## 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 |

of continuity | 9 |

2 Fundamental topological concepts | 17 |

6 Connected spaces Regions in C | 39 |

ComplexDifferential Calculus | 45 |

3 Holomorphic functions | 56 |

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

Holomorphy and Conformality Biholomorphic Mappings | 71 |

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 |

Isolated Singularities Meromorphic Functions | 303 |

2 Biholomorphic mappings | 80 |

Modes of Convergence in Function Theory | 91 |

2 Convergence criteria | 101 |

Chapter 4 Power Series | 109 |

2 Examples of convergent power series | 115 |

3 Holomorphy of power series | 123 |

4 Logarithm functions | 154 |

5 Discussion of logarithm functions | 160 |

Part B The Cauchy Theory | 167 |

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 |

5 Special Taylor series Bernoulli numbers | 220 |

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 |

435 | |

443 | |

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### Common terms and phrases

absolutely convergent addition theorem algebra analysis angle-preserving automorphisms Bernoulli biholomorphic mapping boundary Br(c calculation called Cauchy integral formula Cauchy integral theorem Cauchy-Riemann Cauchy's centered circular sector closed path coefficients compact complex numbers consequently continuation theorem continuous function convergent series converges compactly converges normally converges uniformly criterion defined denote derivatives differential equation domain entire function equivalent Euler example Exercises Exercise exists f is holomorphic finite follows Fourier func function f function theory Gauss holomorphic functions Identity Theorem inequality infinitely injective integral theorem Laurent development Laurent series lemma Let f limit function locally constant logarithm function Math mathematical mathematician meromorphic functions neighborhood normally convergent open disc plane pole polynomial Proof proved R-linear radius of convergence real numbers real-differentiable region G representation Riemann satisfies Show singularity subset Taylor series tion uniform convergence unit disc Weierstrass Werke zero-free zeros