Lecture Notes on Complex Analysis

by Ivan Francis Wilde

Overview

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.

Product Details

ISBN-13:	9781860946431
Publisher:	Imperial College Press
Publication date:	04/28/2006
Pages:	260
Product dimensions:	5.90(w) x 8.90(h) x 0.50(d)

Table of Contents

Preface     vii
Complex Numbers     1
Informal Introduction     1
Complex Plane     2
Properties of the Modulus     4
The Argument of a Complex Number     8
Formal Construction of Complex Numbers     12
The Riemann Sphere and the Extended Complex Plane     14
Sequences and Series     17
Complex Sequences     17
Subsequences     17
Convergence of Sequences     18
Cauchy Sequences     21
Complex Series     23
Absolute Convergence     24
nth-Root Test     25
Ratio Test     26
Metric Space Properties of the Complex Plane     29
Open Discs and Interior Points     29
Closed Sets     32
Limit Points     34
Closure of a Set     36
Boundary of a Set     38
Cantor's Theorem     40
Compact Sets     41
Polygons and Paths in C     49
Connectedness     51
Domains     56
Analytic Functions     59
Complex-Valued Functions     59
Continuous Functions     59
Complex Differentiable Functions     61
Cauchy-Riemann Equations     66
Analytic Functions     70
Power Series     73
The Derived Series     74
Identity Theorem for Power Series     77
The Complex Exponential and Trigonometric Functions     79
The Functions exp z, sin z and cos z     79
Complex Hyperbolic Functions     80
Properties of exp z     80
Properties of sin z and cos z     83
Addition Formulae     84
The Appearance of [pi]     86
Inverse Trigonometric Functions     89
More on exp z and the Zeros of sin z and cos z     91
The Argument Revisited     92
Arg z is Continuous in the Cut-Plane     94
The Complex Logarithm     97
Introduction     97
The Complex Logarithm and its Properties     98
Complex Powers     100
Branches of the Logarithm     103
Complex Integration     111
Paths and Contours     111
The Length of a Contour     113
Integration along a Contour     115
Basic Estimate     120
Fundamental Theorem of Calculus      121
Primitives     123
Cauchy's Theorem     127
Cauchy's Theorem for a Triangle     127
Cauchy's Theorem for Star-Domains     133
Deformation Lemma     136
Cauchy's Integral Formula     138
Taylor Series Expansion     139
Cauchy's Integral Formulae for Derivatives     142
Morera's Theorem     145
Cauchy's Inequality and Liouville's Theorem     146
Identity Theorem     149
Preservation of Angles     154
The Laurent Expansion     157
Laurent Expansion     157
Uniqueness of the Laurent Expansion     163
Singularities and Meromorphic Functions     167
Isolated Singularities     167
Behaviour near an Isolated Singularity     169
Behaviour as [vertical bar] z [vertical bar] to [infinity]     172
Casorati-Weierstrass Theorem     174
Theory of Residues     175
Residues     175
Winding Number (Index)     177
Cauchy's Residue Theorem     179
The Argument Principle     185
Zeros and Poles     185
Argument Principle     187
Rouche's Theorem      189
Open Mapping Theorem     193
Maximum Modulus Principle     195
Mean Value Property     195
Maximum Modulus Principle     196
Minimum Modulus Principle     200
Functions on the Unit Disc     201
Hadamard's Theorem and the Three Lines Lemma     204
Mobius Transformations     207
Special Transformations     207
Inversion     209
Mobius Transformations     210
Mobius Transformations in the Extended Complex Plane     215
Harmonic Functions     219
Harmonic Functions     219
Local Existence of a Harmonic Conjugate     220
Maximum and Minimum Principle     221
Local Properties of Analytic Functions     223
Local Uniform Convergence     223
Hurwitz's Theorem     226
Vitali's Theorem     229
Some Results from Real Analysis     231
Completeness of R     231
Bolzano-Weierstrass Theorem     233
Comparison Test for Convergence of Series     235
Dirichlet's Test     235
Alternating Series Test     236
Continuous Functions on [a, b] Attain their Bounds      236
Intermediate Value Theorem     238
Rolle's Theorem     238
Mean Value Theorem     239
Bibliography     241
Index     243