The Riemann-Hilbert Boundary Value Problem Of Helmolts Equation In A Multiply Connected Region

This paper mainly discusses the Riemann-Hilbert boundary value problem in multi-connected region.and the Helmoltz equation by integral representation of solution.The paper is consists of four chapters:In chapter 1. We investigate the history of quaternion and the Helmoltz equation modified form in complex analysis. We also have some basic knowledge about the special mathematical symbols and other prepare work.In chapter 2. We have the integral representation form of com-plex equation in first order, and the Riemann-Hilbert boundary value problems in single connected domain and in multi-connected region.In chapter 3. This chapter mainly discuses Riemann-Hilbert boundary value problem in simple-connected domain.In chapter 4. We discuss the complex equation in first order of Riemann-Hilbert boundary value problem, we have some ba-sic conclusion after discusions and in the end we have solved the Riemann-Hilbert boundary value problem of Helmoltz equation in multi-connected regions.