| 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 |
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