## Theory of Complex FunctionsThe material from function theory, up to the residue calculus, is developed in a lively and vivid style, well motivated throughout by examples and practice exercises. Additionally, there is ample discussion of the historical evolution of the theory, biographical sketches of important contributors, and citations (original language together with English translation) from their classical works. Yet the book is far from being a mere history of function theory. Even experts will find here few new or long forgotten gems, like Eisenstein's novel approach to the circular functions. This book is destined to accompany many students making their way into a classical area of mathematics which represents the most fruitful example to date of the intimate connection between algebra and analysis. For exam preparation it offers quick access to the essential results and an abundance of interesting inducements. Teachers and interested mathematicians in finance, industry and science will also find reading it profitable, again and again referring to it with pleasure. |

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### Contents

Historical Introduction | 1 |

2 Fundamental topological concepts | 17 |

6 Connected spaces Regions in C | 39 |

1 Complexdifferentiable functions | 47 |

3 Holomorphic functions | 56 |

4 Partial differentiation with respect to x y z and z | 65 |

Holomorphy and Conformality Biholomorphic Mappings | 71 |

2 Biholomorphic mappings | 80 |

5 Special Taylor series Bernoulli numbers | 220 |

CauchyWeierstrassRiemann Function Theory | 227 |

3 The Cauchy estimates and inequalities for Taylor coefficients | 241 |

4 Convergence theorems of WeIERSTRASS | 248 |

5 The open mapping theorem and the maximum principle | 256 |

Miscellany | 265 |

3 Holomorphic logarithms and holomorphic roots | 276 |

6 Asymptotic power series developments | 294 |

Modes of Convergence in Function Theory | 91 |

2 Convergence criteria | 101 |

Chapter 4 Power Series | 109 |

2 Examples of convergent power series | 117 |

3 Holomorphy of power series | 123 |

Elementary Transcendental Functions | 133 |

2 The epimorphism theorem for exp z and its consequences | 141 |

3 Polar coordinates roots of unity and natural boundaries | 148 |

4 Logarithm functions | 154 |

5 Discussion of logarithm functions | 160 |

Part B The Cauchy Theory | 167 |

2 Properties of complex path integrals | 178 |

The Integral Theorem Integral Formula and Power Series | 191 |

2 Cauchys Integral Formula for discs | 201 |

3 The development of holomorphic functions into power series | 208 |

4 Discussion of the representation theorem | 214 |

mappings 3 The local normal form 4 Geometric interpretation | 303 |

2 Automorphisms of punctured domains | 310 |

Convergent Series of Meromorphic Functions | 321 |

4 The Eisenstein theory of the trigonometric functions | 335 |

Laurent Series and Fourier Series | 343 |

2 Properties of Laurent series | 356 |

4 The theta function | 365 |

The Residue Calculus | 377 |

2 Consequences of the residue theorem | 387 |

Chapter 14 Definite Integrals and the Residue Calculus | 395 |

Short Biographies oAbel Cauchy Eisenstein Euler Riemann | 417 |

Literature | 423 |

Symbol Index | 435 |

Subject Index | 443 |

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### Common terms and phrases

according addition algebra analysis biholomorphic boundary bounded calculus called Cauchy Cauchy's centered Chapter circular closed compact compactly complex complex-differentiable concept connected consequently consider constant contains continuous criterion defined definition derivatives determined differentiable disc domain equality equation equivalent Euler example Exercise exists fact field finite follows formula function f function theory give given holds holomorphic functions identity immediately infinitely integral integral formula introduced Laurent lemma lies limit locally logarithm mapping Math mathematics means meromorphic functions metric natural neighborhood normally convergent numbers partial particular path period pole polynomial positive power series principal Proof properties proved radius radius of convergence reader region remarks representation residue respectively Riemann rule satisfies says sequence Show simple singularity space subset Suppose Taylor theorem throughout uniformly unit Weierstrass write zeros