The Riemann Boundaray Value Problem Of Non-normal Type On The Real Axis

The most researches on Riemann boundary value problem are confined to normal type. The Riemann boundary value problem of non-normal type on smooth closed contours first arised in the book [1] written by F.D.Gakhov. Its method of solution requires that the input functions have enough high order continuous Holder derivatives. Prof.Lin Yubo systematically explored the Riemann boudanry value problem of non-normal type in [4-9] in home. Prof.Du Jinyuan constructed a generalized interpolatory polynomial by introducing Peano derivatives and got a new method of solution; thereafter, H.Begehr constructed Hermite interpolatory polynomial in the meaning of non-tangential limit derivatives and got another method of solution. As preparation, these will be presented in the first chapter of this paper.The second chapter is the main part of this paper, in which the formulation of the Riemann boundary value problem of non-normal type on the real axis, the solution method of homogeneous problem, the relation between the two kinds of different derivatives and the inhomogeneous problem will be thoroughly given. In this paper, the solution and the solvability of the Riemann boundary value problem of non-normal type on the real axis will be given. Furthermore, it is shown that the twokinds of derivatives of the function Ψ(z) are existing and equivalent in the case ofthe solution about the original problem, therefore, we get uniformly Hermite interpolatory polynomial. The relation between the two kinds of different derivativesof the function Ψ(z) are similar for smooth closed contours by means of the same proof.