Lecture Notes on Complex Analysis
by Wilde, Ivan FrancisFormat: Paperback
Pub. Date: 2006-06-15
Publisher(s): World Scientific Pub Co Inc
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Summary
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.
Table of Contents
| Preface | p. vii |
| Complex Numbers | p. 1 |
| Informal Introduction | p. 1 |
| Complex Plane | p. 2 |
| Properties of the Modulus | p. 4 |
| The Argument of a Complex Number | p. 8 |
| Formal Construction of Complex Numbers | p. 12 |
| The Riemann Sphere and the Extended Complex Plane | p. 14 |
| Sequences and Series | p. 17 |
| Complex Sequences | p. 17 |
| Subsequences | p. 17 |
| Convergence of Sequences | p. 18 |
| Cauchy Sequences | p. 21 |
| Complex Series | p. 23 |
| Absolute Convergence | p. 24 |
| nth-Root Test | p. 25 |
| Ratio Test | p. 26 |
| Metric Space Properties of the Complex Plane | p. 29 |
| Open Discs and Interior Points | p. 29 |
| Closed Sets | p. 32 |
| Limit Points | p. 34 |
| Closure of a Set | p. 36 |
| Boundary of a Set | p. 38 |
| Cantor's Theorem | p. 40 |
| Compact Sets | p. 41 |
| Polygons and Paths in C | p. 49 |
| Connectedness | p. 51 |
| Domains | p. 56 |
| Analytic Functions | p. 59 |
| Complex-Valued Functions | p. 59 |
| Continuous Functions | p. 59 |
| Complex Differentiable Functions | p. 61 |
| Cauchy-Riemann Equations | p. 66 |
| Analytic Functions | p. 70 |
| Power Series | p. 73 |
| The Derived Series | p. 74 |
| Identity Theorem for Power Series | p. 77 |
| The Complex Exponential and Trigonometric Functions | p. 79 |
| The Functions exp z, sin z and cos z | p. 79 |
| Complex Hyperbolic Functions | p. 80 |
| Properties of exp z | p. 80 |
| Properties of sin z and cos z | p. 83 |
| Addition Formulae | p. 84 |
| The Appearance of [pi] | p. 86 |
| Inverse Trigonometric Functions | p. 89 |
| More on exp z and the Zeros of sin z and cos z | p. 91 |
| The Argument Revisited | p. 92 |
| Arg z is Continuous in the Cut-Plane | p. 94 |
| The Complex Logarithm | p. 97 |
| Introduction | p. 97 |
| The Complex Logarithm and its Properties | p. 98 |
| Complex Powers | p. 100 |
| Branches of the Logarithm | p. 103 |
| Complex Integration | p. 111 |
| Paths and Contours | p. 111 |
| The Length of a Contour | p. 113 |
| Integration along a Contour | p. 115 |
| Basic Estimate | p. 120 |
| Fundamental Theorem of Calculus | p. 121 |
| Primitives | p. 123 |
| Cauchy's Theorem | p. 127 |
| Cauchy's Theorem for a Triangle | p. 127 |
| Cauchy's Theorem for Star-Domains | p. 133 |
| Deformation Lemma | p. 136 |
| Cauchy's Integral Formula | p. 138 |
| Taylor Series Expansion | p. 139 |
| Cauchy's Integral Formulae for Derivatives | p. 142 |
| Morera's Theorem | p. 145 |
| Cauchy's Inequality and Liouville's Theorem | p. 146 |
| Identity Theorem | p. 149 |
| Preservation of Angles | p. 154 |
| The Laurent Expansion | p. 157 |
| Laurent Expansion | p. 157 |
| Uniqueness of the Laurent Expansion | p. 163 |
| Singularities and Meromorphic Functions | p. 167 |
| Isolated Singularities | p. 167 |
| Behaviour near an Isolated Singularity | p. 169 |
| Behaviour as [vertical bar] z [vertical bar] to [infinity] | p. 172 |
| Casorati-Weierstrass Theorem | p. 174 |
| Theory of Residues | p. 175 |
| Residues | p. 175 |
| Winding Number (Index) | p. 177 |
| Cauchy's Residue Theorem | p. 179 |
| The Argument Principle | p. 185 |
| Zeros and Poles | p. 185 |
| Argument Principle | p. 187 |
| Rouche's Theorem | p. 189 |
| Open Mapping Theorem | p. 193 |
| Maximum Modulus Principle | p. 195 |
| Mean Value Property | p. 195 |
| Maximum Modulus Principle | p. 196 |
| Minimum Modulus Principle | p. 200 |
| Functions on the Unit Disc | p. 201 |
| Hadamard's Theorem and the Three Lines Lemma | p. 204 |
| Mobius Transformations | p. 207 |
| Special Transformations | p. 207 |
| Inversion | p. 209 |
| Mobius Transformations | p. 210 |
| Mobius Transformations in the Extended Complex Plane | p. 215 |
| Harmonic Functions | p. 219 |
| Harmonic Functions | p. 219 |
| Local Existence of a Harmonic Conjugate | p. 220 |
| Maximum and Minimum Principle | p. 221 |
| Local Properties of Analytic Functions | p. 223 |
| Local Uniform Convergence | p. 223 |
| Hurwitz's Theorem | p. 226 |
| Vitali's Theorem | p. 229 |
| Some Results from Real Analysis | p. 231 |
| Completeness of R | p. 231 |
| Bolzano-Weierstrass Theorem | p. 233 |
| Comparison Test for Convergence of Series | p. 235 |
| Dirichlet's Test | p. 235 |
| Alternating Series Test | p. 236 |
| Continuous Functions on [a, b] Attain their Bounds | p. 236 |
| Intermediate Value Theorem | p. 238 |
| Rolle's Theorem | p. 238 |
| Mean Value Theorem | p. 239 |
| Bibliography | p. 241 |
| Index | p. 243 |
| Table of Contents provided by Ingram. All Rights Reserved. |
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