Introduction: What are Principal Functions?

Abstract

The existence of certain harmonic functions with prescribed singularities and prescribed boundary behavior is central to a variety of areas of complex function theory. It provides the unifying factor for all the topics of this book. Such functions will be called principal functions. For further orientation let us consider how they are constructed and some of their uses.

The purpose of this Introduction is to give a sampling of principal functions and their various applications. It is not needed for a systematic reading of the ensuing chapters, and some readers may prefer to return to it after acquainting themselves with the concepts and proofs in the book proper. However, we hope that this bird’s-eye view, while leaving many details dim, will help the reader discern the simple common idea of principal functions in its various apparently different realizations.