With their focus on classical complex analysis, Hibschweiler (mathematics and statistics, U. of New Hampshire) and MacGregor (mathematics emeritus, State U. of New York) offer five decades of research and new results in this introduction to fractional Cauchy transforms. They cover basic properties, integral means and relationships between fractional Cauchy transforms and Hardy and Dirichlet spaces, radial and nontangential limits, multipliers, composition, univalent functions, and a characterization of Cauchy transforms, including one-to-one mapping between measures and functions, a key lemma for representation by a real measure, and sufficient conditions for representation of a Cauchy transform. The authors provide notes for each chapter which include references as well as additional comments and directions to related work