Introduction

In Chapter 3 we discussed the correspondence between the boundary C of a Jordan domain D and the circumference K of the unit disc U: {z: |z| < 1} under a conformal mapping of the disc onto the simply connected open set D; we also proved (Theorem 3.2) that the mapping induces a one-to-one correspondence between the individual points of K and the individual points of C = FR D and that this correspondence is in fact topological. If w = f(z) is the mapping function and eiθ is an arbitrary point of K, then the cluster set C(f, eiθ) is degenerate in this case, and we may write f(eiθ) = C(f, eiθ), where f(eiθ) is a continuous onevalued function of eiθ in 0 ≤ θ ≤ 2π, giving a topological mapping of K onto FR D. This can no longer be asserted if the condition that FR D be a Jordan curve is removed and the only condition imposed on D is that it be a simply connected domain. By the Riemann mapping theorem † there is a univalent regular function w = f(z) which maps the open disc U conformally onto D in this general case also. The cluster set C(f, eiθ) of the mapping function f(z) then exists for every point eiθ ∊ K, but we have no assurance that it is degenerate at any given point, or indeed, at any point of K.