The mixed boundary value problem can be reduced to an RH problem as follows. Let the boundary values of the multi-valued analytic function F be given by
Although, the function is in general multi-valued, its derivative is a single-valued analytic function on G. The boundary values of the function are given by
(21)
For the Dirichlet conditions, i.e., and , the functions are equal to the known functions (see (6b)). Thus, the function satisfies the RH problem
(22)
The Neumann conditions can also be reduced to an RH problem by using the Cauchy-Riemann equations and integrating along the boundaries , . Let be the unit tangent vector and be the unit external normal vector to Γ at . Let also be the angle between the normal vector and the positive real axis, i.e., . Then
Thus,
(23)
Since , then by the Cauchy-Riemann equations, we have
Thus, the function satisfies the RH problem
(24)
If we define the real piecewise constant function
(25)
the boundary values of the function satisfy on the boundary Γ the condition
(26)
where
(27)
is known and
(28)
The functions for are given by (6b) and (6c). The functions can be then computed for and by integrating the functions . Then it follows from (7), (26) and (27) that the function is a solution of the RH problem
(29)
or briefly,
(30)
where
(31)
for . In view of (8) and (28), the real constants are known for and are given by
(32)
However, for , the real constants are unknown. Thus, the boundary condition (29) can be written as
(33)
where the function is known and is given by
(34)
Obviously, the functions are known explicitly for with . However, for with , it is required to integrate to obtain .
The functions are not necessary 2π-periodic. In order to keep dealing with periodic functions numerically, we do not compute directly by integrating the functions . Instead, we integrate the functions
According to the definitions of the constants and the functions , we have
which implies that the functions are always 2π-periodic. By using the Fourier series for with , the functions can be written as
(35)
Then the functions are given for with by
(36)
where are undetermined real constants and the functions are given by
(37)
Hence, the boundary condition (33) can then be written as
(38)
where is the real piecewise constant function
(39)
and the function is given by
(40)
Let (unknown real constant) and be the analytic function defined on G by
Then is analytic on G with . The function is a solution of the RH problem
(42)
where the function is given by (10) and the function is defined by
(43)