Description

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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