The Hilbert and Riemann-Hilbert Boundary Problems

Abstract

Let S⁺ be a connected region, bounded by smooth contours L_o, L₁ x2026; L_p, not intersecting one another, the first of which encloses all the others (cf. Fig. 8, § 24). The contour L _o may be absent in which case S⁺ is an infinite region (the plane with holes). By L will be denoted the union of L_o,L₁ … L_p(as before the positive direction of L will be such that S⁺ lies to the left when L is described in that direction), by S⁻ that part of the plane which is the complement of S+ + L, by S_o⁻, S₁⁻… S_p⁻ the components of S⁻, bounded respectively by L_o, L₁ … L_p (the first of those regions is absent, if there is no L_o; S_o⁻ is infinite, if L_o exists).