Chapter0 - A First Course in Complex Analysis Version 1.2c...

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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 San Francisco State University San Francisco, CA 94132 beck@math.sfsu.edu Department of Mathematical Sciences Binghamton University (SUNY) Binghamton, NY 13902-6000 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 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 Joshua Palmatier, Collin Bleak and Sharma Pallekonda at Binghamton University (SUNY) for comments after teaching from this book. Contents 1 Complex Numbers 1.0 Introduction . . . . . . . . . . . . . 1.1 Definition and Algebraic Properties 1.2 Geometric Properties . . . . . . . . 1.3 Elementary Topology of the Plane 1.4 Theorems from Calculus . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 1 3 7 9 10 14 14 16 18 21 22 25 25 28 31 33 36 37 42 42 45 47 49 53 53 55 58 60 2 Differentiation 2.1 First Steps . . . . . . . . . . . . . . 2.2 Differentiability and Holomorphicity 2.3 The CauchyRiemann Equations . . 2.4 Constant Functions . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . 3 Examples of Functions 3.1 Mbius Transformations . . . . . . . . . . . o 3.2 Infinity and the Cross Ratio . . . . . . . . . 3.3 Stereographic Projection . . . . . . . . . . . 3.4 Exponential and Trigonometric Functions . 3.5 The Logarithm and Complex Exponentials Exercises . . . . . . . . . . . . . . . . . . . . . . 4 Integration 4.1 Definition and Basic Properties 4.2 Cauchy's Theorem . . . . . . . 4.3 Cauchy's Integral Formula . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Consequences of Cauchy's Theorem 5.1 Extensions of Cauchy's Formula . . . . 5.2 Taking Cauchy's Formula to the Limit 5.3 Antiderivatives . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . 3 CONTENTS 6 Harmonic Functions 6.1 Definition and Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Mean-Value and Maximum/Minimum Principle . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Power Series 7.1 Sequences and Completeness . . . 7.2 Series . . . . . . . . . . . . . . . . 7.3 Sequences and Series of Functions 7.4 Region of Convergence . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 63 63 65 67 68 68 70 72 75 78 82 82 85 87 90 93 93 97 99 101 104 104 105 105 106 107 109 8 Taylor and Laurent Series 8.1 Power Series and Holomorphic Functions . . . . . 8.2 Classification of Zeros and the Identity Principle 8.3 Laurent Series . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . 9 Isolated Singularities and the Residue Theorem 9.1 Classification of Singularities . . . . . . . . . . . 9.2 Residues . . . . . . . . . . . . . . . . . . . . . . . 9.3 Argument Principle and Rouch's Theorem . . . e Exercises . . . . . . . . . . . . . . . . . . . . . . . . . 10 Discrete Applications of the Residue 10.1 Infinite Sums . . . . . . . . . . . . . 10.2 Binomial Coefficients . . . . . . . . . 10.3 Fibonacci Numbers . . . . . . . . . . 10.4 The `Coin-Exchange Problem' . . . . 10.5 Dedekind sums . . . . . . . . . . . . Solutions to Selected Exercises Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solutions to Selected Exercises Chapter 1 8 2. (b) 19 - 25 i 25 (c) 1 (d) 1 if n = 4k, k Z; i if n = 1 + 4k, k Z; -1 if n = 2 + 4k, k Z; -i if n = 3 + 4k, k Z. 3. (a) 5, -2 - i (b) 5, 5 - 10i 5 i (c) 10 , 3 1( 2 - 1) + 11 ( 2 + 9) 11 1 (d) 8, 8i 4. (a) 2ei 2 i (b) 2e 4 5 (c) 2 3ei 6 5. (a) -1 + i (b) 34i (c) -1 9. (a) z = ei 3 k , k = 0, 1, . . . , 5 (b) z = 2ei 4 + 2 k , k = 0, 1, 2, 3 12. z = ei 4 - 1 and z = ei 5 4 -1 Chapter 2 2. (a) 0 (b) 1 + i 10. (a) differentiable and holomorphic in C with derivative -e-x e-iy (b) nowhere differentiable or holomorphic (c) differentiable on {x + iy C : x = y} with derivative 2x, nowhere holomorphic (d) nowhere differentiable or holomorphic (e) differentiable and holomorphic in C with derivative - sin x cosh y - i cos x sinh y (f) differentiable at 0 with derivative 0, nowhere holomorphic (g) differentiable at 0 with derivative 0, nowhere holomorphic (h) differentiable at 0 with derivative 0, nowhere holomorphic (i) differentiable and holomorphic in C with derivative 2(z - z) Chapter 3 37. (a) differentiable at 0, nowhere holomorphic 109 SOLUTIONS TO SELECTED EXERCISES (b) differentiable and holomorphic on C \ -1, ei 3 , e-i 3 (c) differentiable and holomorphic on C \ {x + iy C : x -1, y = 2} (d) nowhere differentiable or holomorphic (e) differentiable and holomorphic on C \ {x + iy C : x 3, y = 0} (f) differentiable and holomorphic in C (i.e. entire) 38. (a) z = i (b) There is no solution. (c) z = ln + i + 2k , k Z 2 (d) z = + 2k 4i, k Z 2 (e) z = + k, k Z 2 (f) z = k, k Z (g) z = 2i 41. f (z) = c z c-1 Chapter 4 2. -2i 3. (a) 8i (b) 0 (c) 0 (d) 0 20. 0 2 22. 3 110 29 0 for r < |a|; 2i for r > |a| 30 0 for r = 1; - i for r = 3; 0 for r = 5 3 Chapter 5 3. (a) 0 (b) 2i (c) 0 (d) i (e) 0 (f) 0 8. Any simply connected set which does not contain the origin, for example, C \ (-, 0]. Chapter 7 1. (a) divergent (b) convergent (limit 0) (c) divergent i (d) convergent (limit 2 - 2 ) (e) convergent (limit 0) 23. (a) k0 (-4)k z k 1 (b) k0 36k z k SOLUTIONS TO SELECTED EXERCISES 26. (a) k0 (-1)k (z - 1)k k-1 (b) k1 (-1) (z - 1)k k 29. (a) if |a| < 1, 1 if |a| = 1, and 0 if |a| > 1. (b) 1 (c) 1 (careful reasoning!) (d) 1 (careful reasoning!) Chapter 8 1. (a) {z C : |z| < 1}, {z C : |z| r} for any r < 1 (b) C, {z C : |z| r} for any r (c) {z C : |z - 3| > 1}, {z C : r |z - 3| R} for any 1 < r R e 3. k0 k! (z - 1)k 10. The maximum is 3 (attained at z = i), and the minimum is 1 (attained at z = 1). 12. One Laurent series is k0 (-2)k (z - 1)-k-2 , converging for |z - 1| > 2. 13. One Laurent series is k0 (-2)k (z - 2)-k-3 , converging for |z - 2| > 2. 14. One Laurent series is -3(z + 1)-1 + 1, converging for z = -1. 1 7 15. sin z = z -1 + 1 z + 360 z 3 + . . . 6 k 20. (a) k0 (-1) z 2k-2 (2k)! Chapter 9 7. (a) 0 (b) 1 (c) 4 k 9. (a) One Laurent series is k-2 (-1) (z - 2)k , converging for 0 < |z - 2| < 4. 4k+3 (b) - i 8 10. (a) 2i (b) 27i 4 (c) - 2i 7 1 (d) i 3 (e) 2i (f) 0 11. (a) k0 e1 (z + 1)k k! (b) e2i 33! 16. (c) 2 111 ...
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This note was uploaded on 02/27/2012 for the course MATH 417 taught by Professor Staff during the Fall '11 term at SUNY Albany.

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