We have studied the compound periodic boundary problem in the upper half plane above the real axis. Under proper conditions, we obtain a periodic and sectionally holomorphic function in the upper half plane. In addition, we have also solved the compound boundary problem with discontinuities of the first kind of the coefficients in the Hilbert condition.
In this paper, the difficulties on calculation in solving singular integral equations are overcome when the restriction of curve of integration to be a closed contour is cancelled. When the curve is an open arc and the solutions for singular integral equations possess singularities of higher order, the solution and the solvable condition for characteristic equations as well as the generalized Noether theorem for complete equations are given.
