Functions of One Complex Variable I [Hardcover]

This book presents a basic introduction to complex analysis in both an interesting and a rigorous manner. It contains enough material for a full year's course, and the choice of material treated is reasonably standard and should be satisfactory for most first courses in complex analysis. The approach to each topic appears to be carefully thought out both as to mathematical treatment and pedagogical presentation, and the end result is a very satisfactory book. --MATHSCINET

This book is intended as a textbook for a first course in the theory of functions of one complex variable for students who are mathematically mature enough to understand and execute E - 8 arguments. The actual pre? requisites for reading this book are quite minimal; not much more than a stiff course in basic calculus and a few facts about partial derivatives. The topics from advanced calculus that are used (e.g., Leibniz's rule for differ? entiating under the integral sign) are proved in detail. Complex Variables is a subject which has something for all mathematicians. In addition to having applications to other parts of analysis, it can rightly claim to be an ancestor of many areas of mathematics (e.g., homotopy theory, manifolds). This view of Complex Analysis as An Introduction to Mathe? matics has influenced the writing and selection of subject matter for this book. The other guiding principle followed is that all definitions, theorems, etc.I. The Complex Number System.- ?1. The real numbers.- ?2. The field of complex numbers.- ?3. The complex plane.- ?4. Polar representation and roots of complex numbers.- ?5. Lines and half planes in the complex plane.- ?6. The extended plane and its spherical representation.- II. Metric Spaces and the Topology of ?.- ?1. Definition and examples of metric spaces.- ?2. Connectedness.- ?3. Sequences and completeness.- ?4. Compactness.- ?5. Continuity.- ?6. Uniform convergence.- III. Elementary Properties and Examples of Analytic Functions.-l£Ý