## Functions of One Complex Variable IThis book is intended as a textbook for a first course in the theory of functions of one complex variable for students who are mathematically mature enough to understand and execute E - 8 arguments. The actual pre requisites for reading this book are quite minimal; not much more than a stiff course in basic calculus and a few facts about partial derivatives. The topics from advanced calculus that are used (e.g., Leibniz's rule for differ entiating under the integral sign) are proved in detail. Complex Variables is a subject which has something for all mathematicians. In addition to having applications to other parts of analysis, it can rightly claim to be an ancestor of many areas of mathematics (e.g., homotopy theory, manifolds). This view of Complex Analysis as "An Introduction to Mathe matics" has influenced the writing and selection of subject matter for this book. The other guiding principle followed is that all definitions, theorems, etc. |

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Pl available this book as like as part II

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It is a great book. Clear writing and good exercise collection but a bit hard to follow.

### Contents

Chapter I | 1 |

Ho | 7 |

Chapter II | 11 |

Chapter III | 30 |

Γ2 | 54 |

Chapter IV | 58 |

G C is analytic | 100 |

Chapter V | 103 |

Chapter VIII | 195 |

Before stating Runges Theorem let us agree to say that | 198 |

Chapter IX | 210 |

G+ | 212 |

T | 242 |

γ | 246 |

Chapter X | 252 |

зи | 256 |

a | 104 |

over it is not difficult to see that the | 106 |

16 Determine the regions in which the functions f | 112 |

for any closed rectifiable curve y not passing through a | 122 |

Chapter VI | 128 |

Thus | 132 |

Chapter VII | 142 |

for all z in K and n N But | 172 |

equation will give that ƒ and I are everywhere identical | 180 |

applying Dinis Theorem Exercise VII16 Another involves | 262 |

is harmonic in the right half piane and 0 | 272 |

Chapter XI | 279 |

Chapter XII | 292 |

Appendix A | 303 |

Appendix B | 307 |

1 L V AHLFORS Complex Analysis | 311 |

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### Common terms and phrases

according Algebraic analytic function analytic on G apply assume bounded branch called Chapter choose circle closed rectifiable curve compact complex numbers component condition consider constant contains continuous function converges convex Corollary covering defined Definition derivative differentiable disk easy entire function equation equicontinuous equivalent example Exercise exists fact finite fixed follows formula function f ƒ is analytic G₁ given gives harmonic function Hence implies infinite integer Lemma Let f Let G lim sup Maximum meromorphic metric space multiplicity neighborhood Notice obtained one-one open set particular path plane pole polynomial positive possible power series preceding Principle Proof properties Proposition prove radius radius of convergence reader respectively result satisfies says sequence shown side simply connected singularity subset sufficiently suppose Theorem Theory topological space uniformly zero