An Introduction to Classical Complex Analysis: Vol. 1

to Classical Complex Analysis Vol. 1 by Robert B. Burckel Kansas State University 1979 BIRKHAUSER VERLAG BASEL UND STUTTGART CIP-Kurztitelaufnahme der Deutschen Bibliothek Burckel, Robert B.: An introduction to classical complex analysis I by Robert B. Burckel. - Basel. Stuttgart: Birkhiiuser. Vol. I. - 1979. (Lehrbilcher und Monographien aus dem Gebiete der exakten Wissenschaften: Math. Reihe; Bd. 64) All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the Copyright owner. (c) Birkhiiuser Verlag Basel, 1979 North and South America Edition published by ACADEMIC PRESS. INC. III Fifth Avenue, New York, New York 10003 (Pure and Applied Mathematics, A Series of Monographs and Textbooks, Volume 82) ISBN-13: 978-3-0348-9376-3 e-ISBN-13: 978-3-0348-9374-9 DOl: 10.1 007/978-3-0348-9374-9 Library of Congress Catalog Card Number 78-67403 5 Contents Volume I PREFACE 9 Chapter 0 PREREQUISITES AND PRELIMINARIES 13 1 Set Theory 13 2 Algebra 14 3 The Battlefield 14 4 Metric Spaces 15 5 Limsup and All That 18 6 Continuous Functions 20 7 Calculus 21 Chapter I CURVES, CONNECTEDNESS AND CONVEXITY 22 1 Elementary Results on Connectedness 22 2 Connectedness of Intervals, Curves and Convex Sets 23 3 The Basic Connectedness Lemma 28 4 Components and Compact Exhaustions 29 5 Connectivity of a Set 33 6 Extension Theorems 37 Notes to Chapter I 39

Table of Contents:
0 Prerequisites and Preliminaries.- § 1 Set Theory.- § 2 Algebra.- § 3 The Battlefield.- § 4 Metric Spaces.- § 5 Limsup and All That.- § 6 Continuous Functions.- § 7 Calculus.- I Curves, Connectedness and Convexity.- § 1 Elementary Results on Connectedness.- § 2 Connectedness of Intervals, Curves and Convex Sets.- § 3 The Basic Connectedness Lemma.- § 4 Components and Compact Exhaustions.- § 5 Connectivity of a Set.- § 6 Extension Theorems.- Notes to Chapter I.- II (Complex) Derivative and (Curvilinear) Integrals.- § 1 Holomorphic and Harmonic Functions.- § 2 Integrals along Curves.- § 3 Differentiating under the Integral.- § 4 A Useful Sufficient Condition for Differentiability.- Notes to Chapter II.- III Power Series and the Exponential Function.- § 1 Introduction.- § 2 Power Series.- § 3 The Complex Exponential Function.- § 4 Bernoulli Polynomials, Numbers and Functions.- § 5 Cauchy’s Theorem Adumbrated.- § 6 Holomorphic Logarithms Previewed.- Notes to Chapter III.- IV The Index and some Plane Topology.- § 1 Introduction.- § 2 Curves Winding around Points.- § 3 Homotopy and the Index.- § 4 Existence of Continuous Logarithms.- § 5 The Jordan Curve Theorem.- § 6 Applications of the Foregoing Technology.- § 7 Continuous and Holomorphic Logarithms in Open Sets.- § 8 Simple Connectivity for Open Sets.- Notes to Chapter IV.- V Consequences of the Cauchy-Goursat Theorem—Maximum Principles and the Local Theory.- § 1 Goursat’s Lemma and Cauchy’s Theorem for Starlike Regions.- § 2 Maximum Principles.- § 3 The Dirichlet Problem for Disks.- § 4 Existence of Power Series Expansions.- § 5 Harmonic Majorization.- § 6 Uniqueness Theorems.- § 7 Local Theory.- Notes to Chapter V.- VI Schwarz’ Lemma and its Many Applications.- § 1Schwarz’ Lemma and the Conformal Automorphisms of Disks.- § 2 Many-to-one Maps of Disks onto Disks.- § 3 Applications to Half-planes, Strips and Annuli.- § 4 The Theorem of CarathSodory, Julia, Wolff, et al.- § 5 Subordination.- Notes to Chapter VI.- VII Convergent Sequences of Holomorphic Functions.- § 1 Convergence in H(U).- § 2 Applications of the Convergence Theorems; Boundedness Criteria.- § 3 Prescribing Zeros.- § 4 Elementary Iteration Theory.- Notes to Chapter VII.- VIII Polynomial and Rational Approximation—Runge Theory.- § 1 The Basic Integral Representation Theorem.- § 2 Applications to Approximation.- § 3 Other Applications of the Integral Representation.- § 4 Some Special Kinds of Approximation.- § 5 Carleman’s Approximation Theorem.- § 6 Harmonic Functions in a Half-plane.- Notes to Chapter VIII.- IX The Riemann Mapping Theorem.- § 1 Introduction.- § 2 The Proof of Caratheodory and Koebe.- § 3 Fejer and Riesz’ Proof.- § 4 Boundary Behavior for Jordan Regions.- § 5 A Few Applications of the Osgood-Taylor-Caratheodory Theorem.- § 6 More on Jordan Regions and Boundary Behavior.- § 7 Harmonic Functions and the General Dirichlet Problem.- § 8 The Dirichlet Problem and the Riemann Mapping Theorem.- Notes to Chapter IX.- X Simple and Double Connectivity.- § 1 Simple Connectivity.- § 2 Double Connectivity.- Notes to Chapter X.- XI Isolated Singularities.- § 1 Laurent Series and Classification of Singularities.- § 2 Rational Functions.- § 3 Isolated Singularities on the Circle of Convergence.- § 4 The Residue Theorem and Some Applications.- § 5 Specifying Principal Parts—Mittag-Leffler’s Theorem.- § 6 Meromorphic Functions.- § 7 Poisson’s Formula in an Annulus and Isolated Singularities of Harmonic Functions.- Notes toChapter XI.- XII Omitted Values and Normal Families.- § 1 Logarithmic Means and Jensen’s Inequality.- § 2 Miranda’s Theorem.- § 3 Immediate Applications of Miranda.- §4 Normal Families and Julia’s Extension of Picard’s Great Theorem.- § 5 Sectorial Limit Theorems.- § 6 Applications to Iteration Theory.- § 7 Ostrowski’s Proof of Schottky’s Theorem.- Notes to Chapter XII.- Name Index.- Symbol Index.- Series Summed.- Integrals Evaluated.