推广了解析函数和共轭解析函数中的相应结论.

Riemann Boundary Value Problem and Hilbert Boundary Value Problem for K-Analytic Function on an Infinite Straight Line

In this paper, we first introduce the concept of K-analytic function of Cauchy type K-integral on an infinite straight line (fragmentation) and use the K-symmetry transformation method to study some properties of the Cauchy type K-integral. Then, with the help of the index that functions on the infinite straight line and the properties of the Cauchy type K-integral, we obtain the solvable conditions and its expression of Riemann boundary value problem of the K-analytic function on the infinite straight line as well as the relationship between them and the index. Finally, we use the K-symmetry expansion function in a half plane to transform the Hilbert boundary value problem into Riemann boundary value problems on the infinite straight line X, thus obtaining the solvable conditions and its expression of the Hilbert boundary value problem. Both the analytic function and the conjugate analytic function are special cases of the K-analytic function. The results obtained in this paper generalize the analytic function and the conjugate analytic function in the corresponding conclusions.