A Kind of Doubly Periodic Riemann Boundary Value Problem on Two Parallel Curves

We proposed a kind of doubly periodic Riemann boundary value problem on two parallel curves. By using the method of complex functions, we investigated the method for solving this kind of doubly periodic Riemann boundary value problem of normal type and gave the general solutions and the solvable conditions for it.


Introduction
Various kinds of Riemann boundary value problems (BVPs) for analytic functions on closed curves or on open arcs, doubly periodic Riemann BVPs, doubly periodic or quasi-periodic Riemann BVPs and Dirichlet Problems, and BVPs for polyanalytic functions 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 boundary value problems to their corresponding boundary value problems for analytic functions.Recently, inverse Riemann BVPs for generalized analytic functions or bianalytic functions have been investigated in papers [9]- [13].
In this paper, we consider a kind of doubly periodic Riemann boundary value problem on two parallel curves.By using the method of complex functions, we investigate the method for solving kind of doubly periodic Riemann boundary value problem of normal type and give the general solutions and the solvable conditions for it.

A Kind of Doubly Periodic Riemann Boundary Value Problem 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 τ ∈ , and be doubly periodic with

Preliminary Notes
Since ( ) τ ≠ =, by taking logarithm of ( ) D τ for some branch on 0 j L , we may obtain a continuous single-valued function such as ( ) ( )  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) of Normal Type
Problem (1) can be transferred by using ( to the two sides of the first identity in equations ( 6), and multiplying The function is doubly periodic, then by Lemma 1 we have ( ) Then by formulas ( 9) and ( 10), we may rewrite (7) as , e e e e Now we introduce the function has n -order at z = 0, and has singularities at most less than one order near the endpoints a j and . Thus we can get the following results. 1 ° When m > 0, problem (1) is solvable without any restrictive conditions and the general solution is given by where c is arbitrary constant.
3 ° When m < 0, if and only if the restrictive conditions (13) and (when 1 m = − , the condition (15) is unnecessary) are necessary, problem (1) is solvable and the solution can still be given by ( 14) but with  .Thus now, we can transform (6) to By ( 17) and (18), we can rewrite (16) as Now we will meet two kinds of situations in solving problem (1) 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 m < 0, if and only if the restrictive conditions ( Φ must be at most m + 1 ordered at z = 0, and has singularities less than one order at z = a j (j = 1, 2). 1 ° When 0 m ≥ , problem (1) is solvable without any restrictive conditions and the general solution is given by 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 2π

Figure 1 .
Figure 1.Parallel curves in the fundamental period parallelogram P.
if and only if sides of the second identity in Equations (6), gives at most less than one order near the endpoints j a belong to class H or class H * on L 01 and L 02 , respectively.Case 1.If formula (5) holds, that is, ( ) e z γ

2 °
When m = 0, problem (1) is solvable if and only if the restrictive conditions and now the solution is given by

Case 2 .
If formula (5) fails to hold, then by Lemma 1 we see that 0 D * ≠ .Let to class H or class H * on L 01 and L 02 , respectively.Write and the general solution for (1) is given by , the condition (23) is unnecessary) are both necessary, problem (1) is solvable and the solution can still be given by (22) but with one order at z = 0, has singularities at most less than one order near the endpoints j a and has a zero of order one at z D * = .Write ( ) and now the solution is given by , (28) is unnecessary) are necessary, and the solution is given by O, with endpoints being periodic congruent and having the same tangent lines at the periodic congruent points.Let 1 in DR m .
, problem (1) is solvable if and only if the restrictive conditions , problem (1) is solvable if and only if both conditions (26) and the following conditions