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## Über dieses Buch

The purpose of this book is to provide an integrated course in real and complex analysis for those who have already taken a preliminary course in real analysis. It particularly emphasises the interplay between analysis and topology.

Beginning with the theory of the Riemann integral (and its improper extension) on the real line, the fundamentals of metric spaces are then developed, with special attention being paid to connectedness, simple connectedness and various forms of homotopy. The final chapter develops the theory of complex analysis, in which emphasis is placed on the argument, the winding number, and a general (homology) version of Cauchy's theorem which is proved using the approach due to Dixon.

Special features are the inclusion of proofs of Montel's theorem, the Riemann mapping theorem and the Jordan curve theorem that arise naturally from the earlier development. Extensive exercises are included in each of the chapters, detailed solutions of the majority of which are given at the end. From Real to Complex Analysis is aimed at senior undergraduates and beginning graduate students in mathematics. It offers a sound grounding in analysis; in particular, it gives a solid base in complex analysis from which progress to more advanced topics may be made.

## Inhaltsverzeichnis

### Chapter 1. The Riemann Integral

Abstract
An account is given of the Riemann integral for real-valued functions defined on intervals of the real line, a rapid development of the topic made possible by use of the Darboux approach in place of that originally adopted by Riemann. The sense in which integration is the inverse of differentiation is investigated. To cope with the demands of the later chapters the improper Riemann integral is introduced. Uniform convergence of sequences and series is defined and its usefulness in interchanging integration and limits established; to help with circumstances in which uniform convergence is not present, Arzelà’s theorem is proved.
R. H. Dyer, D. E. Edmunds

### Chapter 2. Metric Spaces

Abstract
Metric spaces are introduced and many examples given of these structures. The core properties of completeness, compactness, connectedness and simple connectedness are examined; compactness and connectedness are motivated in a variety of ways. Special attention is paid to various forms of homotopy, and a natural link between simple connectedness and the fundamental group is established. Applications to differential equations are given.
R. H. Dyer, D. E. Edmunds

### Chapter 3. Complex Analysis

Abstract
An account is given of the Riemann integral for real-valued functions defined on intervals of the real line, a rapid development of the topic made possible by use of the Darboux approach in place of that originally adopted by Riemann. The sense in which integration is the inverse of differentiation is investigated. To cope with the demands of the later chapters the improper Riemann integral is introduced. Uniform convergence of sequences and series is defined and its usefulness in interchanging integration and limits established; to help with circumstances in which uniform convergence is not present, Arzelà’s theorem is proved.
R. H. Dyer, D. E. Edmunds

### Backmatter

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