## 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. The first half, more or less, can be used for a one-semester course addressed to undergraduates. The second half can be used for a second semester, at either level. Somewhat more material has been included than can be covered at leisure in one or two terms, to give opportunities for the instructor to exercise individual taste, and to lead the course in whatever directions strikes the instructor's fancy at the time as well as extra read ing material for students on their own. A large number of routine exer cises 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 recommend to anyone to look through them. More recent texts have emphasized connections with real analysis, which is important, but at the cost of exhibiting succinctly and clearly what is peculiar about complex analysis: the power series expansion, the uniqueness of analytic continuation, and the calculus of residues. |

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### Contents

3 | |

9 | |

17 | |

5 Complex Differentiability | 27 |

CHAPTER VIII | 31 |

7 Angles Under Holomorphic Maps | 33 |

2 Convergent Power Series | 47 |

3 Relations Between Formal and Convergent Series | 60 |

5 Fractional Linear Transformations | 227 |

Harmonic Functions | 237 |

2 Examples | 247 |

3 Basic Properties of Harmonic Functions | 254 |

4 The Poisson Formula | 264 |

PART | 276 |

3 Application of Schwarz Reflection | 287 |

2 Compact Sets in Function Spaces | 293 |

4 Analytic Functions | 68 |

6 The Inverse and Open Mapping Theorems | 76 |

7 The Local Maximum Modulus Principle | 83 |

2 Integrals Over Paths | 94 |

3 Local Primitive for a Holomorphic Function | 104 |

4 Another Description of the Integral Along a Path | 110 |

5 The Homotopy Form of Cauchys Theorem | 116 |

7 The Local Cauchy Formula | 126 |

CHAPTER IV | 133 |

3 Artins Proof | 149 |

CHAPTER V | 156 |

3 Isolated Singularities | 165 |

CHAPTER VI | 173 |

2 Evaluation of Definite Integrals | 191 |

CHAPTER VII | 208 |

3 The Upper Half Plane | 215 |

4 Behavior at the Boundary | 299 |

CHAPTER XI | 307 |

2 The Dilogarithm | 315 |

PART THREE | 321 |

2 The PicardBorel Theorem | 330 |

3 Bounds by the Real Part BorelCarathéodory Theorem | 338 |

5 Entire Functions with Rational Values | 344 |

CHAPTER XIII | 351 |

3 Functions of Finite Order | 366 |

3 The Addition Theorem | 383 |

2 The Gamma Function | 396 |

3 The Lerch Formula | 412 |

2 The Main Lemma and its Application | 428 |

3 Analytic Differential Equations | 442 |

455 | |

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### Common terms and phrases

analytic function apply assume boundary bounded called Cauchy's centered Chapter circle closed path coefficients compact complex numbers conclude consider constant contained converges absolutely curve define definition derivative determine differentiable disc entire function equal equation estimate Example EXERCISES exists expression f be analytic f(zo Figure finite follows formal formula function f give given harmonic Hence immediately inside integral interval inverse Lemma Let f Let f(z limit means meromorphic Note obtained open set origin path pole polynomial positive power series power series expansion primitive Proof prove radius of convergence real numbers rectangle Remark residue sequence Show shown side simple sufficiently Suppose tends Theorem Theorem 3.2 uniformly unit disc upper half plane usual write zeros