The homogeneous Riemann boundary-value problem with infinite index on a curvilinear contour. I

Abstract

The Riemann boundary-value problem with infinite index appears to have first appeared in an article by Akhiezer. The general theory of Riemann boundary-value problems with infinite exponential index was constructed in the 60`s by Govorov; it is presented in he monograph. Govorov`s theory was generalized by a number of authors in different directions, As a rule, in this theory and its generalizations, the contour on which the boundary condition is defined is assumed to be rectilinear. Exceptions appear in the work of Govorov himself and Alekhno, where the contour is assumed to be smooth and to have a tangent at infinity; another exception may be found in Danilov, where the contour is a logarithmic spiral. In this investigations it was also assumed that the argument of the coefficient of the problem was exponential asymptotics at infinity. In the present paper we consider a homogeneous boundary-value problem for a more general contour and a coefficient argument that is less well behaved.