Periodic Riemann boundary value problem

In this paper, we discuss the Riemann boundary value problems for the analytic functions with period a pi (a > 0) in the situation where the solution can be unbounded at z = +/-infinity i. When the jump curve is a closed contour, by using the tangent transformation zeta = tan(z/a), we transform the singularities +/-infinity i on the z plane to the singularities +/- i on the zeta plane. When the jump curve is the real axis, by using the transformation omega= e(iz), we transform the singularities +/-infinity i on the z plane to the singularities 0 and infinity on the omega plane. Thus, the original periodic Riemann boundary value problems are reduced to the classical Riemann boundary value problems whose solutions and the corresponding solvability conditions are well known.