双曲线上的Riemann边值问题

Riemann Boundary Value Problems on the Hyperbola

摘要: 本文给出了双曲线上的Riemann边值问题的解法。首先我们通过共形映射的方法将沿封闭曲线剖开的分区全纯函数在无穷远点处Cauchy型积分的性质，Plemelj公式以及Cauchy主值积分在无穷远处的性质推广到双曲线上，利用共形映射将无限长的双曲线上的Riemann边值问题转化为封闭曲线上的Riemann边值问题，利用已有的封闭曲线上的Riemann边值问题的解法对不同情况下的双曲线上的Riemann边值问题进行求解，验证。

Abstract: This article gives the solution to the Riemann boundary value problem on hyperbolas. Firstly, we generalize the properties of Cauchy-type integrals at infinity of the partition holomorphic function cut along the closed curve by the method of conformal mapping, the Plemelj formula and the properties of Cauchy principal integrals at infinity, and use conformal mapping to transform the Riemann boundary value problem on the infinite hyperbola into the Riemann boundary value problem on the closed curve. The solution method of the Riemann boundary value problem on the existing closed curve is used to solve the Riemann boundary value problem on the hyperbola and verify.