An Introduction to Complex Analysis and Geometry
                       John P. D’Angelo
                       Dept. of Mathematics, Univ. of Illinois, 1409 W. Green St., Urbana IL 61801
                       jpda@math.uiuc.edu
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2009 by John P. D’Angelo
                                                            Contents
                            Chapter 1.  From the real numbers to the complex numbers                   11
                              1.  Introduction                                                         11
                              2.  Number systems                                                       11
                              3.  Inequalities and ordered fields                                       16
                              4.  The complex numbers                                                  24
                              5.  Alternative definitions of C                                          26
                              6.  Aglimpse at metric spaces                                            30
                            Chapter 2.  Complex numbers                                                35
                              1.  Complex conjugation                                                  35
                              2.  Existence of square roots                                            37
                              3.  Limits                                                               39
                              4.  Convergent infinite series                                            41
                              5.  Uniform convergence and consequences                                 44
                              6.  The unit circle and trigonometry                                     50
                              7.  The geometry of addition and multiplication                          53
                              8.  Logarithms                                                           54
                            Chapter 3.  Complex numbers and geometry                                   59
                              1.  Lines, circles, and balls                                            59
                              2.  Analytic geometry                                                    62
                              3.  Quadratic polynomials                                                63
                              4.  Linear fractional transformations                                    69
                              5.  The Riemann sphere                                                   73
                            Chapter 4.  Power series expansions                                        75
                              1.  Geometric series                                                     75
                              2.  The radius of convergence                                            78
                              3.  Generating functions                                                 80
                              4.  Fibonacci numbers                                                    82
                              5.  An application of power series                                       85
                              6.  Rationality                                                          87
                            Chapter 5.  Complex differentiation                                         91
                              1.  Definitions of complex analytic function                              91
                              2.  Complex differentiation                                               92
                              3.  The Cauchy-Riemann equations                                         94
                              4.  Orthogonal trajectories and harmonic functions                       97
                              5.  Aglimpse at harmonic functions                                       98
                              6.  What is a differential form?                                         103
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                            4                                 CONTENTS
                            Chapter 6.  Complex integration                                           107
                              1.  Complex-valued functions                                            107
                              2.  Line integrals                                                      109
                              3.  Goursat’s proof                                                     116
                              4.  The Cauchy integral formula                                         119
                              5.  Areturn to the definition of complex analytic function               124
                            Chapter 7.  Applications of complex integration                           127
                              1.  Singularities and residues                                          127
                              2.  Evaluating real integrals using complex variables methods           129
                              3.  Fourier transforms                                                  136
                              4.  The Gamma function                                                  138
                            Chapter 8.  Additional Topics                                             143
                              1.  The minimum-maximum theorem                                         143
                              2.  The fundamental theorem of algebra                                  144
                              3.  Winding numbers, zeroes, and poles                                  147
                              4.  Pythagorean triples                                                 152
                              5.  Elementary mappings                                                 155
                              6.  Quaternions                                                         158
                              7.  Higher dimensional complex analysis                                 160
                              Further reading                                                         163
                            Bibliography                                                              165
                            Index                                                                     167