Linear Fractional Prologue

Abstract

Linear fractional transformations appear in this book at several different levels. Most frequently they play the traditional role of conformal changes of variable, allowing discs to be replaced by half-planes, and arbitrary points to be regarded as the origin. At a deeper level the linear fractional mappings that take the unit disc into itself provide a class of tractable examples diverse enough to reflect much of the character of more general situations. In fact, the Linear Fractional Model Theorem proclaims that every univalent self-map of the disc is conjugate (in a sense to be made precise in Chapters 6 and 8) to a linear fractional map.