作　　者： (王莹）; (段萍）; (杜金元）;

机构地区： 中南财经政法大学统汁与数学学院,武汉430073

出　　处： 《中国科学：数学》 2017年第8期887-918,共32页

摘　　要： 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. In this paper, some Riemann boundary value problems on the positive real axis are presented.Firstly, we introduce the concepts of principle part and order at infinity and zero point for the holomorphic function on the complex plane cut along the positive real axis, which are more extensive than those in the classic sense. Then, the behaviors of Cauchy-type integral and Cauchy principal value integral on the positive real axis at infinity and zero point are discussed, respectively. Based on those, Riemann boundary value problems for sectionally holomorphic functions with the positive real axis as their jump curve are established explicitly, which are different from Riemann problems on the finite curves and more complicated than those on the whole real axis. Finally, some boundary value problems for matrix-valued functions are also constructed, which play a very important role in the asymptotic analysis for orthogonal polynomials on the positive real axis.

关 键 词： 主部 阶 型积分 边值问题 矩阵值函数