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Novinger@Complex Variables 2004.pdf

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Complex Variables by R. B. Ash and W.P. Novinger Preface This book represents a substantial revision of the first edition which was published in 1971. Most of the topics of the original edition have been retained, but in a number of instances the material has been reworked so as to incorporate alternative approaches to these topics that have appeared in the mathematical literature in recent years. The book is intended as a text, appropriate for use by advanced undergraduates or graduate students
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  Complex Variables by R. B. Ash and W.P. NovingerPreface This book represents a substantial revision of the first edition which was published in1971. Most of the topics of the srcinal edition have been retained, but in a number of instances the material has been reworked so as to incorporate alternative approaches tothese topics that have appeared in the mathematical literature in recent years.The book is intended as a text, appropriate for use by advanced undergraduates or gradu-ate students who have taken a course in introductory real analysis, or as it is often called,advanced calculus. No background in complex variables is assumed, thus making the textsuitable for those encountering the subject for the first time. It should be possible tocover the entire book in two semesters.The list below enumerates many of the major changes and/or additions to the first edition.1. The relationship between real-differentiability and the Cauchy-Riemann equations.2. J.D. Dixon’s proof of the homology version of Cauchy’s theorem.3. The use of hexagons in tiling the plane, instead of squares, to characterize simpleconnectedness in terms of winding numbers of cycles. This avoids troublesome detailsthat appear in the proofs where the tiling is done with squares.4. Sandy Grabiner’s simplified proof of Runge’s theorem.5. A self-contained approach to the problem of extending Riemann maps of the unit diskto the boundary. In particular, no use is made of the Jordan curve theorem, a difficulttheorem which we believe to be peripheral to a course in complex analysis. Severalapplications of the result on extending maps are given.6. D.J. Newman’s proof of the prime number theorem, as modified by J. Korevaar, ispresented in the last chapter as a means of collecting and applying many of the ideas andresults appearing in earlier chapters, while at the same time providing an introduction toseveral topics from analytic number theory.For the most part, each section is dependent on the previous ones, and we recommendthat the material be covered in the order in which it appears. Problem sets follow mostsections, with solutions provided (in a separate section).1  2We have attempted to provide careful and complete explanations of the material, whileat the same time maintaining a writing style which is succinct and to the point.c  Copyright 2004 by R.B. Ash and W.P. Novinger. Paper or electronic copies for non-commercial use may be made freely without explicit permission of the authors. All otherrights are reserved.  Complex Variables by Robert B. Ash and W.P. NovingerTable Of Contents Chapter 1: Introduction 1.1 Basic Definitions1.2 Further Topology of the Plane1.3 Analytic Functions1.4 Real-Differentiability and the Cauchy-Riemann Equations1.5 The Exponential Function1.6 Harmonic Functions Chapter 2: The Elementary Theory 2.1 Integration on Paths2.2 Power Series2.3 The Exponential Function and the Complex Trigonometric Functions2.4 Further Applications Chapter 3: The General Cauchy Theorem 3.1 Logarithms and Arguments3.2 The Index of a Point with Respect to a Closed Curve3.3 Cauchy’s Theorem3.4 Another Version of Cauchy’s Theorem Chapter 4: Applications of the Cauchy Theory 4.1 Singularities4.2 Residue Theory4.3 The Open mapping Theorem for Analytic Functions4.4 Linear Fractional Transformations4.5 Conformal Mapping4.6 Analytic Mappings of One Disk to Another1  24.7 The Poisson Integral formula and its Applications4.8 The Jensen and Poisson-Jensen Formulas4.9 Analytic Continuation Chapter 5: Families of Analytic Functions 5.1 The Spaces  A (Ω) and  C (Ω)5.2 The Riemann Mapping Theorem5.3 Extending Conformal Maps to the Boundary Chapter 6: Factorization of Analytic Functions 6.1 Infinite Products6.2 Weierstrass Products6.3 Mittag-Leffler’s Theorem and Applications The Prime Number Theorem 7.1 The Riemann Zeta Function7.2 An Equivalent Version of the Prime Number Theorem7.3 Proof of the Prime Number Theorem
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