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A First Course in Complex Analysis A First Course in Complex Analysis Version 1.4 Matthias Beck Gerald Marchesi Department of Mathematics Department of Mathematical Sciences San Francisco State University Binghamton University (SUNY) San Francisco, CA 94132 Binghamton, NY 13902-6000 [email protected] [email protected] Dennis Pixton Lucas Sabalka Department of Mathematical Sciences Department of Mathematics & Computer Science Binghamton University (SUNY) Saint Louis University Binghamton, NY 13902-6000 St Louis, MO 63112 [email protected] [email protected] Copyright 2002–2012 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. Spe- cial thanks go to Joshua Palmatier, Collin Bleak, Sharma Pallekonda, and Dmytro Savchuk at Binghamton University (SUNY) for comments after teaching from this book. Contents 1 Complex Numbers 1 1.0 Introduction . 1 1.1 Definitions and Algebraic Properties . 2 1.2 From Algebra to Geometry and Back . 3 1.3 Geometric Properties . 6 1.4 Elementary Topology of the Plane . 8 1.5 Theorems from Calculus . 11 Exercises . 12 Optional Lab . 15 2 Differentiation 17 2.1 First Steps . 17 2.2 Differentiability and Holomorphicity . 19 2.3 Constant Functions . 22 2.4 The Cauchy–Riemann Equations . 23 Exercises . 25 3 Examples of Functions 28 3.1 Möbius Transformations . 28 3.2 Infinity and the Cross Ratio . 31 3.3 Stereographic Projection . 34 3.4 Exponential and Trigonometric Functions . 36 3.5 The Logarithm and Complex Exponentials . 39 Exercises . 41 4 Integration 46 4.1 Definition and Basic Properties . 46 4.2 Cauchy’s Theorem . 49 4.3 Cauchy’s Integral Formula . 52 Exercises . 54 3 CONTENTS 4 5 Consequences of Cauchy’s Theorem 58 5.1 Extensions of Cauchy’s Formula . 58 5.2 Taking Cauchy’s Formula to the Limit . 60 5.3 Antiderivatives . 63 Exercises . 66 6 Harmonic Functions 69 6.1 Definition and Basic Properties . 69 6.2 Mean-Value and Maximum/Minimum Principle . 71 Exercises . 73 7 Power Series 75 7.1 Sequences and Completeness . 75 7.2 Series . 77 7.3 Sequences and Series of Functions . 79 7.4 Region of Convergence . 82 Exercises . 85 8 Taylor and Laurent Series 90 8.1 Power Series and Holomorphic Functions . 90 8.2 Classification of Zeros and the Identity Principle . 95 8.3 Laurent Series . 97 Exercises . 100 9 Isolated Singularities and the Residue Theorem 103 9.1 Classification of Singularities . 103 9.2 Residues . 107 9.3 Argument Principle and Rouché’s Theorem . 110 Exercises . 113 10 Discrete Applications of the Residue Theorem 116 10.1 Infinite Sums . 116 10.2 Binomial Coefficients . 117 10.3 Fibonacci Numbers . 118 10.4 The ‘Coin-Exchange Problem’ . 118 10.5 Dedekind sums . 120 Solutions to Selected Exercises 121 Chapter 1 Complex Numbers Die ganzen Zahlen hat der liebe Gott geschaffen, alles andere ist Menschenwerk. (God created the integers, everything else is made by humans.) Leopold Kronecker (1823–1891) 1.0 Introduction The real numbers have many nice properties. There are operations such as addition, subtraction, multiplication as well as division by any real number except zero. There are useful laws that govern these operations such as the commutative and distributive laws. You can also take limits and do calculus. But you cannot take the square root of −1. Equivalently, you cannot find a root of the equation x2 + 1 = 0. (1.1) Most of you have heard that there is a “new” number i that is a root of the Equation (1.1). That is, i2 + 1 = 0 or i2 = −1. We will show that when the real numbers are enlarged to a new system called the complex numbers that includes i, not only do we gain a number with interesting properties, but we do not lose any of the nice properties that we had before. Specifically, the complex numbers, like the real numbers, will have the operations of addi- tion, subtraction, multiplication as well as division by any complex number except zero. These operations will follow all the laws that we are used to such as the commutative and distributive laws. We will also be able to take limits and do calculus. And, there will be a root of Equation (1.1). In the next section we show exactly how the complex numbers are set up and in the rest of this chapter we will explore the properties of the complex numbers. These properties will be both algebraic properties (such as the commutative and distributive properties mentioned already) and also geometric properties. You will see, for example, that multiplication can be described geometrically. In the rest of the book, the calculus of complex numbers will be built on the properties that we develop in this chapter. 1 CHAPTER 1. COMPLEX NUMBERS 2 1.1 Definitions and Algebraic Properties There are many equivalent ways to think about a complex number, each of which is useful in its own right. In this section, we begin with the formal definition of a complex number. We then interpret this formal definition into more useful and easier to work with algebraic language. Then, in the next section, we will see three more ways of thinking about complex numbers. The complex numbers can be defined as pairs of real numbers, C = f(x, y) : x, y 2 Rg , equipped with the addition (x, y) + (a, b) = (x + a, y + b) and the multiplication (x, y) · (a, b) = (xa − yb, xb + ya) . One reason to believe that the definitions of these binary operations are “good" is that C is an extension of R, in the sense that the complex numbers of the form (x, 0) behave just like real numbers; that is, (x, 0) + (y, 0) = (x + y, 0) and (x, 0) · (y, 0) = (x · y, 0). So we can think of the real numbers being embedded in C as those complex numbers whose second coordinate is zero. The following basic theorem states the algebraic structure that we established with our defi- nitions. Its proof is straightforward but nevertheless a good exercise. Theorem 1.1. (C, +, ·) is a field; that is: 8 (x, y), (a, b) 2 C : (x, y) + (a, b) 2 C (1.2) 8 (x, y), (a, b), (c, d) 2 C : (x, y) + (a, b) + (c, d) = (x, y) + (a, b) + (c, d) (1.3) 8 (x, y), (a, b) 2 C : (x, y) + (a, b) = (a, b) + (x, y) (1.4) 8 (x, y) 2 C : (x, y) + (0, 0) = (x, y) (1.5) 8 (x, y) 2 C : (x, y) + (−x, −y) = (0, 0) (1.6) 8 (x, y), (a, b), (c, d) 2 C : (x, y) · (a, b) + (c, d) = (x, y) · (a, b) + (x, y) · (c, d) (1.7) 8 (x, y), (a, b) 2 C : (x, y) · (a, b) 2 C (1.8) 8 (x, y), (a, b), (c, d) 2 C : (x, y) · (a, b) · (c, d) = (x, y) · (a, b) · (c, d) (1.9) 8 (x, y), (a, b) 2 C : (x, y) · (a, b) = (a, b) · (x, y) (1.10) 8 (x, y) 2 C : (x, y) · (1, 0) = (x, y) (1.11) 8 ( ) 2 C n f( )g ( ) · x −y = ( ) x, y 0, 0 : x, y x2+y2 , x2+y2 1, 0 (1.12) Remark. What we are stating here can be compressed in the language of algebra: equations (1.2)–(1.6) say that (C, +) is an Abelian group with unit element (0, 0), equations (1.8)–(1.12) that (C n f(0, 0)g, ·) is an abelian group with unit element (1, 0). (If you don’t know what these terms mean—don’t worry, we will not have to deal with them.) CHAPTER 1. COMPLEX NUMBERS 3 The definition of our multiplication implies the innocent looking statement (0, 1) · (0, 1) = (−1, 0) . (1.13) This identity together with the fact that (a, 0) · (x, y) = (ax, ay) allows an alternative notation for complex numbers. The latter implies that we can write (x, y) = (x, 0) + (0, y) = (x, 0) · (1, 0) + (y, 0) · (0, 1) . If we think—in the spirit of our remark on the embedding of R in C—of (x, 0) and (y, 0) as the real numbers x and y, then this means that we can write any complex number (x, y) as a linear combination of (1, 0) and (0, 1), with the real coefficients x and y. (1, 0), in turn, can be thought of as the real number 1. So if we give (0, 1) a special name, say i, then the complex number that we used to call (x, y) can be written as x · 1 + y · i, or in short, x + iy .
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