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Introduction to classical complex analysis. Vol. 1.

Author: R B Burckel
Publisher: Basel ; Stuttgart : Birkhäuser, 1979.
Series: Mathematische Reihe, 64; Lehrbücher und Monographien aus dem Gegiete der exakten Wissenschaften
Edition/Format:   Print book : EnglishView all editions and formats

Covers some of the salient features of classical, one variable complex function theory. This book develops and exploits methods of all three of the principal schools (those of Cauchy, Riemann and  Read more...


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Document Type: Book
All Authors / Contributors: R B Burckel
ISBN: 376430989X 9783764309893 0121417018 9780121417017
OCLC Number: 25326942
Description: 570 p. ; 24 cm.
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.- 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 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 to Chapter 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.
Series Title: Mathematische Reihe, 64; Lehrbücher und Monographien aus dem Gegiete der exakten Wissenschaften
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