UID:
almahu_9947366256002882
Umfang:
1 online resource (571 p.)
ISBN:
1-281-76327-6
,
9786611763275
,
0-08-087398-7
Serie:
Pure and applied mathematics, a series of monographs and textbooks ; v. 82
Inhalt:
An introduction to classical complex analysis. Volume I
Anmerkung:
Description based upon print version of record.
,
Front Cover; An Introduction to Classical Complex Analysis; Copyright Page; Contents; Preface; Chapter 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; Chapter 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
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Chapter 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; Chapter 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; Chapter IV. The Index and Some Plane Topology; 1 Introduction
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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; Chapter 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
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6 Uniqueness Theorems 7 Local Theory; Notes to Chapter V; Chapter VI. Schwarz' Lemma and its Many Applications; 1 Schwarz' 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 Carathéodory, Julia, Wolff, et al.; 5 Subordination; Notes to Chapter VI; Chapter 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
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Chapter 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; Chapter IX. The Riemann Mapping Theorem; 1 Introduction; 2 The Proof of Carathéodory and Koebe; 3 Fejér and Riesz' Proof; 4 Boundary Behavior for Jordan Regions; 5 A Few Applications of the Osgood-Taylor-Carathéodory Theorem
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6 More on Jordan Regions and Boundary Behavior
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English
Weitere Ausg.:
ISBN 0-12-141701-8
Sprache:
Englisch
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