Doubly Periodic Riemann Boundary Value Problem of Non-Normal Type for Analytic Functions on Two Parallel Curves

In this paper, we present and study a kind of Riemann boundary value problem of non-normal type for analytic functions on two parallel curves. Making use of the method of complex functions, we give the method for solving this kind of doubly periodic Riemann boundary value problem of non-normal type and obtain the explicit expressions of solutions and the solvable conditions for it.


Introduction
Classical Riemann boundary value problems (RBVPs), doubly periodic or quasi-periodic RBVPs and Dirichlet Problems for analytic functions or for polyanalytic functions, on closed curves or on open arcs, have been widely investigated in papers [1]- [8].The main approach is to use the decomposition of polyanalytic functions and their generalization to transform the given boundary value problems to their corresponding boundary value problems for analytic functions, and the fundamental and important tool for which is the Plemelj formula.Professor L. Xing proposed the Periodic Riemann Boundary Value Inverse Problems in paper [9], and then various inverse RBVPs for generalized analytic functions or bianalytic functions have been investigated in papers [10]- [13].
In present paper, we present a kind of doubly periodic RBVP of non-normal type for analytic functions on two parallel curves.On the basis of the results for normal type in paper [14], we give the method for solving this kind of doubly periodic RBVP of non-normal type and obtain the explicit expressions of solutions and the solvable conditions for it.

Doubly Periodic RBVP of Non-Normal Type on Two Parallel Curves
Suppose that 1 ω , 2 ω are complex constants with ( ) Im 0 ω ω ≠ , and P denotes the fundamental period pa- rallelogram with vertices where , , , .
) ( ) ( ) Since j a plays the same roles as other points on 0 j L ( ) , it is natural to require that the unknown functions are bounded at j z a = , that is, the unknown functions ( ) Since j κ can only be 0 and 1 − , the index κ can only take 0, 1, 2 ( ) ( ) ( ) We can easily see that 1 e j z γ will have singularities at most less than one order near the endpoints j a and Lemma 1. Formula ( 5) is valid if and only if

Solution for Problem (1)
Problem (1) can be transferred as is doubly periodic, then by Lemma 1 we have ( ) The function then ( 6) can be rewritten as  ( ) . By the definitions of ( ) ( ) where are satisfied, and now the solution is given by ( ) where c is arbitrary constant. 3 , and due to this the solution for problem (1) has m − order at the point 0 z = .Now the solution for problem (1) can still be given by ( 13), but the following two restrictive conditions are necessary:   .Thus now, we can transform (6) to Now we will meet two kinds of situations in solving problem (1) , the function ( ) is an entire function.And we can write it without counting nonzero constant as where 1 2 , l l are determined by the identity 1) is solvable without any restrictive conditions and the general solution is given by ( ) where c is arbitrary constant.
3˚ When 0 m n λ − − < , problem (1) is solvable if and only if the restrictive conditions with the restrictive condition that ( ) which is to ensure that the solution be finite at z D * = , where 1 2 , , , which is finite at z D * = owing to its structure.
of two parallel curves, lying entirely in the fundamental period parallelogram P, not passing the origin

is analytic in 2 D
, belonging to the class ( ) j h a on 0 j L , satisfying the boundary conditions (1).While

(
doubly periodic if and only if less than one order near the endpoints j a belong to class H or class H * on L 01 and L 02 , respectively.Set And now we will meet three kinds of situations in solving problem (1) in m DR , according to the value of m n , problem (1) is solvable without any restrictive conditions and the general solution is given by

2 .
− , the condition (15) is unnecessary).Case If formula (5) fails to hold, then by Lemma 1 we see that 0 D * ≠ .Let , problem (1) is solvable if and only if the restrictive conditions satisfied, and the general solution is given by satisfied, and the general solution can still be given by (22) but with one order at z = 0, and has singularities at most less than one order near the endpoints j a and ≥ , problem (1) is solvable and the general solution is given by

(
and now the solution is given by < − , if and only if both conditions (26) and the following conditions O, with endpoints being periodic congruent and having the same tangent lines at the periodic congruent points.Let D 1 , D 2 , D 3 denote the domains entirely in the fundamental period parallelogram P, cut by L 01 and L 02 , respectively.Without loss of generality, we suppose that )