柯西型积分和平面上边值问题

Integral of Cauchy Type and Various Boundary-Value Problems in Plane

综述平面各种边值问题的发展状况:以Cauchy型主值奇异积分为主线,用Plemelj公式求解基本的依跳跃问题,然后从齐次Riemann边值问题的解公式和典则函数得到非齐次Riemann边值问题的解;将Hilbert边值问题化为Riemann边值问题求解.进一步对周期、双周期、群不变的边值、带位移边值及它们相互之间的复合等各种问题,提供转化为典型问题的进展和文献.

We provide the theory and development for various boundary-value problems in the plane.By Starting from a integral of Cauchy type and Plemelj formula,we in turn establish these solutions of jump boundary-value problem,homogeneous Riemann＇s problem and Riemann＇s problem defined in a planar simple domain.Then we can transform famous Hilbert＇s problem into Riemann＇s problem by means of a symmetric expansion w.r.t.the unit circle.Finally,we present the literatures and developments for these boundary value problem of period,double period,invariant Mobius＇s group and their compostions each other.