This book discusses a variety of problems which are usually treated in a second course on the theory of functions of one complex variable, the level being gauged for graduate students. It treats several topics in geometric function theory as well as potential theory in the plane, covering in particular: conformal equivalence for simply connected regions, conformal equivalence for finitely connected regions, analytic covering maps, de Branges' proof of the Bieberbach conjecture, harmonic functions, Hardy spaces on the disk, potential theory in the plane. A knowledge of integration theory and functional analysis is assumed.

1 The Complex Number System
2 Metric Spaces and the Topology of C
3 Elementary Properties and Examples of Analytic Functions
4 Complex Integration
5 Singularties
6 The Maximum Modulus Theorem 
7 Compactness and Convergence in the Space of Analytic Functions
8 Runge's Theorem
9 Analytic Continuation and Rimann Surfaces
10 Harmonic Functions
11 Entire Functions
12 The Range of an Analytic Function
TABLE OF CONTENTS, VOLUME Ⅱ
13 Return Basic
14 Conformal Equivalence For Simply Connected Regions
15 Conformal Equivalence For Finitely Connected Regions
16 Analytic Covering Maps
17 De Branges's Proof of the Bieberbach Conjecture
18 Some Fundamental Concepts From Analysis
19 Harmonic Functions Redux
20 Hardy Spaces on the Disk
21 Potential Theory in the Plane
References
List of Symbols
Index

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