Riemann Boundary Problem with an Infinite Index When the Verticity Index is Less Than 1/2

Abstract

Let D be a domain in the complex z-plane such that the boundary ∂D is a simple smooth curve L which begins at the point z = t ₀ and ends at z = ∞. In what follows the letters t and τ will denote the points of the curve L. Denote by ψ(t) the angle between the tangent to the contour L at the point t and the positive real axis. Since L is smooth, the function ψ(t) is continuous at all points t ∈ L including the point t = ∞ (the latter means that ψ(t) approaches a limit as t → ∞, t ∈ L). Everywhere below we assume that lim _t→∞ ψ(t) = 0, ℜt ₀ > 0 and 0 ∉L. These assumptions do not restrict generality, since they may be satisfied by applying an appropriate rotation and shift.