Schwartz Boundary Value Problem On Arc Pentagons

In this paper,we study the solution of Schwarz boundary value problem on arc pentagons.The Schwarz boundary value problem is to find the analytic function in a region so that it satisfies some boundary conditions on the boundary of the region.For the sake of simplicity,we let the real function a(t)=1,b(t)=0.We divide the study of the problem into three parts.The first step is to use the Parqueting-reflection principle to expand the arc pentagon outward until it covers the whole plane,and obtain the boundary value problem on thirty open arcs.In the second step,we classify the 30 open arcs and transform the Hilbert boundary value problem into the Riemann boundary value problem on the open arc according to the known assumed conditions.Finally,by solving the Riemannian boundary value problem on open arcs,we obtain the product decomposition of the boundary value problem on each arc,and then give the integral form of the Hilbert problem.In terms of structure arrangement,this paper first gives the basic knowledge needed to solve the problem,including the definition and related theorems of Cauchy integral,Holdercondition,the formulation and solution of Hilbert problem and the formulation and solution of Riemann boundary value problem on the open arc.Then,the Hilbert boundary value problem on arc pentagons is proposed and transformed into Riemann boundary value problem on open arcs.Finally,the form of solution is given.In this paper,the most important thing to solve the Hilbert boundary value problem on an arc pentagon is to apply the Parqueting-reflection principle to extend a region to the whole plane.Meanwhile,we also use the method in literature [1][2] to get the relationship between the positive and negative boundary values of holomorphic functions on both sides of the straight segment,which is transformed into the boundary value problem on the open arc segment for solving.Finally,the integral representation of Hilbert problem is obtained.