Hilbert Boundary Value Problem with an Unknown Function on Arbitrary Infinite Straight Line ()
1. Introduction
Various kinds of boundary value problems (BVPs) for analytic functions or polyanalytic functions have been widely investigated [1-8]. 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 [9-12].
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.
2. 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 
and 
denote the upper half-plane and the lower halfplane cut by
.
Our objective is to find a pair of functions 
, where 
is holomorphic in the domain 
and continuously extendable to its boundary
, and 
is real-valued and Holder continuous on
, satisfying the following boundary conditions
(1)
where
and
are given functions.
3. 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 
about 
as follows:
, (2)
where 
is the symmetric point of 
about
. For simplicity, we express 
as
. From definition (2), we may establish that 1)
;
2) If 
is defined in
, then 
is also defined in
;
3) If 
is holomorphic in
, then 
is holomorphic in 
because of
;
4) 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)
5) If 
is holomorphic in 
and continuous on
, then
(4)
is a sectionally holomorphic function that jumps on 
with 
finite, and 
possesses the following properties:
(5)
(6)
(7)
6) Let
, where 
and 
are holomorphic in 
(
or
). It is not necessarily true that
.
Problem (1) is normal only if 
on
.
4. 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 
and 
respectively, we obtain the Riemann boundary problem
(8)
or
. (9)
By extending 
to 
about the straight line
, we obtain a sectionally holomorphic function 
as (4) with jump
, satisfying the boundary conditions
.
Thus (9) can be rewritten in the form
. (10)
Due to
, (10) can be written as R problem
, (10)’
where
(11)
and
,
on
.
If 
is a solution of problem (1), then 
extended from 
by (4) must be a solution of (10) or (10)’ in 
(namely
) and satisfies the boundary condition (5). On the other hand, if the solution 
of R problem (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).
Assume that 
is a solution of (10) in
, by making conjugate for (10) we obtain
.
We read from relation (7) that 
is also a solution of (10)’ in
, so that
is a solution of (10)’ in class 
and satisfies the additive condition (5). So that, whenever we find out the solution 
of problem (10)’ in class
, and write out
, then
is actually the solution of problem (1).
Let
.
By 
we know that 
is even.
5. 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.
5.1. Homogeneous Problem
The homogeneous problem of (1) is as follows
. (12)
By canceling the unknown function
, problem (12) becomes
, (13)
which corresponds to the homogeneous problem of R problem (10)’
. (14)
Setting
, we have
.
If we let
, then we know 
on 
with
, and
. By letting
we can rewrite (14) as follows
. (15)
Let us introduce the function
.
Since
, we have
, (16)
where
is real-valued on 
and
. Now the canonical function of R problem (15) or (14) can be taken as
(17)
where 
is an unknown complex constant. We can see from (17) that
, thus R problem (15) can be transferred to the following problem
. (18)
So 
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
, (19)
where 
is an arbitrary polynomial of degree 
with 
if
.
According to (16), we know
(20)
and hence
. (21)
From (17) and (20) it can be seen that
which implies that
. By taking
(22)
we obtain
(23)
and
Consequently, we see that 
if and only if
(24)
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
. (25)
Thus we get the following results.
Theorem 5.1. For the homogeneous problem (12), the following two cases arise.
1) When
, its general solution is
, where 
and 
are given by (19) and (25) respectively, in which condition (24) is satisfied for
, and 
is given by (22) (a real constant factor is permitted for
).
2) When
, it only has zero-solution
.
5.2. 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
. (26)
Therefore 
is actually the particular solution of problem (1), where
(27)
And from (20) we obtain
. (28)
It follows from (3) that 
and from (20) and (21) that
.
While due to 
and (28) we have
, so we obtain
(29)
Therefore, we obtain
(30)
and
(31)
When
, 
has singularity of order 
at
. Now we aim to cancel the singularity of 
at
. From [5], we know that R problem (10)’ is solvable in 
if and only if
(32)
and its unique solution takes the form
. (33)
For the case
, since the solution for (10)’ in 
is unique and 
must be a solution of (10)’ in
, we conclude that
, thus (33) is actually the unique solution of nonhomogeneous problem (8).
Combining the particular solution of nonhomogeneous problem (8) and the general solution of homogeneous problem (14), we known that when
, the general solution of R problem (8) is
(34)
where 
satisfies condition (24) and 
is given by (31); when
, R problem (8) is solvable if and only if (32) is satisfied and the unique solution is given by (33).
Putting the solution 
into the first equation in (1), we obtain
. (35)
Therefore, we derive the following results.
Theorem 5.2. If
, the nonhomogeneous problem (1) is always solvable and its general solution is
, where 
is given by (34) with 
satisfying condition (24) and 
being given by (22) (a real constant factor is permitted for
), while 
is given by (35). If
, under the necessary and sufficient condition (32), the nonhomogeneous problem (1) has unique solution
, where 
and 
are given by (33) and (35) respectively.