## Complex AnalysisThe present book is meant as a text for a course on complex analysis at the advanced undergraduate level, or first-year graduate level. Somewhat more material has been included than can be covered at leisure in one term, to give opportunities for the instructor to exercise his taste, and lead the course in whatever direction strikes his fancy at the time. A large number of routine exercises are included for the more standard portions, and a few harder exercises of striking theoretical interest are also included, but may be omitted in courses addressed to less advanced students. In some sense, I think the classical German prewar texts were the best (Hurwitz-Courant, Knopp, Bieberbach, etc. ) and I would recom mend to anyone to look through them. More recent texts have empha sized connections with real analysis, which is important, but at the cost of exhibiting succinctly and clearly what is peculiar about complex anal ysis: the power series expansion, the uniqueness of analytic continuation, and the calculus of residues. The systematic elementary development of formal and convergent power series was standard fare in the German texts, but only Cartan, in the more recent books, includes this material, which I think is quite essential, e. g. , for differential equations. I have written a short text, exhibiting these features, making it applicable to a wide variety of tastes. The book essentially decomposes into two parts. |

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### Contents

3 | |

1 Definition | 11 |

4 Limits and Compact Sets | 17 |

5 Complex Differentiability | 28 |

CHAPTER VIII | 32 |

7 Angles Under Holomorphic Maps | 34 |

2 Convergent Power Series | 49 |

3 Relations Between Formal and Convergent Series | 62 |

2 Evaluation of Definite Integrals | 180 |

CHAPTER VII | 196 |

3 The Upper Half Plane | 203 |

5 Fractional Linear Transformations | 215 |

Harmonic Functions | 224 |

2 Examples | 234 |

3 Basic Properties of Harmonic Functions | 241 |

5 The Poisson Representation | 249 |

4 Analytic Functions | 69 |

6 The Local Maximum Modulus Principle | 79 |

CHAPTER III | 87 |

2 Integrals Over Paths | 94 |

3 Local Primitive for a Holomorphic Function | 103 |

4 Another Description of the Integral Along a Path | 109 |

5 The Homotopy Form of Cauchys Theorem | 115 |

CHAPTER IV | 123 |

3 Artins Proof | 137 |

CHAPTER V | 144 |

2 Laurent Series | 151 |

4 Dixons Proof of Cauchys Theorem | 162 |

2 The Effect of Small Derivatives | 260 |

4 The PhragmenLindelöf and Hadamard Theorems | 268 |

CHAPTER X | 276 |

3 Functions of Finite Order | 286 |

CHAPTER XI | 292 |

3 The Addition Theorem | 299 |

CHAPTER XII | 307 |

CHAPTER XIII | 324 |

CHAPTER XIV | 340 |

Appendix | 359 |

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### Common terms and phrases

analytic continuation analytic function analytic isomorphism apply assume automorphism boundary bounded calculus Cauchy's theorem Chapter closed disc closed path coefficients complex numbers concludes the proof connected open set constant term contained continuous function converges absolutely converges uniformly defined deleted derivative differentiable disc of radius end point entire function equal equation Example Exercise exists f be analytic f be holomorphic f(zo Figure finite number follows formal power series function f Hence holomorphic function homologous homotopy infinity integral of f interval Laurent series Lemma Let f Let f(z maximum modulus principle meromorphic neighborhood open disc open set partition pole polynomial power series expansion primitive proves the theorem radius of convergence real axis real numbers rectangle residue right-hand side say that f segment sequence simply connected Suppose Theorem 1.2 unit circle unit disc upper half plane whence winding number write