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An Introduction to Classical Complex Analysis : Vol. 1 - R. B. Burckel

An Introduction to Classical Complex Analysis

Vol. 1

Hardcover Published: 1st January 1979
ISBN: 9783764309893
Number Of Pages: 558

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This book is an attempt to cover some of the salient features of classical, one variable complex function theory. The approach is analytic, as opposed to geometric, but the methods of all three of the principal schools (those of Cauchy, Riemann and Weierstrass) are developed and exploited. The book goes deeply into several topics (e.g. convergence theory and plane topology), more than is customary in introductory texts, and extensive chapter notes give the sources of the results, trace lines of subsequent development, make connections with other topics and offer suggestions for further reading. These are keyed to a bibliography of over 1300 books and papers, for each of which volume and page numbers of a review in one of the major reviewing journals is cited. These notes and bibliography should be of considerable value to the expert as well as to the novice. For the latter there are many references to such thoroughly accessible journals as the American Mathematical Monthly and L'Enseignement Math‚matique. Moreover, the actual prerequisites for reading the book are quite modest; for example, the exposition assumes no fore knowledge of manifold theory, and continuity of the Riemann map on the boundary is treated without measure theory. "This is, I believe, the first modern comprehensive treatise on its subject. The author appears to have read everything, he proves everything and he has brought to light many interesting but generally forgotten results and methods. The book should be on the desk of everyone who might ever want to see a proof of anything from the basic theory. ..." (SIAM Review) / " ... An attractive ingenious and many time humorous form increases the accessibility of the book. ..." (Zentralblatt f�r Mathematik) / "Professor Burckel is to be congratulated on writing such an excellent textbook. ... this is certainly a book to give to a good student and he would profit immensely from it. ..." (Bulletin London Mathematical Society)

"This is, I believe, the first modern comprehensive treatise on its subject. The author appears to have read everything, he proves everything, and he has brought to light many interesting but generally forgotten results and methods. The book should be on the desk of everyone who might ever want to see a proof of anything from the basic theory...." (SIAM Review)

" ... An attractive, ingenious, and many time[s] humorous form increases the accessibility of the book...." (Zentralblatt fur Mathematik)

"Professor Burckel is to be congratulated on writing such an excellent textbook.... this is certainly a book to give to a good student [who] would profit immensely from it...." (Bulletin London Mathematical Society)

0 Prerequisites and Preliminaries.- x 1 Set Theory.- x 2 Algebra.- x 3 The Battlefield.- x 4 Metric Spaces.- x 5 Limsup and All That.- x 6 Continuous Functions.- x 7 Calculus.- I Curves, Connectedness and Convexity.- x 1 Elementary Results on Connectedness.- x 2 Connectedness of Intervals, Curves and Convex Sets.- x 3 The Basic Connectedness Lemma.- x 4 Components and Compact Exhaustions.- x 5 Connectivity of a Set.- x 6 Extension Theorems.- Notes to Chapter I.- II (Complex) Derivative and (Curvilinear) Integrals.- x 1 Holomorphic and Harmonic Functions.- x 2 Integrals along Curves.- x 3 Differentiating under the Integral.- x 4 A Useful Sufficient Condition for Differentiability.- Notes to Chapter II.- III Power Series and the Exponential Function.- x 1 Introduction.- x 2 Power Series.- x 3 The Complex Exponential Function.- x 4 Bernoulli Polynomials, Numbers and Functions.- x 5 Cauchy's Theorem Adumbrated.- x 6 Holomorphic Logarithms Previewed.- Notes to Chapter III.- IV The Index and some Plane Topology.- x 1 Introduction.- x 2 Curves Winding around Points.- x 3 Homotopy and the Index.- x 4 Existence of Continuous Logarithms.- x 5 The Jordan Curve Theorem.- x 6 Applications of the Foregoing Technology.- x 7 Continuous and Holomorphic Logarithms in Open Sets.- x 8 Simple Connectivity for Open Sets.- Notes to Chapter IV.- V Consequences of the Cauchy-Goursat Theorem-Maximum Principles and the Local Theory.- x 1 Goursat's Lemma and Cauchy's Theorem for Starlike Regions.- x 2 Maximum Principles.- x 3 The Dirichlet Problem for Disks.- x 4 Existence of Power Series Expansions.- x 5 Harmonic Majorization.- x 6 Uniqueness Theorems.- x 7 Local Theory.- Notes to Chapter V.- VI Schwarz' Lemma and its Many Applications.- x 1 Schwarz' Lemma and the Conformal Automorphisms of Disks.- x 2 Many-to-one Maps of Disks onto Disks.- x 3 Applications to Half-planes, Strips and Annuli.- x 4 The Theorem of CarathSodory, Julia, Wolff, et al.- x 5 Subordination.- Notes to Chapter VI.- VII Convergent Sequences of Holomorphic Functions.- x 1 Convergence in H(U).- x 2 Applications of the Convergence Theorems; Boundedness Criteria.- x 3 Prescribing Zeros.- x 4 Elementary Iteration Theory.- Notes to Chapter VII.- VIII Polynomial and Rational Approximation-Runge Theory.- x 1 The Basic Integral Representation Theorem.- x 2 Applications to Approximation.- x 3 Other Applications of the Integral Representation.- x 4 Some Special Kinds of Approximation.- x 5 Carleman's Approximation Theorem.- x 6 Harmonic Functions in a Half-plane.- Notes to Chapter VIII.- IX The Riemann Mapping Theorem.- x 1 Introduction.- x 2 The Proof of Caratheodory and Koebe.- x 3 Fejer and Riesz' Proof.- x 4 Boundary Behavior for Jordan Regions.- x 5 A Few Applications of the Osgood-Taylor-Caratheodory Theorem.- x 6 More on Jordan Regions and Boundary Behavior.- x 7 Harmonic Functions and the General Dirichlet Problem.- x 8 The Dirichlet Problem and the Riemann Mapping Theorem.- Notes to Chapter IX.- X Simple and Double Connectivity.- x 1 Simple Connectivity.- x 2 Double Connectivity.- Notes to Chapter X.- XI Isolated Singularities.- x 1 Laurent Series and Classification of Singularities.- x 2 Rational Functions.- x 3 Isolated Singularities on the Circle of Convergence.- x 4 The Residue Theorem and Some Applications.- x 5 Specifying Principal Parts-Mittag-Leffler's Theorem.- x 6 Meromorphic Functions.- x 7 Poisson's Formula in an Annulus and Isolated Singularities of Harmonic Functions.- Notes to Chapter XI.- XII Omitted Values and Normal Families.- x 1 Logarithmic Means and Jensen's Inequality.- x 2 Miranda's Theorem.- x 3 Immediate Applications of Miranda.- x4 Normal Families and Julia's Extension of Picard's Great Theorem.- x 5 Sectorial Limit Theorems.- x 6 Applications to Iteration Theory.- x 7 Ostrowski's Proof of Schottky's Theorem.- Notes to Chapter XII.- Name Index.- Symbol Index.- Series Summed.- Integrals Evaluated.

ISBN: 9783764309893
ISBN-10: 376430989X
Series: Lehrbucher Und Monographien Aus Dem Gebiete der Exakten Wis
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 558
Published: 1st January 1979
Publisher: SPRINGER VERLAG GMBH
Country of Publication: CH
Dimensions (cm): 23.37 x 15.49  x 3.56
Weight (kg): 0.91