Course In Complex Analysis In One Variable, A - Martin Moskowitz

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Course In Complex Analysis In One Variable, A

By: Martin Moskowitz

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Hardcover | 18 April 2002

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160 Pages

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22.86 x 15.24 x 1.91

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Complex analysis presents completely unforeseen results that are of a dramatic, even magical, nature. This manual is designed to convey its excitement and extraordinary character to the student. The exposition presents basic complex analysis in around 130 pages, with about 50 exercises. The material constantly relates to and contrasts with that of its sister subject, real analysis. An unusual feature of this book is a short final chapter containing applications of complex analysis to Lie theory. Since much of the content originated in a one-semester course given at the CUNY Graduate Center, the text should be very suitable for first year graduate students in mathematics who want to learn the basics of this important subject. For advanced undergraduates, there should be enough material for a year-long course or, by concentrating on the first three chapters, for one-semester course.

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Preface and Acknowledgments	p. v
First Concepts	p. 1
Fundamentals of the complex field	p. 1
Holomorphic functions	p. 3
Some important examples	p. 5
The Cauchy-Riemann equations	p. 10
Some elementary differential equations	p. 14
Conformality	p. 16
Power series	p. 18
Integration Along a Contour	p. 21
Curves and their trajectories	p. 21
Change of Parameter and a Fundamental Inequality	p. 24
Some important examples of contour integration	p. 27
The Cauchy theorem in simply connected domains	p. 29
Some immediate consequences of Cauchy's theorem for a simply connected domain	p. 39
The Main Consequences of Cauchy's theorem	p. 43
The Cauchy theorem in multiply connected domains and the pre-residue theorem	p. 43
The Cauchy integral formula and its consequences	p. 45
Analyticity, Taylor's theorem and the identity theorem	p. 53
The area formula and some consequences	p. 61
Application to spaces of square integrable holomorphic functions	p. 64
Spaces of holomorphic functions and Montel's theorem	p. 67
The maximum modulus theorem and Schwarz' lemma	p. 70
Singularities	p. 75
Classification of isolated singularities, the theorems of Riemann and Casorati-Weierstrass	p. 75
The principle of the argument	p. 80
Rouche's theorem and its consequences	p. 86
The study of a transcendental equation	p. 91
Laurent expansion	p. 94
The calculation of residues at an isolated singularity, the residue theorem	p. 99
Application to the calculation of real integrals	p. 103
A more general removable singularities theorem and the Schwarz reflection principle	p. 108
Conformal Mappings	p. 113
Linear fractional transformations, equivalence of the unit disk and the upper half plane	p. 113
Automorphism groups of the disk, upper half plane and entire plane	p. 114
Annuli	p. 120
The Riemann mapping theorem for planar domains	p. 123
Applications of Complex Analysis to Lie Theory	p. 131
Applications of the identity theorem: Complete reducibility of representations according to Hermann Weyl and the functional equation for the exponential map of a real Lie group	p. 131
Application of residues: The surjectivity of the exponential map for U(p,q)	p. 134
Application of Liouville's theorem and the maximum modulus theorem: The Zariski density of cofinite volume subgroups of complex Lie groups	p. 138
Applications of the identity theorem to differential topology and Lie groups	p. 140
Bibliography	p. 143
Index	p. 145
Table of Contents provided by Syndetics. All Rights Reserved.

Course In Complex Analysis In One Variable, A

At a Glance

Hardcover

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