Lecture Notes on Complex Analysis

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Imperial College Press, 2006 - Mathematics - 245 pages
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This book is based on lectures presented over many years to second and third year mathematics students in the Mathematics Departments at Bedford College, London, and King's College, London, as part of the BSc. and MSci. program. Its aim is to provide a gentle yet rigorous first course on complex analysis.Metric space aspects of the complex plane are discussed in detail, making this text an excellent introduction to metric space theory. The complex exponential and trigonometric functions are defined from first principles and great care is taken to derive their familiar properties. In particular, the appearance of π, in this context, is carefully explained.The central results of the subject, such as Cauchy's Theorem and its immediate corollaries, as well as the theory of singularities and the Residue Theorem are carefully treated while avoiding overly complicated generality. Throughout, the theory is illustrated by examples.A number of relevant results from real analysis are collected, complete with proofs, in an appendix.The approach in this book attempts to soften the impact for the student who may feel less than completely comfortable with the logical but often overly concise presentation of mathematical analysis elsewhere.
 

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Contents

Complex Numbers
1
Sequences and Series
17
Metric Space Properties of the Complex Plane
29
Analytic Functions
59
The Complex Exponential and Trigonometric Functions
79
The Complex Logarithm
97
Complex Integration
111
Cauchys Theorem
127
Singularities and Meromorphic Functions
167
Theory of Residues
175
The Argument Principle
185
Maximum Modulus Principle
195
Mobius Transformations
207
Harmonic Functions
219
Appendix A Some Results from Real Analysis
231
Bibliography
241

The Laurent Expansion
157

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