Hilbert Boundary Value Problem with an Unknown Function on Arbitrary Infinite Straight Line

We consider a Hilbert boundary value problem with an unknown parametric function on arbitrary infinite straight line passing through the origin. We propose to transform the Hilbert boundary value problem to Riemann boundary value problem, and address it by defining symmetric extension for holomorphic functions about an arbitrary straight line passing through the origin. Finally, we develop the general solution and the solvable conditions for the Hilbert boundary value problem.


Introduction
 and continuously extendable to its boundary , and Various kinds of boundary value problems (BVPs) for analytic functions or polyanalytic functions have been widely investigated [1][2][3][4][5][6][7][8].The main approach is to use the decomposition of polyanalytic functions and their generalization to transform the boundary value problems to their corresponding boundary value problems for analytic functions.Recently, inverse Riemann BVPs for generalized holomorphic functions or bianalytic functions have been investigated [9][10][11][12].

  
is real-valued and Holder continuous on , satisfying the following boundary conditions

 
In this paper, we consider a kind of Hilbert BVP with an unknown parametric function.We first define the symmetric extension of holomorphic function about an infinite straight line passing through the origin, and discuss its several important properties.And after, we propose a Hilbert BVP with an unknown parametric function on arbitrary half-plane with its boundary passing through the origin.Then, we transform the Hilbert BVP into a Riemann BVP on the infinite straight line using the defined symmetric extension.Finally, we discuss the solvable conditions and the solution for the Hilbert BVP.

A Hilbert Boundary Value Problem with an Unknown Function
Let be an infinite straight line with an inclination in the complex plane, passing through the origin and being oriented in upward direction.Let denote the upper half-plane and the lower halfplane cut by . and L are given functions.

Symmetric Extension of Holomorphic Functions about an Infinite Straight Line
An important step in solving problem (1) is to define a symmetric extension of holomorphic functions about the infinite straight line with an inclination For a holomorphic function in the simplyconnected domain  , we define the symmetric extension of where is the symmetric point of about .For simplicity, we express as . From definition (2), we may establish that 1) ; ) If a holomorphic in can be continuously extended to , then  in can be continuously extended to , and their boundary value on satisfies the following equality (3) is a sectionally holomorphic function that jumps finite, and possesses the following properties:  

6) Let
, where f z and

Transformation of Problem (1)
In this section, we develop a general method to solve boundary value problem (1) or similar problems.Let Multiplying the first and the second equation in (1) by respectively, we obtain the Riemann boundary problem or By extending to  about the straight line , we obtain a sectionally holomorphic function Thus ( 9) can be rewritten in the form   0 t   , (10) can be written as R problem Due to where 4) must be a solution of (10) or (10)' in 0 (namely    ) and satisfies the boundary condition (5).On the other hand, if the solution (10)' in satisfies the boundary condition ( 5), then is really a solution of problem (1).Consequently, problem (1) is equivalent to R problem (10)' in together with the additive condition (5).
, by making conjugate for (10) we obtain We read from relation (7) that is also a solution of (10)' in , so that R is a solution of (10)' in class 0 and satisfies the additive condition (5).So that, whenever we find out the solution (10)' in class , and write out is actually the solution of problem (1).Let

Solution of the Hilbert Boundary Value Problem with an Unknown Function
Here, we only consider the problem (1) in the normal case.The nonnormal case can be solved similarly.

Homogeneous Problem
The homogeneous problem of ( 1) is as follows By canceling the unknown function , problem (12) becomes   which corresponds to the homogeneous problem of R problem (10)' , , , then we know on with

and . By letting In
Let us introduce the function where and .Now the canonical function of R problem (15) or ( 14) can be taken as where A is an unknown complex constant.We can see from (17) that , thus R problem (15) can be transferred to the following problem  is holomorphic on the whole complex plane and has order at  .From [5] we know that the general solution of R problem (14) in takes the form From ( 17) and (20) it can be seen that and Consequently, we see that if and only if Then when condition (24) is satisfied, the solution of H problem (13) is given by (19).Now putting the solution of H problem (13) given by (19) into the first equation (or the second equation) in (12), we get Thus we get the following results.Theorem 5.1.For the homogeneous problem (12), the following two cases arise., and A is given by (22) (a real constant factor is permitted for A ).

  
2) When 0  , it only has zero-solution

Nonhomogeneous Problem
In order to solve the n nonhomogeneous problem (1), we only need to find out a particular solution for problem (1).
According to [5], we know that when the R problem (10)' a particular solution in class as follows 0 is actually the particular solution of problem (1), where And from (20) we obtain  and (28) we have , so we obtain Therefore, we obtain and with jump , satisfying the boundary conditions