Synopses & Reviews
The book provides an introduction to the theory of cluster sets, a branch of topological analysis which has made great strides in recent years. The cluster set of a function at a particular point is the set of limit values of the function at that point which may be either a boundary point or (in the case of a non-analytic function) an interior point of the function's domain. In topological analysis, its main application is to problems arising in the theory of functions of a complex variable, with particular reference to boundary behaviour such as the theory of prime ends under conformal mapping. An important and novel feature of the book is the discussion of more general applications to non-analytic functions, including arbitrary functions. The authors assume a general familiarity with classical function theory but include the more specialised material required for the development of the theory of cluster sets, so making the treatment accessible to graduate students.
An introduction to the theory of cluster sets, a branch of topological analysis.
Table of Contents
Preface; 1. Introduction; 2. Functions analytic in a circular disc; 3. Topics in the theory of conformal mapping; 4. Intrinsic properties of cluster sets; 5. Cluster sets of functions analytic in the unit disc; 6. Boundary theory in the large; 7. Boundary theory in the small; 8. Further boundary properties of functions meromorphic in the disc. classification of singularities; 9. Prime ends; Bibliography; Index of symbols; Index.