Book contents
- Frontmatter
- Contents
- Preface
- Chapter 1 INTRODUCTION
- Chapter 2 FUNCTIONS ANALYTIC IN A CIRCULAR DISC
- Chapter 3 TOPICS IN THE THEORY OF CONFORMAL MAPPING
- Chapter 4 INTRINSIC PROPERTIES OF CLUSTER SETS
- Chapter 5 CLUSTER SETS OF FUNCTIONS ANALYTIC IN THE UNIT DISC
- Chapter 6 BOUNDARY THEORY IN THE LARGE
- Chapter 7 BOUNDARY THEORY IN THE SMALL
- Chapter 8 FURTHER BOUNDARY PROPERTIES OF FUNCTIONS MEROMORPHIC IN THE DISC. CLASSIFICATION OF SINGULARITIES
- Chapter 9 PRIME ENDS
- Bibliography
- Index of symbols
- Index
Chapter 9 - PRIME ENDS
Published online by Cambridge University Press: 06 November 2009
- Frontmatter
- Contents
- Preface
- Chapter 1 INTRODUCTION
- Chapter 2 FUNCTIONS ANALYTIC IN A CIRCULAR DISC
- Chapter 3 TOPICS IN THE THEORY OF CONFORMAL MAPPING
- Chapter 4 INTRINSIC PROPERTIES OF CLUSTER SETS
- Chapter 5 CLUSTER SETS OF FUNCTIONS ANALYTIC IN THE UNIT DISC
- Chapter 6 BOUNDARY THEORY IN THE LARGE
- Chapter 7 BOUNDARY THEORY IN THE SMALL
- Chapter 8 FURTHER BOUNDARY PROPERTIES OF FUNCTIONS MEROMORPHIC IN THE DISC. CLASSIFICATION OF SINGULARITIES
- Chapter 9 PRIME ENDS
- Bibliography
- Index of symbols
- Index
Summary
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.
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- Information
- The Theory of Cluster Sets , pp. 167 - 189Publisher: Cambridge University PressPrint publication year: 1966