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Unformatted text preview: A First Course in Complex Analysis Version 1.2c Matthias Beck, Gerald Marchesi, and Dennis Pixton Department of Mathematics Department of Mathematical Sciences San Francisco State University Binghamton University (SUNY) San Francisco, CA 94132 Binghamton, NY 139026000 beck@math.sfsu.edu marchesi@math.binghamton.edu dennis@math.binghamton.edu Copyright 20022009 by the authors. All rights reserved. The most current version of this book is available at the websites http://www.math.binghamton.edu/dennis/complex.pdf http://math.sfsu.edu/beck/complex.html . This book may be freely reproduced and distributed, provided that it is reproduced in its entirety from the most recent version. This book may not be altered in any way, except for changes in format required for printing or other distribution, without the permission of the authors. 2 These are the lecture notes of a onesemester undergraduate course which we have taught several times at Binghamton University (SUNY) and San Francisco State University. For many of our students, complex analysis is their first rigorous analysis (if not mathematics) class they take, and these notes reflect this very much. We tried to rely on as few concepts from real analysis as possible. In particular, series and sequences are treated from scratch. This also has the (maybe disadvantageous) consequence that power series are introduced very late in the course. We thank our students who made many suggestions for and found errors in the text. Special thanks go to Joshua Palmatier, Collin Bleak and Sharma Pallekonda at Binghamton University (SUNY) for comments after teaching from this book. Contents 1 Complex Numbers 1 1.0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Definition and Algebraic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Geometric Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Elementary Topology of the Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.4 Theorems from Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2 Differentiation 14 2.1 First Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2 Differentiability and Holomorphicity . . . . . . . . . . . . . . . . . . . . . . . . . . . 16Differentiability and Holomorphicity ....
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This note was uploaded on 06/18/2011 for the course MATH 375 taught by Professor Marchesi during the Spring '10 term at Binghamton University.
 Spring '10
 MARCHESI
 Math

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