## Functions of One Complex Variable, Volume 1This 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 - I) 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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### Contents

The Complex Number System | 1 |

Metric Spaces and the Topology of C | 11 |

Elementary Properties and Examples of Analytic Functions | 30 |

Copyright | |

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according analytic function analytic on G apply assume bounded branch called Chapter choose circle closed rectifiable curve complete component condition connected consider constant contains continuous function converges convex Corollary covering defined Definition derivative differentiable disk easy entire function equation equicontinuous example Exercise exists f is analytic fact finite fixed follows formula function f given gives harmonic function Hence implies infinite integer Lemma Let f Let G lim sup Maximum meromorphic metric space multiplicity 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 region respectively result satisfies says sequence shown side simply connected singularity sufficiently suppose takes Theorem topological space uniformly zero