An Introduction to Classical Complex Analysis
Published on 15. July 1980
568 pages
978-0-08-087398-5 (ISBN)
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An Introduction to Classical Complex Analysis
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R.B. Burkel
Introduction to Classical Complex Analysis: v. 1
Book
01/1979
Academic Press
€65.61
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Content
- 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
- 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
- § 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
- § 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
- 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
- § 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
- Chapter X. Simple and Double Connectivity
- § 1 Simple Connectivity
- § 2 Double Connectivity
- Notes to Chapter X
- Chapter 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 to Chapter XI
- Chapter 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
- Bibliography
- Name Index
- Subject Index
- Symbol Index
- Series Summed
- Integrals Evaluated
