Complex Analysis

Description

Complex analysis can be a difficult subject and many introductory texts are just too ambitious for today’s students. This book takes a lower starting point than is traditional and concentrates on explaining the key ideas through worked examples and informal explanations, rather than through "dry" theory. Preface......Page 8 Contents......Page 10 1. What Do I Need to Know......Page 14 1.2 Numbers......Page 15 1.3 Sequences and Series......Page 17 1.4 Functions and Continuity......Page 20 1.5 Differentiation......Page 23 1.6 Integration......Page 25 1.7 Infinite Integrals......Page 27 1.8 Calculus of Two Variables......Page 30 2.1 Are Complex Numbers Necessary......Page 32 2.2 Basic Properties of Complex Numbers......Page 34 3.1 Why is Complex Analysis Possible......Page 48 3.2 Some Useful Terminology......Page 50 3.3 Functions and Continuity......Page 54 3.4 The big-oh and small-oh Notations......Page 59 4.1 Differentiability......Page 64 4.2 Power Series......Page 74 4.3 Logarithms......Page 84 4.4 Cuts and Branch Points......Page 87 4.5 Singularities......Page 88 5.1 The Reine-Borel Theorem......Page 92 5.2 Parametric Representation......Page 96 5.3 Integration......Page 102 5.4 Estimation......Page 111 5.5 Uniform Convergence......Page 116 6.1 Cauchy's Theorem: A First Approach......Page 120 6.2 Cauchy's Theorem: A More General Version......Page 124 6.3 Deformation......Page 128 7.1 Cauchy's Integral Formula......Page 132 7.2 The Fundamental Theorem of Algebra......Page 139 7.3 Logarithms......Page 141 7.4 Taylor Series......Page 144 8.1 Laurent Series......Page 150 8.2 Classification of Singularities......Page 157 8.3 The Residue Theorem......Page 159 9.1 Real Integrals: Semicircular Contours......Page 166 9.2 Integrals Involving Circular Functions......Page 171 9.3 Real Integrals: Jordan's Lemma......Page 174 9.4 Real Integrals: Some Special Contours......Page 180 9.5 Infinite Series......Page 189 10.1 Integration of f'/f; Rouche's Theorem......Page 196 10.2 The Open Mapping Theorem......Page 201 10.3 Winding Numbers......Page 205 11.1 Preservation of Angles......Page 208 11.2 Harmonic Functions......Page 211 11.3 Mobius Transformations......Page 216 11.4 Other Transformations......Page 224 12.1 Riemann's Zeta Function......Page 230 12.2 Complex Iteration......Page 234 13. Solutions to Exercises......Page 238 Bibliography......Page 268 Index......Page 270