# Matthias Beck Gerald Marchesi Dennis Pixton Lucas Sabalka

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Matthias BeckGerald MarchesiDennis PixtonLucas SabalkaVersion 1.54

A First Course inComplex AnalysisVersion 1.54Matthias BeckGerald MarchesiDepartment of MathematicsSan Francisco State UniversitySan Francisco, CA 94132mattbeck@sfsu.eduDepartment of Mathematical SciencesBinghamton University (SUNY)Binghamton, NY 13902marchesi@math.binghamton.eduDennis PixtonLucas SabalkaDepartment of Mathematical SciencesBinghamton University (SUNY)Binghamton, NY 13902dennis@math.binghamton.eduLincoln, NE 68502sabalka@gmail.comCopyright 2002–2018 by the authors. All rights reserved. The most current version of this book isavailable at the websitehttp://math.sfsu.edu/beck/complex.html.This book may be freely reproduced and distributed, provided that it is reproduced in its entiretyfrom the most recent version. This book may not be altered in any way, except for changes informat required for printing or other distribution, without the permission of the authors.The cover illustration, Square Squared by Robert Chaffer, shows two superimposed images. Theforeground image represents the result of applying a transformation, z 7 z2 (see Exercises 3.53and 3.54), to the background image. The locally conformable property of this mapping can beobserved through matching the line segments, angles, and Sierpinski triangle features of thebackground image with their respective images in the foreground figure. (The foreground figure isscaled down to about 40% and repositioned to accommodate artistic and visibility considerations.)The background image fills the square with vertices at 0, 1, 1 i, and i (the positive directionalong the imaginary axis is chosen as downward). It was prepared by using Michael Barnsley’schaos game, capitalizing on the fact that a square is self tiling, and by using a fractal-coloringmethod. (The original art piece is in color.) A subset of the image is seen as a standard Sierpinskitriangle. The chaos game was also re-purposed to create the foreground image.

contains optional longer homework problems that could also be used as group projects at the endof a course.We would be happy to hear from anyone who has adopted our book for their course, as wellas suggestions, corrections, or other comments.Acknowledgements. We thank our students who made many suggestions for and found errorsin the text. Special thanks go to Sheldon Axler, Collin Bleak, Pierre-Alexandre Bliman, MatthewBrin, Andrew Hwang, John McCleary, Sharma Pallekonda, Joshua Palmatier, and Dmytro Savchukfor comments, suggestions, and additions after teaching from this book.We thank Lon Mitchell for his initiative and support for the print version of our book withOrthogonal Publishing, and Bob Chaffer for allowing us to feature his art on the book’s cover.We are grateful to the American Institute of Mathematics for including our book in their OpenTextbook Initiative (aimath.org/textbooks).

Contents12345Complex Numbers1.1 Definitions and Algebraic Properties .1.2 From Algebra to Geometry and Back1.3 Geometric Properties . . . . . . . . . .1.4 Elementary Topology of the Plane . .Exercises . . . . . . . . . . . . . . . . . . . .Optional Lab . . . . . . . . . . . . . . . . . .Differentiation2.1 Limits and Continuity . . . . . . . . .2.2 Differentiability and Holomorphicity2.3 The Cauchy–Riemann Equations . . .2.4 Constant Functions . . . . . . . . . . .Exercises . . . . . . . . . . . . . . . . . . . .Examples of Functions3.1 Möbius Transformations . . . . . . . . . .3.2 Infinity and the Cross Ratio . . . . . . . .3.3 Stereographic Projection . . . . . . . . . .3.4 Exponential and Trigonometric Functions3.5 Logarithms and Complex Exponentials .Exercises . . . . . . . . . . . . . . . . . . . . . .Integration4.1 Definition and Basic Properties4.2 Antiderivatives . . . . . . . . .4.3 Cauchy’s Theorem . . . . . . .4.4 Cauchy’s Integral Formula . . .Exercises . . . . . . . . . . . . . . . .1. 2. 4. 8. 10. 13. uences of Cauchy’s Theorem735.1 Variations of a Theme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735.2 Antiderivatives Again . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.3 Taking Cauchy’s Formulas to the Limit . . . . . . . . . . . . . . . . . . . . . . . . . . 77Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 796Harmonic Functions6.1 Definition and Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6.2 Mean-Value and Maximum/Minimum Principle . . . . . . . . . . . . . . . . . . . . .Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7Power Series7.1 Sequences and Completeness . . .7.2 Series . . . . . . . . . . . . . . . . .7.3 Sequences and Series of Functions7.4 Regions of Convergence . . . . . .Exercises . . . . . . . . . . . . . . . . . .89.83838689.91929499102105Taylor and Laurent Series8.1 Power Series and Holomorphic Functions . . . .8.2 Classification of Zeros and the Identity Principle8.3 Laurent Series . . . . . . . . . . . . . . . . . . . .Exercises . . . . . . . . . . . . . . . . . . . . . . . . . .110110115118122Isolated Singularities and the Residue Theorem9.1 Classification of Singularities . . . . . . . . .9.2 Residues . . . . . . . . . . . . . . . . . . . . .9.3 Argument Principle and Rouché’s Theorem .Exercises . . . . . . . . . . . . . . . . . . . . . . . .128128133136139.14214214314414414610 Discrete Applications of the Residue Theorem10.1 Infinite Sums . . . . . . . . . . . . . . . . . .10.2 Binomial Coefficients . . . . . . . . . . . . .10.3 Fibonacci Numbers . . . . . . . . . . . . . .10.4 The Coin-Exchange Problem . . . . . . . .10.5 Dedekind Sums . . . . . . . . . . . . . . . .Theorems From Calculus147Solutions to Selected Exercises150Index155

Chapter 1Complex NumbersDie 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)The real numbers have many useful properties. There are operations such as addition, subtraction,and multiplication, as well as division by any nonzero number. There are useful laws that governthese operations, such as the commutative and distributive laws. We can take limits and docalculus, differentiating and integrating functions. But you cannot take a square root of 1; thatis, you cannot find a real root of the equationx2 1 0 .(1.1)Most of you have heard that there is a “new” number i that is a root of (1.1); that is, i2 1 0or i2 1. We will show that when the real numbers are enlarged to a new system called thecomplex numbers, which includes i, not only do we gain numbers with interesting properties, butwe do not lose many of the nice properties that we had before.The complex numbers, like the real numbers, will have the operations of addition, subtraction,multiplication, as well as division by any complex number except zero. These operations willfollow all the laws that we are used to, such as the commutative and distributive laws. We willalso be able to take limits and do calculus. And, there will be a root of (1.1).As a brief historical aside, complex numbers did not originate with the search for a square rootof 1; rather, they were introduced in the context of cubic equations. Scipione del Ferro (1465–1526) and Niccolò Tartaglia (1500–1557) discovered a way to find a root of any cubic polynomial,which was publicized by Gerolamo Cardano (1501–1576) and is often referred to as qCardano’sq2p3formula. For the cubic polynomial x3 px q, Cardano’s formula involves the quantity4 27 .It is not hard to come up with examples for p and q for which the argument of this square rootbecomes negative and thus not computable within the real numbers. On the other hand (e.g., byarguing through the graph of a cubic polynomial), every cubic polynomial has at least one real1

CHAPTER 1. COMPLEX NUMBERS2root. This seeming contradiction can be solved using complex numbers, as was probably firstexemplified by Rafael Bombelli (1526–1572).In the next section we show exactly how the complex numbers are set up, and in the rest ofthis chapter we will explore the properties of the complex numbers. These properties will beof both algebraic (such as the commutative and distributive properties mentioned already) andgeometric nature. 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 wedevelop in this chapter.1.1Definitions and Algebraic PropertiesThere are many equivalent ways to think about a complex number, each of which is useful inits own right. In this section, we begin with a formal definition of a complex number. We theninterpret this formal definition in more useful and easier-to-work-with algebraic language. Laterwe will see several more ways of thinking about complex numbers.Definition. The complex numbers are pairs of real numbers,C : {( x, y) : x, y R} ,equipped with the addition( x, y) ( a, b) : ( x a, y b)(1.2)( x, y) · ( a, b) : ( xa yb, xb ya) .(1.3)and the multiplicationOne reason to believe that the definitions of these binary operations are acceptable is that C isan extension of R, in the sense that the complex numbers of the form ( x, 0) behave just like realnumbers:( x, 0) (y, 0) ( x y, 0)and( x, 0) · (y, 0) ( xy, 0) .So we can think of the real numbers being embedded in C as those complex numbers whosesecond coordinate is zero.The following result states the algebraic structure that we established with our definitions.Proposition 1.1. (C, , ·) is a field, that is,for all ( x, y), ( a, b) C : ( x, y) ( a, b) C(1.4)for all ( x, y), ( a, b), (c, d) C : ( x, y) ( a, b)

A First Course in Complex Analysis was written for a one-semester undergradu-ate course developed at Binghamton University (SUNY) and San Francisco State University, and has been adopted at several other institutions. For many of our students, Complex Analysis is their ﬁrst rigorous analysis (if not mathematics) class they take, and this book reﬂects this very much. We tried to rely on as .

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