## Theory of Complex FunctionsA lively and vivid look at the material from function theory, including the residue calculus, supported by examples and practice exercises throughout. There is also ample discussion of the historical evolution of the theory, biographical sketches of important contributors, and citations - in the original language with their English translation - from their classical works. Yet the book is far from being a mere history of function theory, and even experts will find a few new or long forgotten gems here. Destined to accompany students making their way into this classical area of mathematics, the book offers quick access to the essential results for exam preparation. Teachers and interested mathematicians in finance, industry and science will profit from reading this again and again, and will refer back to it with pleasure. |

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### Contents

1 | |

6 Connected Spaces Regions in C | 39 |

ComplexDifferential Calculus | 45 |

3 Holomorphic functions | 56 |

4 Partial differentiation with respect to a y z and 2 | 63 |

Holomorphy and Conformality Biholomorphic Mappings | 71 |

2 Biholomorphic mappings | 80 |

Modes of Convergence in Function Theory | 91 |

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 |

2 Convergence criteria | 101 |

Power Series | 109 |

2 Examples of convergent power series | 115 |

3 Holomorphy of power series | 123 |

Elementary Transcendental Functions | 133 |

2 The epimorphism theorem for expz 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 |

Definite Integrals and the Residue Calculus | 395 |

Short Biographies of ABEL 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 calculus called Cauchy integral formula Cauchy integral theorem Cauchy–Riemann Cauchy's circular sector closed path compact complex numbers complex-differentiable consequently continuation theorem continuous function convergent series converges compactly converges normally converges uniformly cosz criterion defined derivatives domain 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 locally constant logarithm function Math mathematical mathematician meromorphic functions neighborhood normally convergent open disc point c e pole polynomial Proof proved radius of convergence real numbers real-differentiable region G RIEMANN satisfies series XL Show singularity subsection Taylor series uniform convergence unit disc WEIERSTRASS Werke zero-free zeros