Riemann boundary value problems on the positive real axis

Abstract

In this paper, some Riemann boundary value problems on the positive real axis are presented. Firstly, we introduce the concepts of principle part and order at infinity and zero point for the holomorphic function on the complex plane cut along the positive real axis, which are more extensive than those in the classic sense. Then, the behaviors of Cauchy-type integral and Cauchy principal value integral on the positive real axis at infinity and zero point are discussed, respectively. Based on those, Riemann boundary value problems for sectionally holomorphic functions with the positive real axis as their jump curve are established explicitly, which are different from Riemann problems on the finite curves and more complicated than those on the whole real axis. Finally, some boundary value problems for matrix-valued functions are also constructed, which play a very important role in the asymptotic analysis for orthogonal polynomials on the positive real axis.