Complex Analysis - A First Course

Complex Analysis - A First Course - A First Course in...

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Unformatted text preview: A First Course in Complex Analysis Version 1.24 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 13902-6000 beck@math.sfsu.edu marchesi@math.binghamton.edu dennis@math.binghamton.edu Copyright 2002–2009 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 one-semester 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 Collin Bleak, Jon Clauss, Sharma Pallekonda, and Joshua Palmatier for comments after teaching from this book. Contents 1 Complex Numbers 1 1.1 Definition and Algebraic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Geometric Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Elementary Topology of the Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4 Theorems from Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2 Differentiation 13 2.1 First Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2 Differentiability and Analyticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3 The Cauchy–Riemann Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.4 Constants and Connectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3 Examples of Functions 23 3.1 M¨obius Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.2 Infinity and the Cross Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.3 Exponential and Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . 28 3.4 The Logarithm and Complex Exponentials . . . . . . . . . . . . . . . . . . . . . . . 313....
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This note was uploaded on 02/07/2011 for the course PHYS 101 taught by Professor Aster during the Spring '11 term at East Tennessee State University.

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Complex Analysis - A First Course - A First Course in...

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