Lecture Notes on Complex Analysis

"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."-- Publisher's website Contents......Page 10 Preface......Page 8 1.1 Informal Introduction......Page 14 1.2 Complex Plane......Page 15 1.3 Properties of the Modulus......Page 17 1.4 The Argument of a Complex Number......Page 21 1.5 Formal Construction of Complex Numbers......Page 25 1.6 The Riemann Sphere and the Extended Complex Plane......Page 27 2.2 Subsequences......Page 30 2.3 Convergence of Sequences......Page 31 2.4 Cauchy Sequences......Page 34 2.5 Complex Series......Page 36 2.6 Absolute Convergence......Page 37 2.7 nth-Root Test......Page 38 2.8 Ratio Test......Page 39 3.1 Open Discs and Interior Points......Page 42 3.2 Closed Sets......Page 45 3.3 Limit Points......Page 47 3.4 Closure of a Set......Page 49 3.5 Boundary of a Set......Page 51 3.6 Cantor's Theorem......Page 53 3.7 Compact Sets......Page 54 3.8 Polygons and Paths in C......Page 62 3.9 Connectedness......Page 64 3.10 Domains......Page 69 4.2 Continuous Functions......Page 72 4.3 Complex Differentiable Functions......Page 74 4.4 Cauchy-Riemann Equations......Page 79 4.5 Analytic Functions......Page 83 4.6 Power Series......Page 86 4.7 The Derived Series......Page 87 4.8 Identity Theorem for Power Series......Page 90 5.1 The Functions exp z sin z and cos z......Page 92 5.3 Properties of exp z......Page 93 5.4 Properties of sin z and cos z......Page 96 5.5 Addition Formulae......Page 97 5.6 The Appearance of TT......Page 99 5.7 Inverse Trigonometric Functions......Page 102 5.8 More on exp z and the Zeros of sin z and cos z......Page 104 5.9 The Argument Revisited......Page 105 5.10 Arg z is Continuous in the Cut-Plane......Page 107 6.1 Introduction......Page 110 6.2 The Complex Logarithm and its Properties......Page 111 6.3 Complex Powers......Page 113 6.4 Branches of the Logarithm......Page 116 7.1 Paths and Contours......Page 124 7.2 The Length of a Contour......Page 126 7.3 Integration along a Contour......Page 128 7.4 Basic Estimate......Page 133 7.5 Fundamental Theorem of Calculus......Page 134 7.6 Primitives......Page 136 8.1 Cauchy's Theorem for a Triangle......Page 140 8.2 Cauchy's Theorem for Star-Domains......Page 146 8.3 Deformation Lemma......Page 149 8.4 Cauchy's Integral Formula......Page 151 8.5 Taylor Series Expansion......Page 152 8.6 Cauchy's Integral Formulae for Derivatives......Page 155 8.7 Morera's Theorem......Page 158 8.8 Cauchy's Inequality and Liouville's Theorem......Page 159 8.9 Identity Theorem......Page 162 8.10 Preservation of Angles......Page 167 9.1 Laurent Expansion......Page 170 9.2 Uniqueness of the Laurent Expansion......Page 176 10.1 Isolated Singularities......Page 180 10.2 Behaviour near an Isolated Singularity......Page 182 10.3 Behaviour as |z|-> oo......Page 185 10.4 Casorati-Weierstrass Theorem......Page 187 11.1 Residues......Page 188 11.2 Winding Number (Index)......Page 190 11.3 Cauchy's Residue Theorem......Page 192 12.1 Zeros and Poles......Page 198 12.2 Argument Principle......Page 200 12.3 Rouche's Theorem......Page 202 12.4 Open Mapping Theorem......Page 206 13.1 Mean Value Property......Page 208 13.2 Maximum Modulus Principle......Page 209 13.3 Minimum Modulus Principle......Page 213 13.4 Functions on the Unit Disc......Page 214 13.5 Hadamard's Theorem and the Three Lines Lemma......Page 217 14.1 Special Transformations......Page 220 14.2 Inversion......Page 222 14.3 Mobius Transformations......Page 223 14.4 Mobius Transformations in the Extended Complex Plane......Page 228 15.1 Harmonic Functions......Page 232 15.2 Local Existence of a Harmonic Conjugate......Page 233 15.3 Maximum and Minimum Principle......Page 234 16.1 Local Uniform Convergence......Page 236 16.2 Hurwitz's Theorem......Page 239 16.3 Vitali's Theorem......Page 242 A.1 Completeness of R......Page 244 A.2 Bolzano-Weierstrass Theorem......Page 246 A.4 Dirichlet's Test......Page 248 A.6 Continuous Functions on [a b] Attain their Bounds......Page 249 A.8 Rolle's Theorem......Page 251 A.9 Mean Value Theorem......Page 252 Bibliography......Page 254 Index......Page 256

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 [pi], 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.

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. This work aims to provide a course on complex analysis.