{ "query": "in 4 sentences please summarize your context", "namespace": "cdf5e841-bd92-4996-b7f2-7c2a398184c5", "messages": [ { "role": "assistant", "content": "Hallo diverse NF SepTest! Ich bin Simon, der KI-Chatbot, der das Large Language Model gpt-4o-mini-2024-07-18 verwendet, um Ihre Fragen zu beantworten. Ich freue mich, Ihnen bei Ihren Anliegen zu helfen!\n\nHier sind einige Themen, \u00fcber die wir sprechen k\u00f6nnten: \n\n1. \ud83d\udcd8 Einf\u00fchrung in komplexe Zahlen und ihre Eigenschaften\n2. \ud83d\udd0d Differenzierung komplexer Funktionen und die Cauchy-Riemann-Gleichungen\n3. \ud83d\udcc8 Integration komplexer Funktionen und Cauchys Integralsatz\n4. \ud83e\uddee Taylor- und Laurent-Reihen in der komplexen Analyse\n5. \ud83c\udf10 Isolierte Singularit\u00e4ten und der Residuensatz\n6. \ud83d\udcca Anwendungen der komplexen Analyse in der Mathematik\n7. \ud83d\udcdd \u00dcbungen und L\u00f6sungen zu komplexen Zahlen und Funktionen\n8. \ud83d\udcda Die Bedeutung von harmonischen Funktionen in der komplexen Analyse\n9. \ud83d\udd17 Verbindungen zwischen komplexer und reeller Analyse\n10. \ud83e\udde9 Topologische Konzepte in der komplexen Analyse\n\nGibt es ein bestimmtes Thema auf dieser Liste, das Sie n\u00e4her erkunden m\u00f6chten?" }, { "role": "user", "content": "halo" }, { "role": "assistant", "content": "Hallo! Wie kann ich Ihnen helfen? Gibt es eine spezifische Frage oder ein Thema, \u00fcber das Sie mehr wissen m\u00f6chten?" }, { "role": "user", "content": "what are numbers?" }, { "role": "assistant", "content": "Numbers are abstract concepts used to represent quantities, values, or positions in a sequence. They can be classified into various types, including:\n\n1. **Natural Numbers**: The set of positive integers (1, 2, 3, ...).\n2. **Whole Numbers**: Natural numbers including zero (0, 1, 2, 3, ...).\n3. **Integers**: Whole numbers that can be positive, negative, or zero (..., -2, -1, 0, 1, 2, ...).\n4. **Rational Numbers**: Numbers that can be expressed as a fraction of two integers (e.g., 1/2, 3/4).\n5. **Irrational Numbers**: Numbers that cannot be expressed as a simple fraction (e.g., \u221a2, \u03c0).\n6. **Real Numbers**: All rational and irrational numbers.\n7. **Complex Numbers**: Numbers that include a real part and an imaginary part (e.g., 3 + 4i).\n\nIf you have a specific aspect of numbers you would like to explore further, please let me know!" }, { "role": "user", "content": "thank you" }, { "role": "assistant", "content": "Could you please clarify your question or provide more details about what you're looking for? I'm here to help!" }, { "role": "user", "content": "what is integration?" }, { "role": "assistant", "content": "Integration is a fundamental concept in calculus that refers to the process of finding the integral of a function. It can be understood in two main contexts:\n\n1. **Definite Integration**: This involves calculating the area under a curve defined by a function over a specific interval. The result is a numerical value that represents this area. For example, the definite integral of a function \\( f(x) \\) from \\( a \\) to \\( b \\) is denoted as:\n \\[\n \\int_{a}^{b} f(x) \\, dx\n \\]\n\n2. **Indefinite Integration**: This refers to finding the antiderivative of a function, which is a function whose derivative is the original function. The result includes a constant of integration (C) because the derivative of a constant is zero. For example, the indefinite integral of \\( f(x) \\) is denoted as:\n \\[\n \\int f(x) \\, dx = F(x) + C\n \\]\n where \\( F(x) \\) is the antiderivative of \\( f(x) \\).\n\nIntegration has numerous applications in mathematics, physics, engineering, and other fields, such as calculating areas, volumes, and solving differential equations.\n\nIf you have any specific questions about integration or its applications, feel free to ask!" }, { "role": "user", "content": "in 5 sentences please summarize your context" }, { "role": "assistant", "content": "The document titled *A First Course in Complex Analysis* is designed for undergraduate students and covers fundamental concepts in complex analysis, including complex numbers, differentiation, integration, and the properties of holomorphic functions. It emphasizes a rigorous approach, introducing series and sequences from scratch, and aims to lead students toward understanding the Residue Theorem and its applications in both continuous and discrete mathematics. The authors, Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka, have structured the content to be accessible while minimizing reliance on prior real analysis concepts. The book includes numerous examples, exercises, and proofs to reinforce learning and understanding of complex analysis principles. Additionally, it provides insights into the historical context and significance of complex analysis in mathematics." }, { "role": "user", "content": "in 6 sentences please summarize your context" }, { "role": "assistant", "content": "The document titled *A First Course in Complex Analysis* serves as a comprehensive textbook for undergraduate students, focusing on the fundamental concepts of complex analysis. It covers a range of topics, including complex numbers, differentiation, integration, power series, and the properties of holomorphic functions, while minimizing reliance on prior real analysis knowledge. The authors, Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka, aim to guide students toward understanding significant results such as the Residue Theorem and its applications in both continuous and discrete mathematics. The book is structured to include numerous examples, exercises, and proofs, reinforcing the learning process and providing a rigorous approach to the subject. Additionally, it emphasizes the historical context and significance of complex analysis within the broader field of mathematics. The text is designed to be accessible and engaging, making it suitable for students encountering rigorous analysis for the first time." }, { "role": "assistant", "content": "The document titled *A First Course in Complex Analysis* is designed for undergraduate students and covers essential topics in complex analysis, including complex numbers, differentiation, integration, and the properties of holomorphic functions. It emphasizes a rigorous approach, introducing series and sequences from scratch, and aims to lead students toward understanding significant results such as the Residue Theorem and its applications in both continuous and discrete mathematics. The authors, Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka, have structured the content to be accessible while minimizing reliance on prior real analysis concepts. The book includes numerous examples, exercises, and proofs to reinforce learning and understanding of complex analysis principles. Additionally, it provides insights into the historical context and significance of complex analysis within the broader field of mathematics. The text is designed to be engaging and informative, making it suitable for students encountering rigorous analysis for the first time." }, { "role": "user", "content": "in 4 sentences please summarize your context" }, { "role": "assistant", "content": "" } ], "stream": true, "language_level": "", "chat_channel": ":3990:1726562101179", "language": "German", "tone": "neutral", "writing_style": "standard", "model": "gpt-4o-mini-2024-07-18", "knowledgebase": "ki-dev-large", "seed": 1055, "client_id": 1726562101179, "all_context": false, "follow_up_for": null, "knowledgebase_files_count": 1, "override_command": null, "disable_clarity_check": false, "custom_primer": "", "logging": true, "query_route": "" } INITIALIZATION Knowledgebase: ki-dev-large Base Query: in 4 sentences please summarize your context Model: gpt-4o-mini-2024-07-18 Use Curl?: None ================================================== **Elapsed Time: 0.00 seconds** ================================================== ROUTING Query type: summary ================================================== **Elapsed Time: 1.79 seconds** ================================================== RAG PARAMETERS Max Context To Include: 100 Lowest Score to Consider: 0.1 ================================================== **Elapsed Time: 0.14 seconds** ================================================== VECTOR SEARCH ALGORITHM TO USE Use MMR search?: True Use Similarity search?: False ================================================== **Elapsed Time: 0.10 seconds** ================================================== VECTOR SEARCH DONE ================================================== **Elapsed Time: 6.17 seconds** ================================================== PRIMER Primer: You are Simon, a highly intelligent personal assistant in a system called KIOS. You are a chatbot that can read knowledgebases through the "CONTEXT" that is included in the user's chat message. Your role is to act as an expert at summarization and analysis. In your responses to enterprise users, prioritize clarity, trustworthiness, and appropriate formality. Be honest by admitting when a topic falls outside your scope of knowledge, and suggest alternative avenues for obtaining information when necessary. Make effective use of chat history to avoid redundancy and enhance response relevance, continuously adapting to integrate all necessary details in your interactions. Use as much tokens as possible to provide a detailed response. ================================================== **Elapsed Time: 0.18 seconds** ================================================== FINAL QUERY Final Query: CONTEXT: ########## File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 3 Context: # "And what is the use of a book," thought Alice, "without pictures or conversations?" Lewis Carroll (Alice in Wonderland) ## About this book *A First Course in Complex Analysis* was written for a one-semester undergraduate 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 few concepts from real analysis as possible. In particular, series and sequences are treated from scratch, which has the consequence that power series are introduced late in the course. The goal our book works toward is the Residue Theorem, including some nontraditional applications from both continuous and discrete mathematics. A printed paperback version of this open textbook is available from Orthogonal Publishing (www.orthogonalpublishing.com) or your favorite online bookseller. ## About the authors **Matthias Beck** is a professor in the Mathematics Department at San Francisco State University. His research interests are in geometric combinatorics and analytic number theory. He is the author of three other books, *Computing the Continuous Discretion: Integer-point Enumeration in Polyhedra* (with Sinal Robins, Springer 2007), *The Art of Proof: Basic Training for Mathematics* (with Ross Geoghegan, Springer 2010), and *Combinatorial Reciprocity Theorem: An Invitation to Enumerative Geometric Combinatorics* (with Raman Sanyal, AMS 2018). **Gerald M. Green** is a lecturer in the Department of Mathematical Sciences at Binghamton University (SUNY). **Dennis P. Stinson** is a professor emeritus in the Department of Mathematical Sciences at Binghamton University (SUNY). His research interests are in dynamical systems and formal languages. **Lucas Sabalka** is an applied mathematician at a technology company in Lincoln, Nebraska. He works on 3-dimensional computer vision applications. He was formerly a professor of mathematics at St. Louis University, after postdoctoral positions at UC Davis and Binghamton University (SUNY). His mathematical research interests are in geometric group theory, low dimensional topology, and computational algebra. #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 5 Context: # Contents 1. **Complex Numbers** 1.1. Definitions and Algebraic Properties . . . . . . . . 2 1.2. From Algebra to Geometry and Back . . . . . . . . 5 1.3. Geometric Properties . . . . . . . . . . . . . . . . 9 1.4. Elementary Topology of the Plane . . . . . . . . . 12 Optional Lab . . . . . . . . . . . . . . . . . . . . . . 21 2. **Differentiation** 2.1. Limits and Continuity . . . . . . . . . . . . . . . . 23 2.2. Differentiability and Holomorphicity . . . . . . . 28 2.3. The Cauchy–Riemann Equations . . . . . . . . . . . 32 2.4. Constant Functions . . . . . . . . . . . . . . . . . 36 3. **Examples of Functions** 3.1. Möbius Transformations . . . . . . . . . . . . . . . 43 3.2. Infinity and the Cross Ratio . . . . . . . . . . . 46 3.3. Stereographic Projection . . . . . . . . . . . . . . 48 3.4. Exponential and Trigonometric Functions . . . . . 55 3.5. Logarithms and Complex Exponentials . . . . . . . 59 4. **Integration** 4.1. Definition and Basic Properties . . . . . . . . . . 71 4.2. Antiderivatives . . . . . . . . . . . . . . . . . . 76 4.3. Cauchy’s Theorem . . . . . . . . . . . . . . . . . 80 4.4. Cauchy’s Integral Formula . . . . . . . . . . . . 85 5. **Consequences of Cauchy’s Theorem** 5.1. Variations of a Theme . . . . . . . . . . . . . . . 97 5.2. Antiderivatives Again . . . . . . . . . . . . . . 100 5.3. Taking Cauchy's Formulas to the Limit . . . . . . 102 #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 7 Context: # 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) 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 govern these operations, such as the commutative and distributive laws. We can take limits and do calculus, differentiating and integrating functions. But you cannot take a square root of \(-1\); that is, you cannot find a real root of the equation $$ x^2 + 1 = 0. \tag{1.1} $$ Most of you have heard that there is a "new" number that is a root of (1.1); that is, \(x^2 + 1 = 0\) or \(x^2 = -1\). We will show that when the real numbers are enlarged to a new system called the **complex numbers**, which includes \(i\), not only do we gain numbers with interesting properties, but we 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 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 (1.1). As a brief historical aside, complex numbers did not originate with the search for a square root of \(-1\); rather, they were introduced in the context of cubic equations. Scipione del Ferro (1465–1526) and Niccolò Tartaglia (1499–1557) 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 **Cardano's formula**. For the cubic polynomial \(x^3 + px + q\), Cardano's formula involves the quantity $$ \sqrt[3]{-\frac{q}{2} + \sqrt{\left(\frac{q}{2}\right)^2 + \left(\frac{p}{3}\right)^3}} + \sqrt[3]{-\frac{q}{2} - \sqrt{\left(\frac{q}{2}\right)^2 + \left(\frac{p}{3}\right)^3}}. $$ #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 9 Context: So we can think of the real numbers being embedded in \( \mathbb{C} \) as those complex numbers whose second coordinate is zero. The following result states the algebraic structure that we established with our definitions. ## Proposition 1.1 Let \( (x, y) \) be a field, that is, for all \( (a, b) \in \mathbb{C} \): 1. \( (x, y) + (a, b) \in \mathbb{C} \) (1.4) 2. \( (x, y) + (a, d) + (b, c) = (x, y) + ((a + b), c) \) (1.5) 3. \( (x, y) + (0, 0) = (x, y) \) (1.6) 4. \( (x, y) + (-x, -y) = (0, 0) \) (1.7) 5. \( (a, b) + (c, d) = (x, y) \cdot (a, b) + (c, d) \) (1.8) 6. \( (x, y) \in \mathbb{C} \) (1.9) 7. \( (x, y) \cdot (a, b) = (x \cdot a, y \cdot b) \) (1.10) 8. \( (x, y) \cdot (b, d) = (x \cdot b, (a \cdot b)(c, d)) \) (1.11) 9. \( (x, y) \cdot (a, b) = (x, y) \) (1.12) 10. \( (x, y) \cdot (1, 0) = (x, y) \) (1.13) 11. For all \( (x, y) \in \mathbb{C} \setminus \{(0, 0)\} \): \( (x, y) \cdot \left( \frac{(y, -x)}{x^2 + y^2} \right) = (1, 0) \) (1.14) What we are stating here can be compressed in the language of algebra: equations (1.4)–(1.8) say that \( (\mathbb{C}, +) \) is an Abelian group with identity \( (0, 0) \); equations (1.10)–(1.14) say that \( (\mathbb{C} \setminus \{(0, 0)\}, \cdot) \) is an Abelian group with identity \( (1, 0) \). The proof of Proposition 1.1 is straightforward but nevertheless makes for good practice (Exercise 1.14). We give one sample: **Proof (1.18)**. By our definition for complex addition and properties of additive inverse in \( \mathbb{R} \): \[ (x, y) + (-x, -y) = (x + (-x), y + (-y)) = (0, 0). \] The definition of our multiplication implies the innocent-looking statement: \[ (0, 1) \cdot (1, 0) = (1, 0). \quad (1.15) \] #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 10 Context: ``` # Complex Numbers This identity together with the fact that $$ (x_0, \cdots, x_n) \cdot (x, y) = (ax, az) $$ allows an alternative notation for complex numbers. The latter implies that we can write $$ (x, y) = (x_0, 0) + (0, y) = (x_0, 0) + (0, 1)(0, y). $$ If we think—in the spirit of our remark about embedding R into 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. Now (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 $$ x + iy. $$ ### Definition The number x is called the **real part** and y the **imaginary part** of the complex number x + iy, often denoted as Re(x + iy) = x and Im(x + iy) = y. The identity (1.15) then reads $$ i^2 = -1. $$ In fact, much more can now be said with the introduction of the square root of -1. It is not just that (1.1) has a root, but every nonconstant polynomial has roots in C: ### Fundamental Theorem of Algebra (see Theorem 5.11). Every nonconstant polynomial of degree d has d roots (counting multiplicity) in C. The proof of this theorem requires some (important) machinery, so we defer its proof and an extended discussion of it to Chapter 5. We invite you to check that the definitions of our binary operations and Proposition 1.1 are coherent with the usual real arithmetic rules if we think of complex numbers as given in the form x + iy. *The names have historical reasons: people thought of complex numbers as unreal, imagined.* ``` #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 12 Context: # COMPLEX NUMBERS Aside from the exceptional case of 0, for any complex number \( z \), the arguments of \( z \) all differ by a multiple of \( 2\pi \), just as we saw for the example \( z = 1 \). The absolute value of the difference of two vectors has a nice geometric interpretation: ## Proposition 1.2 Let \( z_1, z_2 \in \mathbb{C} \) be two complex numbers, thought of as vectors in \( \mathbb{R}^2 \), and let \( d(z_1, z_2) \) denote the distance between the endpoints of the two vectors in \( \mathbb{R}^2 \) (see Figure 1.2). Then \[ d(z_1, z_2) = |z_1 - z_2| = |z_2 - z_1|. \] **Proof:** Let \( z_1 = x_1 + iy_1 \) and \( z_2 = x_2 + iy_2 \). From geometry, we know that \[ d(z_1, z_2) = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}. \] This is the definition of \( |z_1 - z_2| \). Since \( (x_1 - x_2) = (x_2 - x_1) \) and \( (y_1 - y_2) = (y_2 - y_1) \), this is also equal to \( |z_2 - z_1| \. \qed ![Figure 1.2: Geometry behind the distance between two complex numbers.](path_to_image) That \( |z_1 - z_2| = |z_2 - z_1| \) simply says that the vector from \( z_2 \) to \( z_1 \) has the same length as the vector from \( z_1 \) to \( z_2 \). One reason to introduce the absolute value and argument of a complex number is that they allow us to give a geometric interpretation for the multiplication of two complex numbers. Let’s say we have two complex numbers \( z_1 = x_1 + iy_1 \), with absolute value \( r_1 \) and argument \( \phi_1 \), and \( z_2 = x_2 + iy_2 \), with absolute value \( r_2 \) and argument \( \phi_2 \). #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 13 Context: # FROM ALGEBRA TO GEOMETRY AND BACK ## 7 ϕ₂. This means we can write \(x_1 + iy_1 = (r_1 \cos \phi_1) + i(r_1 \sin \phi_1)\) and \(x_2 + iy_2 = (r_2 \cos \phi_2) + i(r_2 \sin \phi_2)\). To compute the product, we make use of some classic trigonometric identities: \[ (x_1 + iy_1)(x_2 + iy_2) = (r_1 \cos \phi_1 + ir_1 \sin \phi_1)(r_2 \cos \phi_2 + ir_2 \sin \phi_2) \] \[ = r_1 r_2 (\cos \phi_1 \cos \phi_2 - \sin \phi_1 \sin \phi_2) + i(r_1 r_2 \cos \phi_2 \sin \phi_1 + r_1 r_2 \sin \phi_2 \cos \phi_1) \] \[ = r_1 r_2 \left( \cos(\phi_1 + \phi_2) + i \sin(\phi_1 + \phi_2) \right). \] So the absolute value of the product is \(r_1 r_2\) and one of its arguments is \(ϕ_1 + ϕ_2\). Geometrically, we are multiplying the lengths of the two vectors representing our two complex numbers and adding their angles measured with respect to the positive real axis. ![Figure 1.3: Multiplication of complex numbers](#) In view of the above calculation, it should come as no surprise that we will have to deal with quantities of the form \(x + iy\) (where \(y\) is some real number) quite a bit. To save space, bytes, ink, etc., and because "Mathematics is for lazy people" we introduce a shortcut notation and define: \[ e^{i\phi} = \cos \phi + i \sin \phi. \] ^1 You should convince yourself that there is no problem with this fact since there are many possible arguments for complex numbers, as both cosine and sine are periodic functions with period \(2\pi\). ^2 Peter Halton (unpublished articles, Hudson River Undergraduate Mathematics Conference 2000). #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 14 Context: # COMPLEX NUMBERS Figure 1.4 shows three examples. At this point, this exponential notation is indeed ![Figure 1.4: Three sample complex numbers of the form \( e^{x} \).](image-path) purely a notation. We will later see in Chapter 3 that it has an intimate connection to the complex exponential function. For now, we motivate this maybe strange seeming definition by collecting some of its properties: ## Proposition 1.3. For any \( x_1, x_2 \in \mathbb{R} \): 1. \( e^{(x_1 + x_2 i)} = e^{x_1} e^{x_2 i} \) 2. \( e^{0} = 1 \) 3. \( \frac{d}{dx} e^{x} = e^{x} \) 4. \( |e^{x}| = 1 \) 5. \( e^{i\pi} = -1 \) You are encouraged to prove them (Exercise 1.16); again, we give a sample. ## Proof of \( e^{x} \). By definition of \( e^{x} \): \[ \frac{d}{dx} e^{(x + i)} = -\sin{(x)} + \cos{(x)} = i ( \cos{(x)} + i\sin{(x)}) = i e^{(x+i)} \] #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 15 Context: # Geometric Properties Proposition 1.3 implies that \( z^{n} = 1 \) for any integers \( n \) and \( r > 0 \). Thus, numbers of the form \( z^{n} \) with \( z \in \mathbb{Q} \) play a pivotal role in solving equations of the form \( z^{n} = 1 \)—plenty of reason to give them a special name. **Definition.** A root of unity is of the form \( z^{n} \) for some integers \( n \) and \( r > 0 \). A root of unity \( z \) is a complex number \( z \) such that \( z^{n} = 1 \) for some positive integer \( n \). In this case, we call \( z \) an \( n \)th root of unity. If \( n \) is the smallest positive integer with the property \( z^{n} = 1 \), then \( z \) is a primitive \( n \)th root of unity. **Example 1.4.** The 4th roots of unity are \( 1 \) and \( i \) and \( e^{i\frac{\pi}{2}} \). The latter two are primitive 4th roots of unity. With our new notation, the sentence "the complex number \( x + iy \) has absolute value \( r \) and argument \( \theta \) now becomes the identity \[ x + iy = r e^{i\theta}. \] The left-hand side is often called the **rectangular form**, the right-hand side the **polar form** of this complex number. We now have five different ways of thinking about a complex number: the formal definition, in rectangular form, in polar form, and geometrically, using Cartesian coordinates or polar coordinates. Each of these five ways is useful in different situations, and translating between them is an essential ingredient in complex analysis. The five ways and their corresponding notation are listed in Figure 1.5. This list is not exhaustive; see, e.g., Exercise 1.21. ## 1.3 Geometric Properties From the chain of basic inequalities \[ -\sqrt{x^{2} + y^{2}} \leq -|z| \leq \sqrt{x^{2} + y^{2}} \leq |z| \leq \sqrt{x^{2} + y^{2}}, \] (or, alternatively, by arguing with basic geometric properties of triangles), we obtain the inequalities \[ -|z| \leq \text{Re}(z) \leq |z| \] and \[ -|z| \leq \text{Im}(z) \leq |z|. \tag{1.16} \] The square of the absolute value has the nice property \[ |x + iy|^{2} = x^{2} + y^{2} = (x + iy)(x - iy). \] #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 16 Context: # COMPLEX NUMBERS | | Algebraic | Geometric | Exponential | |------------|------------------|------------------|---------------| | Formal | \(x + iy\) | Cartesian | \(re^{i\theta}\) | | | Rectangular | \(x\) | Polar | | | | \(y\) | | **Figure 1.5:** Five ways of thinking about a complex number. This is one of many reasons to give the process of passing from \(x + iy\) to \(x - iy\) a special name. ## Definition The number \(x - iy\) is the (complex) conjugate of \(x + iy\). We denote the conjugate by \[ x + iy = x - iy. \] Geometrically, conjugating \(z\) means reflecting the vector corresponding to \(z\) with respect to the real axis. The following collects some basic properties of the conjugate. ## Proposition 1.5 For any \(z_1, z_2, z_3 \in \mathbb{C}\): 1. \(z_1 \overline{z_2} = \overline{z_1 z_2}\) 2. \(\overline{z_1 z_2} = \overline{z_2} \overline{z_1}\) 3. \(\overline{\left(\frac{z_1}{z_2}\right)} = \frac{\overline{z_2}}{\overline{z_1}}\) 4. \(\overline{z} = z\) 5. \(|z| = |z|\) The proofs of these properties are easy (Exercise 1.22); once more we give a sample. ### Proof of (b) Let \(z_1 = x_1 + iy_1\) and \(z_2 = x_2 + iy_2\). Then \[ z_1 z_2 = (x_1 + iy_1)(x_2 + iy_2) = (x_1x_2 - y_1y_2) + i(x_1y_2 + y_1x_2) = \overline{z_1} \overline{z_2}. \] Image Analysis: ### 1. Localization and Attribution - **Page**: This is a page from a document or book, numbered at the top left as "10." - **Image 1**: An illustrated table at the top of the page labeled as "Figure 1.5." - **Text Block**: Multiple paragraphs of text underneath the table, including definitions, propositions, and proofs. ### 2. Object Detection and Classification - **Image 1:** - **Objects**: A table with three rows and three columns. - **Categories**: Types of complex number representations. - **Key Features**: - **Row 1**: "Formal" corresponds to (x, y). - **Row 2**: "Algebraic" includes rectangular (x + iy) and exponential (re^iθ). - **Row 3**: "Geometric" includes cartesian and polar representations. ### 3. Scene and Activity Analysis - **Scene**: - The scene depicts a page from a textbook or academic document. - The key activity is educational content delivery, specifically discussing complex numbers. ### 4. Text Analysis - **Detected Text**: - Title: "COMPLEX NUMBERS." - Sections: - "Figure 1.5: Five ways of thinking..." - Definitions and propositions related to complex numbers. - Proof of a property. - **Analysis**: - The page discusses different representations of complex numbers, highlights their properties, and presents relevant mathematical proofs. ### 5. Diagram and Chart Analysis - **Image 1:** - **Content**: The table categorizes the representations of complex numbers. - **Axes/Scales**: Not applicable. - **Legends**: Explanatory text within the cells. - **Key Insights**: - Complex numbers can be represented in multiple forms: formal, algebraic, geometric. - The table is a clear and concise way to present these different perspectives. ### 6. Product Analysis - **Not applicable**: No products were depicted in the image. ### 7. Anomaly Detection - **Not applicable**: No anomalies in the visual content. ### 8. Color Analysis - **Dominant Colors**: Black text on a white background. - **Impact**: Standard colors for printed or digital educational content, aiding readability and focus. ### 9. Perspective and Composition - **Perspective**: Straight-on view, typical for document pages. - **Composition**: - The table (Image 1) is at the top, drawing initial attention. - Subsequent text follows a structured format, typical of academic documents. ### 10. Contextual Significance - **Contribution**: - The image and text provide fundamental knowledge about complex numbers. - It enriches the educational material by categorizing and defining complex number representations. ### 11. Metadata Analysis - **Not available**: No metadata provided from the image. ### 12. Graph and Trend Analysis - **Not applicable**: No graphs present. ### 13. Graph Numbers - **Not applicable**: No graphs present. ### Additional Aspects - **Process Flows and Descriptions**: - A defined process for understanding the conjugate of a complex number. - Steps to prove a proposition about complex numbers. - **Type Designations**: - Designates formal, algebraic, and geometric types of complex number representations. - **Trend and Interpretation**: - Emphasizes the versatility and various intuitions behind complex number representations. - **Tables**: - The table encapsulates the five types of complex number representations succinctly. This comprehensive examination focuses on the detailed aspects and provides a thorough understanding of the educational content on the described page. #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 19 Context: # ELEMENTARY TOPOLOGY OF THE PLANE ## 1. Types of Points (a) A point \( a \in G \) is an **interior point** of \( G \) if some open disk centered at \( a \) is a subset of \( G \). (b) A point \( b \in E \) is a **boundary point** of \( G \) if every open disk centered at \( b \) contains a point in \( G \) and also a point that is not in \( G \). (c) A point \( c \in C \) is an **accumulation point** of \( G \) if every open disk centered at \( c \) contains a point of \( G \) different from \( c \). (d) A point \( d \in G \) is an **isolated point** of \( G \) if some open disk centered at \( d \) contains no point of \( G \) other than \( d \). The idea is that if you don't move too far from an interior point of \( G \) then you remain in \( G \); but a boundary point you can make an arbitrarily small move and get to a point inside \( G \) and you can also make an arbitrarily small move and get to a point outside \( G \). ## Definition A set is **open** if all its points are interior points. A set is **closed** if it contains all its boundary points. ## Example 1.18 For \( r > 0 \) and \( a \in \mathbb{C} \), the sets \( \{ z \in \mathbb{C} : |z - a| < r \} \) and \( \{ z \in \mathbb{C} : |z - a| = r \} \) are open. The closed disk \[ \overline{D[a, r]} := \{ z \in \mathbb{C} : |z - a| \leq r \} \] is an example of a closed set. A given set might be neither open nor closed. The complex plane \( \mathbb{C} \) and the empty set \( \emptyset \) are the only sets that are both open and closed. ## Definition The **boundary** \( \partial G \) of a set \( G \) is the set of all boundary points of \( G \). The interior of \( G \) is the set of all interior points of \( G \). The closure of \( G \) is the set \( \overline{G} = G \cup \partial G \). ## Example 1.19 The closure of the open disk \( D[a, r] \) is \( \overline{D[a, r]} \). The boundary of \( D[a, r] \) is the circle \( \partial D[a, r] \). ## Definition The set \( G \) is **bounded** if \( G \subset D[0, R] \) for some \( R \). One notion that is somewhat subtle in the complex domain is the idea of **connectedness**. Intuitively, a set is connected if it is "in one piece." In a set \( R \), a set is connected if and only if it is an interval, so there is little reason to discuss the matter. However, in the plane there is a vast variety of connected subsets. #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 21 Context: # ELEMENTARY TOPOLOGY OF THE PLANE This definition uses the calculus notions of continuity and differentiability; that is, \( \gamma: [a, b] \to C \) being continuous means that for all \( t_0 \in [a, b] \) \[ \lim_{t \to t_0} \gamma(t) = \gamma(t_0) \] and the derivative of \( \gamma \) at \( t_0 \) is defined by \[ \gamma'(t_0) = \lim_{t \to t_0} \frac{\gamma(t) - \gamma(t_0)}{t - t_0} \] \[ \gamma_1(t) = -2 + 2e^{it}, \quad 0 \leq t \leq 2\pi \] \[ \gamma_2(t) = \begin{cases} 3 + i(t - 2) & \text{if } 0 \leq t \leq 3 \\ 6 - i + i(t - 1) & \text{if } 3 < t \leq 5 \end{cases} \] Figure 1.8: Two paths and their parameterizations: \( \gamma_1 \) is smooth and \( \gamma_2 \) is continuous and piecewise smooth. Figure 1.8 shows two examples. We remark that each path comes with an orientation, i.e., a sense of direction. For example, the path \( \gamma_1 \) in Figure 1.8 is different from \[ \gamma_1(t) = -2 + 2e^{it}, \quad 0 \leq t \leq \frac{3\pi}{2} \] even though both \( \gamma_1 \) and \( \gamma_3 \) yield the same picture; \( \gamma_1 \) features a counter-clockwise orientation, where as that of \( \gamma_2 \) is clockwise. It is a customary and practical abuse of notation to use the same letter for the path and its parameterization. We emphasize that a path must have a parameterization, and that the parameterization must be defined and continuous on a closed and bounded interval \([a, b]\). Since topologically we may identify \( C \) with \( \mathbb{R}^2 \), a path can be specified. #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 22 Context: ``` # Complex Numbers by giving two continuous real-valued functions of a real variable, \( x(t) \) and \( y(t) \), and setting \( y(t) = x(t) + iy(t) \). ## Definition The path \( \gamma: [a, b] \to C \) is simple if \( \gamma(t_1) = \gamma(t_2) \) is one-to-one, with the possible exception that \( \gamma(a) = \gamma(b) \) in plain English: the path does not cross itself. A path \( \gamma: [a, b] \to C \) is closed if \( \gamma(a) = \gamma(b) \). ## Example 1.11 The unit circle \( C[0,1] \), parametrized, e.g., by \( \gamma(t) = e^{it} \), \( 0 \leq t < 2\pi \), is a simple closed path. As seems intuitively clear, any path is connected; however, a proof of this fact requires a bit more preparation in topology. The same goes for the following result, which gives a useful property of open connected sets. ## Theorem 1.12 If any two points in \( G \subset C \) can be connected by a path in \( G \), then \( G \) is connected. Conversely, if \( G \subset C \) is open and connected, then any two points of \( G \) can be connected by a path in \( G \); in fact, we can connect any two points of \( G \) by a chain of horizontal and vertical segments lying in \( G \). Here a chain of segments in \( G \) means the following: there are points \( z_{0}, z_{1}, \ldots, z_{n} \) such that \( z_{0} \) and \( z_{n} \) are the endpoints of a horizontal or vertical segment in \( G \) for all \( k = 0, 1, \ldots, n-1 \) (it is not hard to parametrize such a chain, so it determines a path). ## Example 1.13 Consider the open unit disk \( D[0, 1) \). Any two points in \( D[0, 1) \) can be connected by a chain of at most two segments in \( D[0, 1) \) and so \( D[0, 1) \) is connected. Now let \( G = (D[0, 1) \setminus \{0\}) \). This is the punctured disk obtained by removing the center from \( D[0, 1) \). Then \( G \) is open and it is connected, but now you may need more than two segments to connect points. For example, you need three segments to connect \( z_{0} \to z_{1} \to z_{2} \) since we cannot go through \( 0 \). We remark that the second part of Theorem 1.12 is not generally true if \( G \) is not open. For example, circles are connected but there is no way to connect two distinct points of a circle by a chain of segments that are subsets of the circle. A more extreme example, discussed in topology texts, is the "topologist's sine curve," which is a connected set \( S \) that contains points that cannot be connected by a path of any sort within \( S \). ``` #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 25 Context: # Elementary Topology of the Plane ## 1.16 Prove Proposition 1.3. ## 1.17 Fix a positive integer \( n \). Prove that the solutions to the equation \( z^n = 1 \) are precisely \( z = e^{2 \pi i k/n} \) where \( k \in \mathbb{Z} \). (Hint: To show that every solution of \( z^n = 1 \) is of this form, first prove that it must be of the form \( z = e^{i \theta} \) for some \( \theta \in \mathbb{R} \), then write \( z = m + ib \) for some integer \( m \) and some real number \( 0 < b < 1 \), and then argue that \( b \) has to be zero.) ## 1.18 Show that \[ z^2 - 1 = (z - 1)\left(z^2 + 2 \cos \frac{\pi}{3}(z^2 - 2z \cos \frac{\pi}{3} + 1\right) \] and deduce from this closed formula for \( \cos \frac{\pi}{3} \) and \( \cos \frac{2\pi}{3} \). ## 1.19 Fix a positive integer \( n \) and a complex number \( w \). Find all solutions to \( z^n = w \). (Hint: Write \( w \) in terms of polar coordinates.) ## 1.20 Use Proposition 1.3 to derive the triple angle formulas: (a) \(\cos(3\phi) = 4\cos^3\phi - 3\cos\phi \sin^2\phi\) (b) \(\sin(3\phi) = 3\cos^2\phi \sin\phi - \sin^3\phi\) ## 1.21 Given \( x, y \in \mathbb{R} \), define the matrix \( M(x, y) = \begin{bmatrix} x & -y \\ y & x \end{bmatrix} \). Show that \[ M(x, y) + M(a, b) = M(x + a, y + b) \] and \[ M(x, y)M(a, b) = M(xa - yb, xb + ya) \] (This means that the set \( \{ M(x, y) : x, y \in \mathbb{R} \} \) equipped with the usual addition and multiplication of matrices, behaves exactly like \( C = \{ (x, y) : x, y \in \mathbb{R} \} \).) ## 1.22 Prove Proposition 1.5. #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 26 Context: # COMPLEX NUMBERS ## 1.23 Sketch the following sets in the complex plane: - (a) \( \{ z \in \mathbb{C} : |z - 1 + i| = 1 \} \) - (b) \( \{ z \in \mathbb{C} : |z - 1 + i| \leq 2 \} \) - (c) \( \{ z \in \mathbb{C} : \text{Re}(z^2 + 2 - 2i) = 3 \} \) - (d) \( \{ z \in \mathbb{C} : |z - i| + |z + i| = 3 \} \) - (e) \( \{ z \in \mathbb{C} : |z| = |z - 1| \} \) - (f) \( \{ z \in \mathbb{C} : |z - 1| = 2 |z + 1| \} \) - (g) \( \{ z \in \mathbb{C} : \text{Re}(z) = 1 \} \) - (h) \( \{ z \in \mathbb{C} : \text{Im}(z) = 1 \} \) ## 1.24 Suppose \( \rho \) is a polynomial with real coefficients. Prove that: - (a) \( \rho(\overline{z}) = \overline{\rho(z)} \). - (b) \( \rho(z) = 0 \) if and only if \( \rho(\overline{z}) = 0 \). ## 1.25 Prove the reverse triangle inequality (Proposition 1.7(b)): \[ |z_1 - z_2| \leq |z_1| + |z_2|. \] ## 1.26 Use the previous exercise to show that \[ \left| \frac{1}{z - 2} \right| < \frac{1}{3} \] for every \( z \) on the circle \( C[0,2] \). ## 1.27 Sketch the sets defined by the following constraints and determine whether they are open, closed, or neither; bounded; connected. - (a) \( |z + 3| < 2 \) - (b) \( \text{Im}(z) < 1 \) - (c) \( 0 < |z - 1| < 2 \) - (d) \( |z| = |1 + i| + 2 \) - (e) \( |z| = |1 + i| + 3 \) ## 1.28 What are the boundaries of the sets in the previous exercise? ## 1.29 Let \( G \) be the set of points \( z \in \mathbb{C} \) satisfying either \( z = r \) and \( -2 < r < -1 \), or \( |z| < 1 \), or \( z = \sigma \) where \( \sigma = -2 \). - (a) Sketch the set \( G \), being careful to indicate exactly the points that are in \( G \). - (b) Determine the interior points of \( G \). #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 27 Context: # Elementary Topology of the Plane 21 (c) Determine the boundary points of \( G \). (d) Determine the isolated points of \( G \). ## 1.30 The set \( G \) in the previous exercise can be written in three different ways as the union of two disjoint nonempty separated subsets. Describe them, and in each case say briefly why the subsets are separated. ## 1.31 Show that the union of two regions with nonempty intersection is itself a region. ## 1.32 Show that if \( A \subset B \) and \( B \) is closed, then \( A \subset B \). Similarly, if \( A \subset B \) and \( A \) is open, show that \( A \) is contained in the interior of \( B \). ## 1.33 Find a parameterization for each of the following paths: (a) The circle \( C(1 + i, 1) \), oriented counter-clockwise (b) The line segment from \( -1 - i \) to \( 2i \) (c) The top half of the circle \( C(0, \frac{3}{4}) \), oriented clockwise (d) The rectangle with vertices \( \pm 1 \pm 2i \), oriented counter-clockwise (e) The ellipse \( \{ z \in \mathbb{C} : |z - 1| + |z + 1| = 4 \} \), oriented counter-clockwise ## 1.34 Draw the path parameterized by \[ \gamma(t) = \cos(t) \cdot \cosh(t) + i \cdot \sin(t) \cdot \sinh(t), \quad 0 \leq t \leq 2\pi. \] ## 1.35 Let \( G \) be the annulus determined by the inequalities \( 2 < |z| < 3 \). This is a connected open set. Find the maximum number of horizontal and vertical segments in \( G \) needed to connect two points of \( G \). --- ### Optional Lab Open your favorite web browser and search for the complex function grapher for the open-source software GeoGebra. #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 28 Context: # Complex Numbers 1. Convert the following complex numbers into their polar representation, i.e., give the absolute value and the argument of the number: \[ 3 + 4i = r e^{i\theta} = 2 + 2i = \frac{1}{\sqrt{3 + i}} = \] After you have finished computing these numbers, check your answers with the program. 2. Convert the following complex numbers given in polar representation into their rectangular representation: \[ 2e^{i\theta} = 3e^{i\phi} = \frac{1}{e^{i\varphi}} = -\sqrt{2}e^{2i\phi} = \] After you have finished computing these numbers, check your answers with the program. 3. Pick your favorite five numbers from the ones that you’ve played around with and put them in the tables below, in both rectangular and polar form. Apply the functions listed to your numbers. Think about which representation is more helpful in each instance. | rectangular | polar | |-------------|------------| | z + 1 | | | z + 2 - i | | | 2z | | | -z | | | z^2 | | | Re(z) | | | Im(z) | | | /Im(z) | | | |z| | | | 1/2 | | 4. Play with other examples until you get a feel for these functions. #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 29 Context: # Chapter 2 ## Differentiation Mathematical study and research are very suggestive of generalizing. Whispers make several efforts before he climbed the Mathematics in the 1860s and even then it cut the life short of this party. Now, however, any source can be looked up for a small act, and perhaps one does not appreciate the difficulty of the original account. So in mathematics, it may be found hard to realize the great initial difficulty of making a little step which may seem so natural and obvious, and it may not be surprising if such a step has been found and lost again. **Louis Nodel (1888–1972)** We will now start our study of complex functions. The fundamental concept on which all of calculus is based is that of a limit—it allows us to develop the central properties of continuity and differentiability of functions. Our goal in this chapter is to do the same for complex functions. ### 2.1 Limits and Continuity **Definition.** A (complex) function \( f \) is a map from a subset \( G \subset \mathbb{C} \) to \( \mathbb{C} \); in this situation we will write \( f : G \to \mathbb{C} \) and call \( G \) the domain of \( f \). This means that each element \( z \in G \) gets mapped to exactly one complex number, called the image of \( z \) and usually denoted by \( f(z) \). So far there is nothing that makes complex functions any more special than, say, functions from \( \mathbb{R} \) to \( \mathbb{R} \). In fact, we can construct many familiar looking functions from the standard calculus repertoire, such as \( f(z) = z \) (the identity map), \( f(z) = z^2 + 1 \), or \( f(z) = \frac{1}{z} \). The former three could be defined on all of \( \mathbb{C} \), whereas for the latter we have to exclude the origin \( z = 0 \) from the domain. On the other hand, we could construct some functions that make use of a certain representation of \( z \), for example, \( f(x, y) = x - 2iy \), \( f(x, y) = y - i \), or \( f(z, \varphi) = 2r e^{i\varphi} \). #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 30 Context: Next we define limits of a function. The philosophy of the following definition is not restricted to complex functions, but for sake of simplicity we limit it only for those functions. ## Definition Suppose \( f : G \to \mathbb{C} \) and \( z_0 \) is an accumulation point of \( G \). If \( \epsilon \) is a complex number such that for every \( \delta > 0 \) we can find \( \delta > 0 \) such that \( 0 < |z - z_0| < \delta \), then \[ \lim_{z \to z_0} f(z) = w_0. \] This definition is the same as is found in most calculus texts. The reason we require that \( z_0 \) is an accumulation point of the domain is just that we need to be sure that there are points \( z \) of the domain that are arbitrarily close to \( z_0 \). Just as in the real case, our definition (i.e., the part that says \( 0 < |z - z_0| \)) does not require that \( z_0 \) is in the domain of \( f \); the definition explicitly ignores the value of \( f(z_0) \). ## Example 2.1 Let's prove that \( \lim_{z \to 2} z^2 = 4 \). Given \( \epsilon > 0 \), we need to determine \( \delta > 0 \) such that \( 0 < |z - 2| < \delta \) implies \( |z^2 - 4| < \epsilon \). We rewrite: \[ |z^2 - 4| = |z - 2||z + 2|. \] If we choose \( \delta < 1 \), then the factor \( |z + 2| \) on the right can be bounded by 3 (draw a picture). This means that any \( \delta < \min\{1, \frac{\epsilon}{3}\} \) should do the trick; in this case, \( 0 < |z - 2| < \delta \) implies: \[ |z^2 - 4| < 3 |z - 2| < 3 \delta < \epsilon. \] This proves \( \lim_{z \to 2} z^2 = 4 \). #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 32 Context: ``` # Differentiation Proof of **Theorem**. Assume that \( x_0 \in G \) (otherwise there is nothing to prove), and let \( L = \lim_{x \to x_0} f(x) \) and \( M = \lim_{x \to x_0} g(x) \). Then we know that given \( \epsilon > 0 \), we can find \( \delta_1, \delta_2 > 0 \) such that \[ 0 < |x - x_0| < \delta_1 \implies |f(x) - L| < \frac{\epsilon}{2} \] and \[ 0 < |x - x_0| < \delta_2 \implies |g(x) - M| < \frac{\epsilon}{2}. \] Thus, choosing \( \delta = \min(\delta_1, \delta_2) \), we infer that \( 0 < |x - x_0| < \delta \) implies \[ |f(x) + g(x) - (L + M)| \leq |f(x) - L| + |g(x) - M| < \epsilon. \] Here we used the triangle inequality (Proposition 1.6). This proves that \[ \lim_{x \to x_0} (f(x) + g(x)) = L + M, \] which was our claim. Because the definition of the limit is somewhat elaborate, the following fundamental definition looks almost trivial. ## Definition Suppose \( f : G \to \mathbb{C} \). If \( x_0 \in G \) and either \( x_0 \) is an isolated point of \( G \) or \[ \lim_{x \to x_0} f(x) = f(x_0) \] then \( f \) is continuous at \( x_0 \). More generally, \( f \) is continuous on \( E \subset G \) if it is continuous at every \( x \in E \). However, in almost all proofs using continuity it is necessary to interpret this in terms of \( \epsilon \) and \( \delta \). ## Definition Suppose \( f : G \to \mathbb{C} \) and \( x_0 \in G \). Then \( f \) is continuous at \( x_0 \) if, for every positive real number \( \epsilon \) there is a positive real number \( \delta > 0 \) such that \[ |f(x) - f(x_0)| < \epsilon \quad \text{for all } x \in E \text{ satisfying } |x - x_0| < \delta. \] See Exercise 2.11 for a proof that these definitions are equivalent. ``` #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 33 Context: # LIMITS AND CONTINUITY ## Example 2.5 We already proved (in Example 2.1) that the function \( f : C \to C \) given by \( f(x) = x^2 \) is continuous at \( x = 1 \). You're invited (Exercise 2.8) to extend our proof to show that, in fact, this function is continuous on \( C \). On the other hand, let \( g : C \to C \) be given by \[ g(x) = \begin{cases} \frac{1}{x} & \text{if } x \neq 0 \\ 1 & \text{if } x = 0 \end{cases} \] In Example 2.3 we proved that \( g \) is not continuous at \( x = 0 \). However, this is its only point of discontinuity (Exercise 2.9). Just as in the real case, we can "take the limit inside" a continuous function, by considering composition of functions. ## Definition The image of the function \( g : G \to C \) is the set \( \{ g(x) : x \in G \} \). If the image of \( g \) is contained in the domain of another function \( f : H \to C \), we define the composition \( f \circ g : G \to C \) through \[ (f \circ g)(x) = f(g(x)). \] ## Proposition 2.6 Let \( g : G \to C \) with image contained in \( H \), and let \( f : H \to C \). Suppose \( x_n \) is an accumulation point of \( G \), \( \lim_{n \to \infty} g(x_n) = w_0 \in H \), and \( f \) is continuous at \( w_0 \). Then \[ \lim_{n \to \infty} f(g(x_n)) = f\left(\lim_{n \to \infty} g(x_n)\right), \] in short, \[ \lim_{n \to \infty} f(g(x_n)) = f\left( \lim_{n \to \infty} g(x_n) \right). \] **Proof.** Given \( \epsilon > 0 \), we know there is \( \delta > 0 \) such that \[ |w - w_0| < \delta \implies |f(w) - f(w_0)| < \epsilon. \] For this, we also know there is a \( \eta > 0 \) such that \[ 0 < |x - z_0| < \eta \implies |g(x) - w| < \delta. \] Stringing these two implications together gives that \[ 0 < |x - z_0| < \eta \implies |f(g(x)) - f(w_0)| < \epsilon. \] #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 36 Context: # Differentiation The basic properties for derivatives are similar to those we know from real calculus. In fact, the following rules follow mostly from properties of the limit. ## Proposition 2.10 Suppose \( f \) and \( g \) are differentiable at \( c \in \mathbb{C} \) and \( g \) is differentiable at \( g(c) \). Then: 1. \( (f + g)'(z) = f'(z) + g'(z) \) for any \( z \in \mathbb{C} \) 2. \( (f \cdot g)'(z) = f'(z)g(z) + f(z)g'(z) \) 3. \( \left( \frac{f}{g} \right)'(z) = \frac{f'(z)g(z) - f(z)g'(z)}{(g(z))^2} \) provided that \( g(z) \neq 0 \) 4. \( (z^n)' = n z^{n-1} \) for any nonzero integer \( n \) 5. \( g \) is continuous at \( c \) 6. \( (h \circ g)'(z) = h'(g(z))g'(z) \) **Proof of (2):** \[ (fg)'(z) = \lim_{h \to 0} \frac{f(z + h)g(z + h) - f(z)g(z)}{h} \] \[ = \lim_{h \to 0} \left( f(z + h)g(z + h) - f(z)g(z + h) + f(z)g(z + h) - f(z)g(z) \right) \frac{1}{h} \] \[ = \lim_{h \to 0} \left( \frac{f(z + h) - f(z)}{h}g(z + h) + f(z)\frac{g(z + h) - g(z)}{h} \right) \] \[ = \lim_{h \to 0} \frac{f(z + h) - f(z)}{h} \cdot g(z) + f(z) \cdot \lim_{h \to 0} \frac{g(z + h) - g(z)}{h} \] \[ = f'(z)g(z) + f(z)g'(z) \] Note that we have used the definition of the derivative and Proposition 2.4(a) & (b) (the addition and multiplication rules for limits). A prominent application of the differentiation rules is the composition of a complex function \( f(z) \) with a path \( \gamma(t) \). The proof of the following result gives the reverse. ## Proposition 2.11 Suppose \( f \) is holomorphic at \( c \in \mathbb{C} \) with \( f(\gamma(0)) \) and \( \gamma_1 \) and \( \gamma_2 \) are two smooth paths that pass through \( c \), making an angle of \(\theta\) with each other. Then if \( \gamma_1 \) transforms \( z \) and \( \gamma_2 \) into smooth paths which meet \( f(\gamma) \), the transformed paths make an angle of \( \alpha \) with each other. In words, a holomorphic function with nonzero derivative preserves angles. Functions that preserve angles in this way are called conformal. #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 38 Context: # 2 Differentiation Now define \( z_0 = g(x_0) \) and set \[ \phi(w) = \begin{cases} \frac{f(w) - f(w_0)}{w - w_0} & \text{if } w \neq w_0 \\ f'(w_0) & \text{if } w = w_0 \end{cases} \] This is continuous at \( w_0 \) and \(\lim_{w \to w_0} \phi(w) = f'(w_0)\), so we can apply Proposition 2.6: \[ g'(x_0) = \lim_{h \to 0} \frac{1}{\phi(g(x_0 + h))} \cdot \frac{1}{f'(w_0)} \] ## 2.3 The Cauchy–Riemann Equations When considering a real-valued function \( f: \mathbb{R}^2 \to \mathbb{R} \) of two variables, there is no notion of the derivative of a function. For such a function, we instead only have partial derivatives \( \frac{\partial f}{\partial x}(x_0, y_0) \) and \( \frac{\partial f}{\partial y}(x_0, y_0) \) (and also directional derivatives) which depend on the way in which we approach a point \( (x_0, y_0) \in \mathbb{R}^2 \). For a complex-valued function \( f(z) \), we now have a new concept of the derivative \( f'(z_0) \), which definition cannot depend on the way in which we approach a point \( z_0 = (x_0, y_0) \in \mathbb{C} \). It is logical, then, that there should be a relationship between the complex derivative \( f'(z_0) \) and the partial derivatives \[ \frac{\partial f}{\partial x}(z_0) = \lim_{h \to 0} \frac{f(x_0 + h) - f(x_0, y_0)}{h} \] and \[ \frac{\partial f}{\partial y}(z_0) = \lim_{k \to 0} \frac{f(x_0, y_0 + k) - f(x_0, y_0)}{k} \] (this definition is exactly as in the real-valued case). This relationship between the complex derivative and partial derivatives is very strong, and it is a powerful computational tool. It is described by the Cauchy–Riemann equations, named after Augustin Louis Cauchy (1789–1857) and Georg Friedrich Bernhard Riemann (1826–1866), even though the equations appeared already in the works of Jean le Rond d'Alembert (1717–1783) and Leonhard Euler (1707–1783). #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 41 Context: In the second case, \( \Delta z = i \Delta y \) and \[ f'(z_0) = \lim_{y_0 \to y_0} \frac{f(x_0 + i y) - f(x_0)}{i y} = \lim_{y_0 \to 0} \frac{f(x_0 + i y) - f(x_0)}{y} \] Thus, we have shown that \( f'(z_0) = f_x(z_0) - i f_y(z_0) \). (b) Suppose the Cauchy-Riemann equation (2.2) holds and the partial derivatives \( f_x \) and \( f_y \) are continuous in an open disk centered at \( z_0 \). Our goal is to prove that \( f'(z_0) = f_x(z_0) + i f_y(z_0) \). By (2.2), \[ f(z_0) = x_0 + i y_0 \quad f(z_0 + \Delta z) = f(z_0 + \Delta x + i \Delta y) \] On the other hand, we can rewrite the difference quotient for \( f'(z_0) \) as \[ f(z_0 + \Delta z) - f(z_0) = \frac{f(z_0 + \Delta z) - f(z_0)}{\Delta z} = \frac{f(z_0 + \Delta x + i \Delta y) - f(z_0)}{\Delta z} = \frac{f(z_0 + \Delta x) - f(z_0)}{\Delta z} + \frac{f(z_0 + \Delta y) - f(z_0)}{\Delta z} \] Thus, \[ \lim_{\Delta z \to 0} \frac{f(z_0 + \Delta z) - f(z_0)}{\Delta z} = \lim_{\Delta y \to 0} \frac{f(z_0 + i \Delta y) - f(z_0)}{\Delta y} = \lim_{\Delta x \to 0} \left( \frac{f(z_0 + \Delta x + i \Delta y) - f(z_0)}{\Delta z} \right). \] We claim that both limits on the right-hand side are 0, so we have achieved our set goal. The fractions \( \frac{\Delta y}{\Delta z} \) and \( \frac{\Delta x}{\Delta z} \) are bounded in absolute value by 1, so we just need to see that the differences in parentheses are 0. The second term on the right-hand side of (2.5) has a limit of 0 since, by definition, \[ f(z) = \lim_{\Delta z \to 0} \frac{f(z + \Delta z) - f(z)}{\Delta z} \] and taking the limit here as \( \Delta z \to 0 \) is the same as taking the limit as \( \Delta x \to 0 \). #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 42 Context: ```markdown 36 # Differentiation We cannot do something equivalent for the first term in (2.5), since now both \( \Delta x \) and \( \Delta y \) are involved, and both change as \( \Delta x \to 0 \). Instead, we apply the Mean-Value Theorem A.2 for real functions\(^1\) to the real and imaginary parts \( u(x, y) \) and \( v(x, y) \) of \( f(z) \). Theorem A.2 gives real numbers \( \xi \) and \( \eta \) such that \[ \frac{u(x_0 + \Delta x, y_0 + \Delta y) - u(x_0, y_0)}{\Delta y} = u_y(x_0, y_0 + \eta) \] and \[ \frac{v(x_0 + \Delta x, y_0 + \Delta y) - v(x_0, y_0)}{\Delta y} = v_y(x_0, y_0 + \xi). \] Thus, \[ \frac{f(x_0 + \Delta x + i \Delta y) - f(x_0 + i y)}{\Delta y} = \frac{u(x_0 + \Delta x, y_0 + \Delta y) - u(x_0, y_0)}{\Delta y} - iv_x(x_0, y_0) \] \[ = \left( \frac{u(x_0 + \Delta x, y_0 + \Delta y) - u(x_0, y_0)}{\Delta y} \right) - i v_y(x_0, y_0) + \left( \frac{u(x_0 + \Delta x, y_0 + \Delta y) - u(x_0, y_0)}{\Delta y} - iv_y(x_0, y_0) \right) = u_y(x_0, y_0) - iv_y(x_0, y_0). \tag{2.6} \] Because \( u \) and \( v \) are continuous at \( (x_0, y_0) \), \[ \lim_{\Delta y \to 0} u(x_0 + \Delta x, y_0 + \Delta y) = u(x_0, y_0) \] and \[ \lim_{\Delta y \to 0} v(x_0 + \Delta x, y_0 + \Delta y) = v(x_0, y_0), \] and so (2.6) goes to \( \Delta z \to 0 \), which we set out to prove. ## 2.4 Constant Functions As a simple application of the definition of the derivative of a complex function, we consider functions that have a derivative of 0. In a typical calculus course, one of the first applications of the Mean-Value Theorem for real-valued functions (Theorem A.1) is to constant functions. ``` #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 43 Context: # CONSTANT FUNCTIONS ## Proposition 2.16 If \( I \) is an interval and \( f : I \rightarrow \mathbb{R} \) is a real-valued function with \( f'(x) \) defined and equal to 0 for all \( x \in I \), then there is a constant \( c \in \mathbb{R} \) such that \( f(x) = c \) for all \( x \in I \). **Proof:** The Mean-Value Theorem A.2 says that for any \( x, y \in I \), \[ f(y) - f(x) = f' \left( c \right) (y - x) \] for some \( 0 < a < 1 \). Now \( f' (x + a(y - x)) = 0 \), so the above equation yields \( f(y) = f(x) \). Since this is true for any \( x, y \in I \), the function \( f \) must be constant on \( I \). We do not (yet) have a complex version of the Mean-Value Theorem, and so we will use a different argument to prove that a complex function whose derivative is always 0 must be constant. Our proof of Proposition 2.16 required two key features of the function \( f \), both of which are somewhat obviously necessary. The first is that \( f \) be differentiable everywhere in its domain. In fact, if \( f \) is not differentiable everywhere, we can construct functions that have zero derivative almost everywhere but that have infinitely many values in their image. The second key feature is that the interval \( I \) is connected. It is certainly important for the domain to be connected in both the real and complex cases. For instance, if we define the function \[ f(z) = \begin{cases} 1 & \text{if } \text{Re}(z) \geq 0 \\ 2 & \text{if } \text{Re}(z) < 0 \end{cases} \] then \( f(z) \) is 0 for all \( z \in I \), but \( f \) is not constant. This may seem like a silly example, but it illustrates a pivotal to proving a function is constant that we must be careful of. Recall that a region of \( \mathbb{C} \) is an open connected subset. ## Theorem 2.17 If \( G \subset \mathbb{C} \) is a region and \( f : G \rightarrow \mathbb{C} \) is a complex-valued function with \( f'(z) \) defined and equal to 0 for all \( z \in G \), then \( f \) is constant. **Proof:** We will show that \( f \) is constant along horizontal segments and along vertical segments in \( G \). #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 44 Context: # Differentiation Suppose that \( H \) is a horizontal line segment in \( G \). Thus there is some number \( y_0 \in \mathbb{R} \) such that the imaginary part of any \( z \in H \) is \( y_0 \). Now consider the real part \( u(x) \) of the function \( f \) for \( x \in H \). Since \( \text{Im}(z) = y_0 \) is constant on \( H \), we can consider \( f(z) = u(x) + iv(y) \) to be just a function of \( x \) for \( z = x + iy \). By assumption, \( f(z) = 0 \) for \( z \in H \) we have \( u(x) = \text{Re}(f(z)) = 0 \). Thus, by Proposition 2.16, \( u(x) \) is constant on \( H \). We can argue the same way to see that the imaginary part \( v(y) \) of \( f(z) \) is constant on \( H \), since \( v(y) = \text{Im}(f(z)) = 0 \) on \( H \). Since both the real and imaginary parts of \( f(z) \) are constant on \( H \), the function \( f(z) \) itself is constant on \( H \). This same argument works for vertical segments, interchanging the roles of the real and imaginary parts. We have thus proved that if \( f \) is constant along horizontal segments and along vertical segments in \( G \), and if \( x \) and \( y \) are two points in \( G \) that can be connected by a path composed of horizontal and vertical segments, we conclude that \( f(x) = f(y) \). But any two points of a region may be connected by finitely many such segments by Theorem 1.12, so \( f \) has the same value at any two points of \( G \), thus proving the theorem. There are a number of surprising applications of Theorem 2.17; see, e.g., Exercises 2.20 and 2.21. ## Exercises 1. Use the definition of limit to show for any \( z \in G \) that \( \lim_{z \to z_0} (az + b) = az_0 + b \). 2. Evaluate the following limits or explain why they don't exist. (a) \( \lim_{x \to 2} \frac{x^2 - 4}{x - 2} \) (b) \( \lim_{x \to 1} \ln(x + (2x + y)) \) 3. Prove that, if a limit exists, then it is unique. 4. Prove Proposition 2.4. 5. Let \( G \subset \mathbb{C} \) and suppose \( z_0 \) is an accumulation point of \( G \). Show that \( \lim_{z \to z_0} f(z) = 0 \) if and only if \( \lim_{z \to z_0} |f(z)| = 0 \). #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 45 Context: # CONSTANT FUNCTIONS ## 2.6. Proposition 2.2 is useful for showing that limits do not exist, but it is not at all useful for showing that a limit does exist. For example, define $$ f(x,y) = \frac{xy}{x^2 + y^2} \quad \text{where } (x,y) \neq (0,0). $$ Show that the limits of \( f \) along all straight lines through the origin exist and are equal, but \( \lim_{(x,y) \to (0,0)} f(x,y) \) does not exist. (Hint: Consider the limit along the parabola \( y = x^2 \).) ## 2.7. Suppose that \( f(z) = u(x,y) + iv(x,y) \) and \( u_0 = u(x_0,y_0) \) and \( v_0 = v(x_0,y_0) \). Prove that $$ \lim_{(x,y) \to (x_0,y_0)} f(z) = u_0 + iv_0 $$ if and only if $$ \lim_{(x,y) \to (x_0,y_0)} u'(x,y) = u_0 \quad \text{and} \quad \lim_{(x,y) \to (x_0,y_0)} v'(x,y) = v_0. $$ ## 2.8. Show that the function \( f : C \to \mathbb{C} \) given by \( f(z) = z^2 \) is continuous on \( C \). ## 2.9. Show that the function \( g : C \to \mathbb{C} \) given by $$ g(z) = \begin{cases} \frac{1}{z} & \text{if } z \neq 0 \\ 1 & \text{if } z = 0 \end{cases} $$ is continuous on \( C \setminus \{0\} \). ## 2.10. Determine where each of the following functions \( f : C \to C \) is continuous: (a) $$ f(z) = \begin{cases} 0 & \text{if } z = 0 \text{ or } |z| \text{ is irrational} \\ 1 & \text{if } |z| \in \mathbb{Q} \setminus \{0\} \end{cases} $$ (b) $$ f(z) = \begin{cases} 0 & \text{if } z = 0 \\ \sin z & \text{if } z \neq 0 \text{ and } z \neq re^{i\theta} \end{cases} $$ #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 46 Context: ``` ## 2.11. Show that the two definitions of continuity in Section 2.1 are equivalent. Consider separately the cases where \(x_0\) is an accumulation point of \(G\) and where \(x_0\) is an isolated point of \(G\). ## 2.12. Consider the function \(f : C \setminus \{0\} \to G\) given by \(f(z) = \frac{1}{z}\). Apply the definition of the derivative to give a direct proof that \(f'(z) = -\frac{1}{z^2}\). ## 2.13. Prove Proposition 2.6. ## 2.14. Prove Proposition 2.10. ## 2.15. Find the derivative of the function \(T(z) = \frac{a z^2 + b z + c}{e^{z}}\), where \(a, b, c \in C\) with \(ad - bc = 0\). When is \(T'(z) = 0\)? ## 2.16. Prove that if \(f(z)\) is given by a polynomial in \(z\) then \(f\) is entire. What can you say if \(f(z)\) is given by a polynomial in \(z = x + iy\) and \(y = \Im z\)? ## 2.17. Prove or find a counterexample. If \(x\) and \(y\) are real valued and continuous, then \(f(z) = u(x,y) + iv(x,y)\) is continuous; if \(u\) and \(v\) are (real) differentiable then \(f\) is (complex) differentiable. ## 2.18. Where are the following functions differentiable? Where are they holomorphic? Determine their derivatives at points where they are differentiable. 1. \(f(z) = e^{z} \cdot e^{c+iy}\) 2. \(f(z) = 2x + iy\) 3. \(f(z) = x^3 + iy\) 4. \(f(z) = e^{c-iy}\) 5. \(f(z) = \cosh x - i \sin y\) 6. \(f(z) = \Im z\) 7. \(f(z) = |z|^2 = x^2 + y^2\) 8. \(f(z) = z^2\) 9. \(f(z) = \Im z\) 10. \(f(z) = \frac{4}{Re z} - i(Re z)^2\) ``` #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 48 Context: ``` 42 # Differentiation ## 2.27 Consider the general real homogeneous quadratic function \( u(x,y) = ax^2 + bxy + cy^2 \), where \( a, b, \) and \( c \) are real constants. (a) Show that \( u \) is harmonic if and only if \( a + c = 0 \). (b) If \( u \) is harmonic then show that \( u \) is the real part of a function of the form \( f(z) = A z^2 \) for some \( A \in \mathbb{C} \). Give a formula for \( A \) in terms of \( a \) and \( c \). ## 2.28 Re-prove Proposition 2.10 by using the formula for \( f' \) given in Theorem 2.13. ## 2.29 Prove that, if \( G \subset \mathbb{C} \) is a region and \( f : G \to \mathbb{C} \) is a complex-valued function with \( f''(z) \) defined and equal to 0 for all \( z \in G \), then \( f(z) = az + b \) for some \( a, b \in \mathbb{C} \). (Hint: Use Theorem 2.17 to show that \( f'(z) = a \), and then use Theorem 2.17 again for the function \( f(z) - az \).) ``` #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 49 Context: # Chapter 3 ## Examples of Functions _To many, mathematics is a collection of theorems. For us, mathematics is a collection of examples; a theorem is a statement about a collection of examples and the purpose of proving theorems is to classify and explain the examples..._ — John B. Conway In this chapter we develop a toolkit of complex functions. Our ingredients are familiar from calculus: linear functions, exponentials and logarithms, and trigonometric functions. Yet, when we move these functions into the complex world, they take on—at times drastically different—new features. ### 3.1 Möbius Transformations The first class of functions that we will discuss in some detail are built from linear polynomials. **Definition.** A linear fractional transformation is a function of the form \[ f(z) = \frac{az + b}{cz + d} \] where \(a, b, c, d \in \mathbb{C}\) and \(ad - bc \neq 0\). Then \(f\) is called a Möbius transformation. Exercise 2.16 said that any polynomial \(p\) is an entire function, and so the linear fractional transformation \(f(z) = \frac{az + b}{cz + d}\) is holomorphic on \(\mathbb{C} \setminus \{-\frac{d}{c}\}\), unless \(c = 0\) (in which case \(c\) is entire). If \(c \neq 0\) then \(f(z) = \frac{az + b}{cz + d}\) implies \(ad - bc \neq 0\), which means that a Möbius transformation \(f(z) = \frac{az + b}{cz + d}\) will never take on the value \(-\frac{b}{d}\). Our first proposition in this chapter says that with these small observations about the domain and image of a Möbius transformation, we obtain a class of bijections, which are quite special among complex functions. #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 52 Context: # Examples of Functions For real numbers \( a, b, \gamma, y, r \) and \( s \) that satisfy \( \beta^2 + \gamma^2 > 4\alpha \) (Exercise 3.3). The form (3.1) is more convenient for us, because it includes the possibility that the equation describes a line (precisely when \( a = 0 \)). Suppose \( x + iy \) satisfies (3.1); we need to prove that \( x + iy = \frac{x - iy}{x^2 + y^2} \) satisfies a similar equation. \[ u + iv = \frac{x - iy}{x^2 + y^2}. \] We can rewrite (3.1) as: \[ 0 = \alpha + \frac{y}{x^2 + y^2} + \frac{8}{x^2 + y^2}. \] This leads to: \[ 0 = \alpha + \beta u + \gamma (a + b + r^2). \tag{3.2} \] But this equation, in conjunction with Exercise 3.3, says that \( x + iy \) lies on a circle or line. ## 3.2 Infinity and the Cross Ratio In the context of Möbius transformations, it is useful to introduce a formal way of saying that a function \( f \) "blows up" in absolute value, and this gives rise to a notion of infinity. **Definition.** Suppose \( f : G \to \mathbb{C} \). 1. \( \lim_{x \to a} f(x) = \infty \) means that for every \( M > 0 \) we can find \( \delta > 0 \) so that, for all \( x \in G \) satisfying \( 0 < |x - a| < \delta \), we have \( |f(x)| > M \). 2. \( \lim_{x \to b} f(x) = L \) means that for every \( \epsilon > 0 \) we can find \( N > 0 \) so that, for all \( x \in G \) satisfying \( |x| > N \), we have \( |f(x) - L| < \epsilon \). 3. \( \lim_{x \to \infty} f(x) = \infty \) means that for every \( M > 0 \) we can find \( N > 0 \) so that, for all \( x \in G \) satisfying \( |x| > N \), we have \( |f(x)| > M \). In the first definition we require that \( a \) be an accumulation point of \( G \), while in the second and third we require that \( b \) be an "extended accumulation point" of \( G \), in the sense that for every \( B > 0 \) there is some \( z \in G \) with \( |z| > B \). As in Section 2.1, the limit, in any of these senses, is unique if it exists. #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 55 Context: We can take this remark a step further, based on the idea that three distinct points in \( \mathbb{C} \) determine a unique circle passing through them. If the three points are in \( \mathbb{C} \) and non-collinear, this fact comes straight from Euclidean geometry: if the three points are on a straight line or if one of the points is at infinity, then the circle is a straight line plus one. ### Example 3.10 The Möbius transformation \( f : \mathbb{C} \to \mathbb{C} \) given by \( f(z) = \frac{az + b}{cz + d} \) maps: - \( 1 \to 0 \) - \( i \to 1 \) - \( -1 \to \infty \) The points \( 1, i, -1 \) uniquely determine the unit circle and the points \( 0, 1, \) and \( \infty \) uniquely determine the real line, viewed as a circle in \( \mathbb{C} \). Thus Corollary 3.8 implies that if \( f \) maps the unit circle to \( \mathbb{R} \), which we already concluded in Example 3.5. Conversely, if we know where three distinct points in \( \mathbb{C} \) are transformed by a Möbius transformation, then we should be able to figure out everything about the transformation. There is a computational device that makes this easier to see. ### Definition If \( z_1, z_2, z_3 \) are any four points in \( \mathbb{C} \) with \( z_1, z_2, \) and \( z_3 \) distinct, then their cross ratio is defined as: \[ [z_1, z_2, z_3, z_4] = \frac{(z_1 - z_2)(z_3 - z_4)}{(z_1 - z_4)(z_3 - z_2)} \] This includes the implicit definitions \([z_1, z_2, z_3, \infty] = 0\) and, if one of \( z_1, z_2, \) or \( z_3 \) is \( 0 \), then the two terms containing \( 0 \) are canceled; e.g., \([0, z_2, z_3, \infty] = \frac{z_2 - z_3}{0 - z_2} = \frac{1}{-1}\). ### Example 3.11 Our running example \( f(z) = \frac{1}{z} \) can be written as \( f(z) = [1, i, -1] \). ### Proposition 3.12 The function \( f : \mathbb{C} \to \mathbb{C} \) defined by \( f(z) = [z_1, z_2, z_3, z] \) is a Möbius transformation that satisfies: \[ f(z_1) = 0, \quad f(z_2) = 1, \quad f(z_3) = \infty. \] Moreover, if any Möbius transformation with \( g(z_1) = 0, \, g(z_2) = 1, \) and \( g(z_3) = \infty \), then \( f \) and \( g \) are identical. ### Proof Most of this follows from our definition of \( f \), but we need to prove the uniqueness statement. By Proposition 3.11, the inverse \( f^{-1} \) is a Möbius transformation and, by Exercise 3.10, the composition \( g = f \circ f^{-1} \) is again a Möbius transformation. #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 57 Context: # Stereographic Projection To begin, we think of \(C\) as the \((x,y)\)-plane in \(\mathbb{R}^3\), that is, \(C = \{(x,y,0) \in \mathbb{R}^3\}\). To describe stereographic projection, we will be less concerned with actual complex numbers \(x + iy\) and more concerned with their coordinates. Consider the unit sphere \[ S^2 = \{(x,y,z) \in \mathbb{R}^3 : x^2 + y^2 + z^2 = 1\}. \] The sphere and the complex plane intersect in the set \(\{(x,y,0) : x^2 + y^2 = 1\}\), which corresponds to the equator on the sphere and the unit circle on the complex plane, as depicted in Figure 3.1. Let \(N = (0,0,1)\), the north pole of \(S^2\), and let \(S = (0,0,-1)\), the south pole. ![Figure 3.1: Setting up stereographic projection.](#) ## Definition The stereographic projection of \(S^2\) to \(\hat{C}\) from \(N\) is the map \(\varphi: S^2 \setminus \{N\} \to \hat{C}\) defined as follows. For any point \(P \in S^2 \setminus \{N\}\), let \(x\) be the \(x\)-coordinate of \(P\) and let \(l\) be the line through \(N\) and \(P\). The \(z\)-coordinate of \(P\) is strictly less than \(1\), the line through \(N\) and \(P\) intersects \(C\) in exactly one point \(Q\). Define \(\varphi(P) = Q\). We also declare that \(\varphi(N) = \infty\). ## Proposition 3.14 The map \(\varphi\) is given by \[ \varphi(x,y,z) = \left( \frac{x}{1-z}, \frac{y}{1-z}, 0 \right) \quad \text{if } z \neq 1, \] \[ \varphi(x,y,z) = \infty \quad \text{if } z = 1. \] #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 60 Context: # Examples of Functions Project it to the plane by the stereographic projection \( \pi \), apply \( f \) to the point that results, and then pull this point back to \( S^2 \) by \( \pi^{-1} \). We know \( \varphi(x,y,z) = \left( \frac{x}{1-z}, \frac{y}{1-z} \right) \) which we now regard as the complex number \[ \rho + i \gamma = \frac{-x}{1-z} + i \frac{y}{1-z}. \] We know from a previous calculation that \( p^2 + q^2 = \frac{1}{z^2} \), which gives \( x^2 + y^2 = (1 + z)(1 - z) \). Thus, \[ f\left( \frac{x}{1-z} + i \frac{y}{1-z} \right) = \frac{1 - z}{x + iy} = \frac{1 - z}{(1 - z) x + i (1 - z) y} = \frac{1 - 2X - iY}{x^2 + y^2}. \] Rather than plug this result into the formulas for \( \pi^* \), we can just ask what triple of numbers will be mapped to this particular pair using the formulas \( \varphi(x,y,z) = \left( \frac{x}{1-z}, \frac{y}{1-z} \right) \). Thus, we have shown that the effect of \( f \) on \( S^2 \) is to take \( (x,y,z) \) to \( (x, -y, -z) \). This is a rotation around the \( x \)-axis by 180 degrees. We now have a second argument that \( f(z) \) takes circles and lines to circles and lines. A circle in \( C \) is taken to a circle on \( S^2 \) by \( \pi \). Then \( f(z) = \frac{1}{z} \) rotates the sphere which certainly takes circles to circles. Now \( g \) takes circles back to circles and lines. We can also say that the circles that go to lines under \( f(z) = \frac{1}{z} \) are the circles through 0 because 0 is mapped to \( (0, 0) \) under \( \pi \), and so a circle through 0 in \( C \) goes to a circle through the south pole on \( S^2 \). Now 180-degree rotation about the \( x \)-axis takes the south pole to the north pole, and our circle is now passing through \( N \). But we know that \( g \) will take this circle to a line in \( C \). We end by mentioning that there is, in fact, a way of putting the complex metric on \( S^2 \). It is certainly not the (finite) distance function induced by \( \pi \). Indeed, the origin in the complex plane corresponds to the south pole of \( S^2 \). We have to be able to arbitrarily far away from the origin in \( C \), so the complex distance function has to increasingly grow with the \( z \)-coordinate. The closest points are to the north pole \( N \) corresponding to 0 in \( C \), the larger their distance to the origin, and to each other! In this light, a 'line' in the Riemann sphere \( S^2 \) corresponds to a circle in \( S^2 \) through \( N \). In the regular sphere, the circle has finite lengths, but as a line on the Riemann sphere with the complex metric, it has infinite length. #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 62 Context: # EXAMPLES OF FUNCTIONS They are continuous in \( \mathbb{C} \) and satisfy the Cauchy–Riemann equation (2.2): \[ \frac{\partial f}{\partial x} (z) = \frac{\partial f}{\partial y} (z) \] for all \( z \in \mathbb{C} \). Thus Theorem 2.13 says that \( f(z) = \exp(z) \) is entire with derivative \[ f'(z) = \frac{\partial f}{\partial x} (z) = \exp(z). \] We should make sure that the complex exponential function specializes to the real exponential function for real arguments: indeed, if \( z = x \in \mathbb{R} \), then \[ \exp(x) = e^x = \left( \cos(0) + i\sin(0) \right) = e^x. \] The trigonometric functions—sine, cosine, tangent, cotangent, etc.—also have complex analogues; however, they do not play the same prominent role as in the real case. In fact, we can define them as merely being special combinations of the exponential function. ## Definition The (complex) sine and cosine are defined as: \[ \sin z = \frac{1}{2i} \left( \exp(iz) - \exp(-iz) \right) \] and \[ \cos z = \frac{1}{2} \left( \exp(iz) + \exp(-iz) \right), \] respectively. The tangent and cotangent are defined as: \[ \tan z = \frac{\sin z}{\cos z} = \frac{e^{iz} - e^{-iz}}{i(e^{iz} + e^{-iz}) - 1} \] and \[ \cot z = \frac{\cos z}{\sin z} = \frac{e^{iz} + 1}{e^{iz} - e^{-iz}}, \] respectively. Note that to write tangent and cotangent in terms of the exponential function, we used the fact that \( \exp(iy) - \exp(-iy) = 0 \). Because \( \exp \) is entire, so are sine and cosine. #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 63 Context: # Exponential and Trigonometric Functions ![Image properties of the exponential function](path/to/image.png) Figure 3.2: Image properties of the exponential function. As with the exponential function, we should make sure that we're not redefining the real sine and cosine: if \( z \in \mathbb{R} \) then \[ \sin x = \frac{1}{2i} \left( \exp(ix) - \exp(-ix) \right) \] \[ = \frac{1}{2} \left( \cos x + i \sin x - \cos(-x) - i \sin(-x) \right) = \sin x. \] #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 66 Context: Example 3.18. Here are a few evaluations of Log to illustrate this function: - Log(2) = ln(2) + Arg(2) = ln(2) - Log(-i) = ln(1) + Arg(-1) = \( \frac{3\pi}{2} \) - Log(-3) = ln(3) + Arg(-3) = ln(3) + \( \pi i \) - Log(1 - i) = ln(√2) + Arg(1 - i) = \( \frac{1}{2} \ln(2) - \frac{\pi i}{4} \) The principal logarithm is not continuous on the negative part of the real line, and so Log is a branch of the logarithm on \( \mathbb{C} \setminus \mathbb{R}^+ \). Any branch of the logarithm on a region G can be similarly extended to a function defined on \( G \setminus \{0\} \). Furthermore, the evaluation of any branch of the logarithm at a specific \( z_n \) can differ from Log(z) only by a multiple of \( 2\pi i \); the reason for this is once more the periodicity of the exponential function. So what about the second equation in (3.3), namely, Log(exp(z)) = z? Let’s try the principal logarithm: if \( z = x + iy \) then Log(exp(z)) = ln|\( z \)| + Arg(exp(z)) = ln|\( z \)| + Arg(exp(z)). The right-hand side is equal to \( z = x + iy \) if and only if \( y \in (-\pi, \pi) \). Something similar will happen with any other branch of Log of the logarithm—there will always be many z's for which Log(exp(z)) ≠ \( z \). A warning sign pointing in a similar direction concerns the much-cherished real logarithm: the ln(xy) = ln(x) + ln(y) has no analogue in \( \mathbb{C} \). For example, Log(i) + Log(-1) = \( \frac{1}{2} \ln(2 + \sqrt{2}) = \ln 2 + \frac{pi}{4} \) but Log(i - 1) = Log(-1) = \( \ln 2 - \frac{1}{2} \cdot 2 i \). The problem is that we need to come up with an argument convention to define a logarithm and then stick to this convention. There is quite a bit of subtlety here; e.g., the multi-valued map arg z := all possible arguments of z. #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 71 Context: # LOGARITHMS AND COMPLEX EXPONENTIALS ### 3.18 Find a Möbius transformation that maps the unit disk to \(\{x + iy \in \mathbb{C} : x + y > 0\}\). ### 3.19 The Jacobian of a transformation \(z = u(x, y)\), \(w = v(x, y)\) is the determinant of the matrix \[ \begin{bmatrix} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} \\ \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y} \end{bmatrix}. \] Show that if \(f = u + iv\) is holomorphic then the Jacobian equals \(|f'(z)|^2\). ### 3.20 Find the fixed points in \(C\) of \(f(z) = \frac{az + b}{cz + d}\). ### 3.21 Find each Möbius transformation \(f\): (a) \(f\) maps \(0 \mapsto 1\), \(1 \mapsto \infty\), \(\infty \mapsto 0\). (b) \(f\) maps \(1 \mapsto -1\), \(-1 \mapsto i\), and \(i \mapsto -1\). (c) \(f\) maps the \(x\)-axis to \(y = x\), the \(y\)-axis to \(y = -x\), and the unit circle to itself. ### 3.22 (a) Find a Möbius transformation that maps the unit circle to \(\{x + iy \in \mathbb{C} : x + y = 0\}\). (b) Find two Möbius transformations that map the unit disk \[ \{z \in \mathbb{C} : |z| < 1\} \] to \(\{x + iy \in \mathbb{C} : x + y > 0\}\) and \(\{x + iy \in \mathbb{C} : x + y < 0\}\), respectively. ### 3.23 Given \(a \in \mathbb{R} \setminus \{0\}\), show that the image of the line \(y = a\) under inversion is the circle with center \(\frac{1}{a}\) and radius \(\frac{1}{|a|}\). ### 3.24 Suppose \(z_1, z_2\), and \(z_3\) are distinct points in \(\mathbb{C}\). Show that \(z\) is on the circle passing through \(z_1, z_2\), and \(z_3\) if and only if \([z, z_1, z_2, z_3]\) is real or \(\infty\). #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 72 Context: # EXAMPLES OF FUNCTIONS ## 3.25 Prove that the stereographic projection of Proposition 3.14 is a bijection by verifying that \( \phi \circ \psi^{-1} \) and \( \psi \circ \phi^{-1} \) are the identity map. ## 3.26 Find the image of the following points under the stereographic projection \( \phi \): - \( (0, 0, -1) \) - \( (0, 0, 1) \) - \( (1, 0, 0) \) - \( (1, 1, 0) \) - \( (1, 1, 1) \) ## 3.27 Consider the plane \( H \) determined by \( x + y - z = 0 \). What is a unit normal vector to \( H \)? Compute the image of \( H \cap S^2 \) under the stereographic projection \( \phi \). ## 3.28 Prove that every circle in the extended complex plane \( \mathbb{C} \) is the image of some circle in \( S^2 \) under the stereographic projection \( \phi \). ## 3.29 Describe the effect of the basic Möbius transformations: rotation, real dilation, and translation on the Riemann sphere. *Hint:* For the first two, consider all circles in \( S^2 \) centered on the \( N \) axis, and all circles through both \( N \) and \( S \). For translation, consider two families of circles through \( N \), orthogonal to and perpendicular to the translation. ## 3.30 Show that \( \sin(i) = \sin(E) \) and \( \cos(i) = \cos(E) \). ## 3.31 Let \( z = x + iy \) and show that: (a) \( \sin z = \sin x \cosh y + i \cos x \sinh y \). (b) \( \cos z = \cos x \cosh y - i \sin x \sinh y \). ## 3.32 Prove that the zeros of \( \sin z \) are all real valued. Conclude that they are precisely the integer multiples of \( \pi \). ## 3.33 Describe the images of the following sets under the exponential function \( \exp(z) \): (a) The line segment defined by \( z = 1 + iy, \, 0 \leq y \leq 2\pi \) (b) The line segment defined by \( z = i + x, \, 0 \leq x \leq 2\pi \) (c) The rectangle defined by \( z = x + iy \in \mathbb{C}: \, 0 \leq x \leq 1, \, 0 \leq y \leq 2\pi \) #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 74 Context: # EXAMPLES OF FUNCTIONS ## 3.41. Convert the following expressions to the form \( z = x + iy \). (Reason carefully.) 1. \( e^z \) 2. \( e^{\cos(\log(3 + 4i))} \) 3. \( i^z \) 4. \( e^{i \pi} \) 5. \( e^{z} \) 6. \( e^{(1 + i) z} \) 7. \( \sqrt{3(1 - i)} \) 8. \( \left( \frac{1}{1} \right)^4 \) ## 3.42. Is \( \arg(E) = -\arg(z) \) true for the multiple-valued argument? What about \( \arg(E) = \Arg(z) \) for the principal branch? ## 3.43. For the multiple-valued logarithm, is there a difference between the set of all values of \( \log(z^2) \) and the set of all values of \( 2 \log(z) \)? (Hint: Try some fixed numbers for \( z \).) ## 3.44. For each of the following functions, determine all complex numbers for which the function is holomorphic. If you run into a logarithm, use the principal value unless otherwise stated. (a) \( \frac{z^2}{2} \) (b) \( \frac{1}{z^2} \) (c) \( \Log(z - 2i + 1) \) where \( \Log(z) = \ln |z| + i \arg(z) \) with \( 0 < \arg(z) < 2\pi \) (d) \( \exp(z) \) (e) \( z - 3i \) (f) \( i^z \) ## 3.45. Find all solutions to the following equations: (a) \( \Log(z) = \frac{1}{2} \) \( \cos(z) = 0 \) (b) \( \Log(z) = \frac{1}{2} \) (c) \( \exp(z) = \pi \) (d) \( \sin(z) = \cos(4) \) (e) \( \exp(z) = \exp(\sqrt{E}) \) (f) \( z = 1 + i \) #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 76 Context: ``` ## 3.54 As in the previous problem, let \( f(z) = z^2 \). Let \( Q \) be the square with vertices at \( 0, 2, 2+2i, \) and \( 2i \). Draw \( f(Q) \) and identify the types of image curves corresponding to the segments from \( 2 \) to \( 2+2i \) and from \( 2+2i \) to \( 2i \). They are not parts of either straight lines or circles. **Hint:** You can write the vertical segment parametrically as \( z(t) = 2 + it \). Eliminate the parameter in \( u + iv = f(t) \) to get a \( (u, v) \) equation for the image curve. Exercises 3.53 and 3.54 are related to the cover picture of this book. ``` #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 77 Context: # Chapter 4 ## Integration >If things are nice there is probably a good reason why they are nice; and if you do not know at least one reason for this good fortune, then you will have work to do. >— Richard Askey We are now ready to start integrating complex functions—and we will not stop doing so for the remainder of this book; it turns out that complex integration is much richer than real integration (in one variable). The initial reason for this is that we have an extra dimension to play with: the calculus integral \(\int_a^b f(z) dz\) has a fixed integration path, from \(a\) to \(b\) along the real line. For complex functions, there are many different ways to go from \(a\) to \(b\). ### 4.1 Definition and Basic Properties At first sight, complex integration is not really different from real integration. Let \(a, b \in \mathbb{R}\) and let \(C = [a, b]\) be continuous. Then we define \[ \int_C g(z) \, dz := \int_a^b \text{Re} \, g(t) \, dt + i \int_a^b \text{Im} \, g(t) \, dt. \tag{4.1} \] This definition is analogous to that of integration of a parametric curve in \(\mathbb{R}^2\). For a function that takes complex numbers as arguments, we typically integrate over a path \(γ\) (in place of a real interval). If you meditate about the substitution rule for real integrals (Theorem A.6), the following definition, which is based on (4.1), should come as no surprise. **Definition.** Suppose \(γ\) is a smooth path parametrized by \(γ(t)\), \(a \leq t \leq b\), and \(f\) is a complex function which is continuous on \(γ\). Then we define the integral of \(f\) on \(γ\) as \[ \int_γ f(z) \, dz = \int_a^b f(γ(t)) \, γ'(t) \, dt. \] #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 80 Context: ``` 74 # INTEGRATION For any parametrization \( \gamma(t) \), \( a \leq t \leq b \). Naturally, the length of a piecewise smooth path \( \gamma(t) \) is the sum of the lengths of its smooth components. ## Example 4.4 Let \( \gamma \) be the line segment from \( 0 \) to \( 1 + i \), which can be parametrized by \( \gamma(t) = t + it \) for \( 0 \leq t \leq 1 \). Then: \[ \text{length}(\gamma) = \int_{0}^{1} \left| 1 + i \frac{d}{dt} \right| dt = \int_{0}^{1} \sqrt{2} dt = \sqrt{2}. \] ## Example 4.5 Let \( \gamma(t) \) be the unit circle, which can be parametrized by \( \gamma(t) = e^{it} \) for \( 0 \leq t \leq 2\pi \). Then: \[ \text{length}(\gamma) = \int_{0}^{2\pi} \left| \frac{d\gamma}{dt} \right| dt = \int_{0}^{2\pi} dt = 2\pi. \] Now we observe some basic facts about how the line integral behaves with respect to function addition, scalar multiplication, inverse parametrization, and path concatenation; we also give an upper bound for the absolute value of an integral, which we will make use of time and again. ## Proposition 4.6 Suppose \( f \) is a piecewise smooth path, \( f \) and \( g \) are complex functions which are continuous on \( \gamma \) and \( \gamma \subset \mathbb{C} \). (a) \[ \int_{\gamma} (f + g) \, dz = \int_{\gamma} f \, dz + \int_{\gamma} g \, dz. \] (b) If \( \gamma \) is parametrized by \( \gamma(t) \), \( a \leq t \leq b \), we define the path \( \gamma \) from \( \gamma(a+b) \), \( a \leq t \leq b \). Then: \[ \int_{\gamma} f \, dz = \int_{\gamma} f \, dz + \int_{\gamma} g \, dz. \] (c) If \( \gamma_1 \) and \( \gamma_2 \) are piecewise smooth paths such that \( \gamma_2 \) starts where \( \gamma_1 \) ends, we define the path \( \gamma_1 \# \gamma_2 \) by following \( \gamma_1 \) to its end and then continuing on \( \gamma_2 \) to its end. Then: \[ \int_{\gamma_1 \# \gamma_2} f \, dz = \int_{\gamma_1} f \, dz + \int_{\gamma_2} f \, dz. \] (d) \[ \left| \int_{\gamma} f \, dz \right| \leq \max |f(z)| \cdot \text{length}(\gamma). \] The path \( -\gamma \) defined in (b) is the path that we obtain by traveling through \( \gamma \) in the opposite direction. ``` #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 81 Context: # Definition and Basic Properties **Proof.** (a) follows directly from the definition of the integral and Theorem A.4, the analogous theorem from calculus. (b) follows with the real change of variables \( I = a + b - c \): \[ f = \int_{y_1}^{y_2} f(y(a + b - c)) \, dt \] \[ = -\int_{y_1}^{y_2} f(y(a + b - c)) \, dt \] \[ = \int_{y_1}^{y_2} f(y)(y(a + b - c)) \, da - \int_{y_1}^{y_2} f(y)(y) \, da = -\int f. \] (c) We need a suitable parametrization \( \gamma(t) \) for \( \gamma_1\), \( \gamma_2\). If \( \gamma_1\) has domain \([a_1, b_1]\) and \( \gamma_2\) has domain \([b_2, d_2]\), then we can use: \[ \gamma(t) = \begin{cases} \gamma_1(t) & \text{if } a_1 \leq t \leq b_1 \\ \gamma_2(t - b_1 + a_2) & \text{if } b_1 < t \leq b_2 - a_2 \end{cases} \] with domain \([a_1, b_1, b_2 - a_2]\). Now we break the integral over \( \gamma_2\) into two pieces and apply the change of variables \( I = e - b_1 + a_2\): \[ \int_{\gamma_2} f = \int_{a_1}^{b_1} f(y(t)) \, dt \] \[ = \int_{a_1}^{b_1} f(y(t)) \, dt + \int_{b_1}^{b_2} f(y(t)) \, dt. \] \[ = \int f + \int f. \] The last step follows since \( y\) restricted to \([a_1, b_1]\) is \( \gamma\) and \( y\) restricted to \([b_1 + b_2 - a_2]\) is a reparametrization of \( \gamma_2\). #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 85 Context: # Antiderivatives Let \( f \) be defined. By the same argument, \[ F(z+h) - F(z) = \int_{z}^{z+h} f = \int_{z}^{z+h} f \] for any path \( \gamma \) from \( z \) to \( z+h \). The constant function 1 has the antiderivative \( z \in C \), and so \( F = 1 \), by Theorem 4.11. Thus, \[ F(z+h) - F(z) = \frac{1}{h} \int_{z}^{z+h} (f(w) - f(z)) \, dw = \frac{1}{h} \int_{z}^{z+h} f(w) - f(z) \, dw \] If \(|h|\) is sufficiently small, then the line segment from \( z \) to \( z+h \) will be contained in \( G \), and so by applying the assumptions of our theorem for the third time, \[ F(z+h) - F(z) = \frac{1}{h} \int_{z}^{z+h} (f(w) - f(z)) \, dw = \frac{1}{h} \int_{z}^{z+h} (f(w) - f(z)) \, dw. \quad (4.2) \] We will show that the right-hand side goes to zero as \( h \to 0 \), which will conclude the theorem. Given \( \epsilon > 0 \), we can choose \( \delta > 0 \) such that \[ |w - z| < \delta \implies |f(w) - f(z)| < \epsilon \] because \( f \) is continuous at \( z \). (We can also choose \( \delta \) small enough so that (4.2) holds.) Thus, if \( |h| < \delta \), we can estimate with Proposition 6.6(1): \[ \left| \frac{1}{h} \int_{z}^{z+h} (f(w) - f(z)) \, dw \right| \leq \frac{1}{|h|} \max_{|w|<|h|} |f(w) - f(z)| \cdot \text{length}(h) = \max_{|w|<|h|} |f(w) - f(z)| < \epsilon. \] There are several variations of Theorem 4.15, as we can play with the assumptions about paths in the statement of the theorem. We give one such variation, namely, for polygonal paths, i.e., paths that are composed of unions of line segments. You should convince yourself that the proof of the following result is identical to that of Theorem 4.15. #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 86 Context: Corollary 4.16. Suppose \( G \subset \mathbb{C} \) is a region and \( z_0 \in G \). Let \( f : G \to \mathbb{C} \) be a continuous function such that \[ \int_{C} f(z) \, dz = 0 \] for any closed polygonal path \( C \subset G \). Then the function \( F : G \to \mathbb{C} \) defined by \[ F(z) := \int_{\gamma_0}^{z} f, \] where \( \gamma \) is any polygonal path in \( G \) from \( z_0 \) to \( z \), is an antiderivative for \( f \) on \( G \). If you compare our proof of Theorem 4.15 to its analogue in \( \mathbb{R} \), you will see similarities, as well as some complications due to the fact that we now have to operate in the plane as opposed to the real line. Still, so far we have essentially been “doing calculus” when computing integrals. We will now take a radical departure from this philosophy by studying complex integrals that stay invariant under certain transformations of the paths we are integrating over. ### 4.3 Cauchy's Theorem The central theorem of complex analysis is based on the following concept. **Definition.** Suppose \( \gamma_0 \) and \( \gamma_1 \) are closed paths in the region \( G \subset \mathbb{C} \), parametrized by \( \gamma_0(t) \), \( 0 \leq t \leq 1 \), and \( \gamma_1(t) \), \( 0 \leq t \leq 1 \), respectively. Then \( \gamma_0 \) is \( G \)-homotopic to \( \gamma_1 \) if there exists a continuous function \( H : [0, 1] \times G \to G \) such that, for all \( t \in [0, 1] \): \[ H(0, t) = \gamma_0(t), \] \[ H(1, t) = \gamma_1(t), \] \[ H(s, 0) = H(s, 1). \] We use the notation \( \gamma_0 \sim_G \gamma_1 \) to mean \( \gamma_1 \) is \( G \)-homotopic to \( \gamma_0 \). The function \( H(t, s) \) is called a homotopy. For each fixed \( s \), a homotopy \( H(t, s) \) is a path parametrized by \( t \), and as \( s \) goes from \( 0 \) to \( 1 \), these paths continuously transform from \( \gamma_0 \) to \( \gamma_1 \). The last condition in (4.3) simply says that each of these paths is also closed. ### Example 4.17. Figure 4.1 attempts to illustrate that the unit circle is \( C(1) \), homotopic to the square with vertices \( \pm 3 \pm i \). Indeed, you should check (Exer- #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 90 Context: # Integration ![Figure 4.2](path/to/image) This ellipse is \( \mathbb{C} \setminus \mathbb{R} \)-contractible. ## Corollary 4.20 Suppose \( G \subseteq \mathbb{C} \) is a region, \( f \) is holomorphic in \( G \), \( \gamma \) is piecewise smooth, and \( \gamma \cap \partial G = \emptyset \). Then \[ \int_{\gamma} f = 0. \] This corollary is worth mediating over. For example, you should compare it with Corollary 4.13: both results give a zero integral, yet they make truly opposite assumptions (one about the existence of an antiderivative, the other about the existence of a derivative). Naturally, Corollary 4.20 gives many evaluations of integrals, such as this: ### Example 4.21 Since \( \log z \) is holomorphic in \( G = \mathbb{C} \setminus \{ 0 \} \) and the ellipse in Figure 4.2 is \( G \)-contractible, Corollary 4.20 gives \[ \int_{\gamma} \log z \, dz = 0. \] ■ ### Exercise 4.25(a) says that any closed path is \( C \)-contractible, which yields the following special case of Corollary 4.20. ## Corollary 4.22 If \( f \) is entire and \( \gamma \) is any piecewise smooth closed path, then \[ \int_{\gamma} f = 0. \] The theorems and corollaries in this section are useful not just for showing that certain integrals are zero: #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 92 Context: # INTEGRATION Corollary 4.25. If \( f \) is holomorphic in an open set containing \( D[w, R] \), then \[ f(w) = \frac{1}{2\pi i} \int_{\mathcal{C}[w, R]} (f + R e^{i\theta}) \, d\theta, \] \[ u(w) = \frac{1}{2\pi i} \int_{0}^{2\pi} (w + R e^{i\theta}) \, d\theta \] and \[ v(w) = \frac{1}{2\pi i} \int_{0}^{2\pi} (w + R e^{i\theta}) \, d\theta. \] **Proof of Theorem 4.24 and Corollary 4.25.** By assumption, \( f \) is holomorphic in an open set \( G \) that contains \( D[w, R] \), and so \( f(z) \) is holomorphic in \( H = G \setminus \{w\} \). For any \( 0 < r < R \), \[ C[w, r] \mapsto C[w, R]. \] And so Cauchy's Theorem 4.18 and Exercise 4.4 give \[ \left| \frac{f(z)}{C[w, R]} \right|_{C[w, R]} = \frac{1}{2\pi i} \int_{C[w, R]} \frac{f(z) \, dx}{C[w, R] - z} \] \[ = \frac{1}{2\pi i} \int_{C[w, R]} \frac{f(z) - f(w)}{C[w, R] - z} \, dz \] \[ \leq S_{\max} \, \text{within} \, C[w, R] \quad (4.6) \] \[ \leq \max_{z \in C[w, R]} |f(z) - f(w)| \cdot \text{length}(C[w, r]) \cdot \frac{1}{2\pi R}. \] Here the inequality comes from Proposition 4.6(4). Now let \( \epsilon > 0 \). Because \( f \) is continuous at \( w \), there exists \( \delta > 0 \) such that \( |z - w| < \delta \) implies \[ |f(z) - f(w)| < \frac{\epsilon}{2\pi R}. \] #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 94 Context: # Integration This all smells like good coffee, except ... we might be just dreaming. The argument may be intuitively clear, but intuition doesn’t prove anything. We'll look at it carefully, fill in the gaps, and then we'll see what we have proved. First, we need a notion of the interior of a simple closed path. The fact that any such path \( \gamma \) divides the complex plane into two connected open sets \( \gamma \) (the bounded one of which we call the inside or interior of \( \gamma \)) is one of the first substantial theorems ever proved in topology, the Jordan Curve Theorem, due to Camille Jordan (1838–1922). In this book, we shall assume the validity of the Jordan Curve Theorem. Second, we need to specify the orientation of \( \gamma \), since if the formula gives \( F(w) \) for one orientation then it will give \( -F(w) \) for the other orientation. **Definition.** A piecewise smooth closed path \( \gamma \) is positively oriented if it is parameterized so that its inside is on the left as our parameterization traverses \( \gamma \). An example is a counter-clockwise oriented circle. Third, if \( \gamma \) is positively oriented and \( D[\gamma, R] \) is a closed disk inside \( \gamma \), then we need a homotopy from \( \gamma \) to the counterclockwise circle \( C[\gamma, R] \) that stays inside \( \gamma \) and away from \( D[\gamma, R] \). This is provided directly by another substantial theorem of topology, the Arzelà–Ascoli Theorem, although there are other methods. Again, in this book we shall assume the existence of this homotopy. These results of topology seem intuitively obvious but are surprisingly difficult to prove. If you'd like to see a proof, we recommend that you take a course in topology. There is still a subtle problem with our proof. We assumed that \( \gamma \) is in \( G \), but we also need the interior of \( \gamma \) to be contained in \( G \). Since we need to apply Cauchy’s Theorem to the homotopy between \( \gamma \) and \( C[\gamma, R] \), we could just add this as an assumption to our theorem, but the following formulation will be more convenient later. **Theorem 4.27 (Cauchy’s Integral Formula).** Suppose \( f \) is holomorphic in the region \( G \) and \( \gamma \) is a positively oriented, simple, closed, piecewise smooth path, such that \( w \) is inside \( \gamma \) and \( r > 0 \). Then \[ f(w) = \frac{1}{2\pi i} \int_{\gamma} \frac{f(z)}{z - w} \, dz. \] \[ \text{This is the Jordan of Jordan normal form fame, but not the one of Gauss–Jordan elimination.} \] #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 96 Context: For the two integrals on the right-hand side, we can use Theorem 4.24 with the function \( f(z) = \exp(z) \), which is entire, and so (note that both 2 and 0 are inside \( \mathbb{C} \)): \[ \int_{C[0,3]} \frac{\exp(z)}{z^2 - 2z} \, dz = \frac{1}{2} \cdot 2\pi i \cdot \exp(2) - \frac{1}{2} \cdot 2\pi i \cdot \exp(0) = \pi (e^2 - 1). \] ## Exercises ### 4.1 Find the length of the following paths: (a) \( \gamma(t) = 3 + it, \ -1 \leq t \leq 1 \) (b) \( \gamma(t) = i + e^{it}, \ 0 \leq t \leq 1 \) (c) \( \gamma(t) = \sin(t), \ -\pi \leq t \leq \pi \) (d) \( \gamma(t) = i - e^{it}, \ 0 \leq t \leq 2\pi \) Draw pictures of each path and convince yourself that the lengths you computed are sensible. (The last path is a cycle; it is the trace of a fixed point on a wheel as it makes one rotation.) ### 4.2 Compute the lengths of the paths from Exercise 1.33: (a) the circle \( C(1 + i, 1) \) (b) the line segment from \( -1 \) to \( 2i \) (c) the top half of the circle \( C(0, 3) \) (d) the rectangle with vertices \( 1 \pm 2i \) ### 4.3 Integrate the function \( f(z) = z \) over the three paths given in Example 4.1. ### 4.4 Integrate \( \int_C f(z) \, dz \) where \( C \) is the unit circle, oriented counterclockwise. More generally, show that for any \( w \in \mathbb{C} \) and \( r > 0 \), \[ \int_{C(w,r)} \frac{dz}{z - w} = 2\pi i. \] ### 4.5 Integrate the following functions over the circle \( C(0,2) \): (a) \( f(z) = z \) (b) \( f(z) = z^2 - 2z + 3 \) (c) \( f(z) = \frac{1}{z} \) (d) \( f(z) = xy \) #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 98 Context: ``` 4.10. Prove the following integration by parts statement: Let \( f \) and \( g \) be holomorphic in \( G \), and suppose \( \gamma \subset G \) is a piecewise smooth path from \( z_0 \) to \( z_1 \). Then \[ \int_{\gamma} f(z) g'(z) \, dz = f(z_1) g(z_1) - f(z_0) g(z_0) - \int_{\gamma} f'(z) g(z) \, dz. \] 4.11. Let \( I(k) := \int_0^{2\pi} e^{ik\theta} \, d\theta \). (a) Show that \( I(0) = 1 \). (b) Show that \( I(k) = 0 \) if \( k \) is a nonzero integer. (c) What is \( I(k)^2 \)? 4.12. Compute \( \int_{C[0,1]} z^2 \, dz \). 4.13. Show that \( \int_{\gamma} z^{-1} \, dz = 0 \) for any closed piecewise smooth \( \gamma \) and any integer \( n \neq -1 \). (If \( n \) is negative, assume that \( \gamma \) does not pass through the origin, since otherwise the integral is not defined.) 4.14. Exercise 4.13 excluded \( n = -1 \) for a good reason: Exercise 4.4 gives a counterexample. Generalizing these, if \( n \) is any integer, find a closed path \( \gamma \) so that \( \int_{\gamma} z^{-1} \, dz = 2 \pi i \). 4.15. Taking the previous two exercises one step further, fix \( z_0 \in \mathbb{C} \) and let \( \gamma \) be a simple, closed, positively oriented, piecewise smooth path such that \( z_0 \) is inside \( \gamma \). Show that, for any integer \( n \), \[ \int_{\gamma} (z - z_0)^{-n} \, dz = \begin{cases} 2 \pi i & \text{if } n = -1 \\ 0 & \text{otherwise}. \end{cases} \] 4.16. Prove that \( \int_\gamma e^{z^2} \, dz = 0 \) for any closed path \( \gamma \). 4.17. Show that \( F(z) := \log(t + i) - \frac{1}{2} \log(-z - i) \) is an antiderivative of \( \frac{1}{\sqrt{z}} \) for \( \mathrm{Re}(z) > 0 \). Is \( F(z) \) equal to \( \arctan z \)? ``` #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 99 Context: # CAUCHY'S INTEGRAL FORMULA ### 4.18. Compute the following integrals, where \( \gamma \) is the line segment from \( 4 \) to \( 4i \): 1. \( \int_{\gamma} \frac{1+z}{z} \, dz \) 2. \( \int_{\gamma} \frac{dz}{z^2 + z} \) 3. \( \int_{\gamma} \frac{z^2}{z} \, dz \) 4. \( \int_{\gamma} \sin(z^2) \, dz \) ### 4.19. Compute the following integrals. (Hint: One of these integrals is considerably easier than the other.) (a) \[ \int_{\gamma} e^{\gamma(t)} \, dt \quad \text{where } \gamma(t) = e^{it}, \quad -\frac{\pi}{2} \leq t \leq \frac{\pi}{2} \] (b) \[ \int_{\gamma} e^{\gamma(t)} \, dt \quad \text{where } \gamma(t) = e^{it}, \quad \frac{\pi}{2} \leq t \leq \frac{3\pi}{2} \] ### 4.20. Show that (4.4) gives a homotopy between the unit circle and the square with vertices \( \pm 1 \pm i \). ### 4.21. Use Exercise 1.34 to give a homotopy that proves an alternative to (4.4) and does not need a piecewise definition. ### 4.22. Suppose \( a \in \mathbb{C} \) and \( \gamma_0 \) and \( \gamma_1 \) are two counterclockwise circles so that \( a \) is inside both of them. Give a homotopy that proves \( \gamma_0 \sim_{h} \gamma_1 \). ### 4.23. Prove that \( \gamma_\infty \) is an equivalence relation. ### 4.24. Suppose that \( r \) is a closed path in a region \( G \), parameterized by \( r(t), \, t \in [0, 1] \), and it is a continuous increasing function from \( [0, 1] \) onto \( [0, 1] \). Show that \( r \) is \( G \)-homotopic to the reparameterized path \( r_0(t) = r(t) + (1-t)r \) for \( 0 \leq s \leq 1 \). ### 4.25. (a) Prove that any closed path is \( C \)-contractible. (b) Prove that any two closed paths are \( C \)-homotopic. #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 101 Context: # Cauchy’s Integral Formula ## 4.31 Prove that for \( 0 < r < 1 \), \[ \frac{1}{2 \pi} \int_{0}^{2 \pi} \frac{1 - r^{2}}{1 - 2r \cos(\phi) + r^{2}} \, d\phi = 1. \] (The function \( P_r(\phi) \) is the Poisson kernel and plays an important role in the world of harmonic functions, as we will see in Exercise 6.13.) ## 4.32 Suppose \( f \) and \( g \) are holomorphic in the region \( G \) and \( y \) is a simple piecewise smooth \( G \)-contractible path. Prove that if \( f(e) = g(e) \) for all \( e \in y \), then \( f(e) = g(e) \) for all \( e \) inside \( y \). ## 4.33 Show that Corollary 4.20, for simple paths, is also a corollary of Theorem 4.27. ## 4.34 Compute \[ I(r) := \int_{|z-1| = r} \frac{dz}{z^{2} + 1} \] for \( r \neq 1 \). ## 4.35 Find \[ \int_{C(0,1)} \frac{dz}{z^{2} - 2z - 8} \] for \( r = 1 \) and \( r = 5 \). (Hint: Compute a partial-fractions expansion of the integrand.) ## 4.36 Use the Cauchy Integral Formula (Theorem 4.24) to evaluate the integral in Exercise 4.35 when \( r = 3 \). ## 4.37 Compute the following integrals. (a) \[ \int_{C(-1, 3)} \frac{z^{2}}{z^{2} - 4} \, dz \] (b) \[ \int_{C(1)} \sin z \, dz \] (c) \[ \int_{C(2)} \frac{\exp(z)}{z^{3}} \, dz \] (d) \[ \int_{C(2)} \frac{\exp(z)}{z^{4}} \, dz \] *Named after Siméon Denis Poisson (1781–1840).* #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 104 Context: we rewrite using Theorem 4.27. $$ \frac{f(z + \Delta z) - f(z)}{\Delta z} = \frac{1}{2\pi i} \int_{\gamma} \frac{f(t)}{t - z} \, dt - \frac{1}{2\pi i} \int_{\gamma} \frac{f(t)}{(t - z)^2} \, dz $$ $$ = \frac{1}{2\pi i} \int_{\gamma} \frac{f(t)}{(t - z)^2} \, dt. \tag{5.1} $$ Theorem 5.1 will follow if we can show that the following expression gets arbitrarily small as \( \Delta z \to 0 \): $$ \frac{f(z + \Delta z) - f(z)}{\Delta z} = \frac{1}{2\pi i} \int_{\gamma} \frac{f(t)}{(t - z)^2} \, dt - \frac{f(z)}{(z - w)^2} \, dz $$ This can be made arbitrarily small if we can show that the integral on the right-hand side stays bounded as \( \Delta z \to 0 \). In fact, by Proposition 4.6, it suffices to show that the integrand stays bounded as \( \Delta z \to 0 \) (because \( z \) and curve length \( \gamma \) are fixed). Let \( M = \max_{t \in \gamma} |f(t)| \) (whose existence is guaranteed by Theorem A1). Choose \( \delta > 0 \) such that \( |z - w| < \delta \): that is, \( |z - w| \geq \epsilon \) for all \( y \in \gamma \). By the reverse triangle inequality (Corollary 1.7(b)), for all \( y \in \gamma \): $$ \frac{|f(t)|}{|t - z|^2} \leq \frac{|f(z)|}{|z - w|^2} \leq \frac{M}{(a - |z - w|)^2}, $$ which certainly stays bounded as \( \Delta z \to 0 \). This proves (5.1) and thus the Cauchy Integral Formula for \( f' \). The proof of the formula for \( f'' \) is very similar and will be left to Exercise 5.2. Theorem 5.1 suggests that there are similar formulas for the higher derivatives of \( f \). This is in fact true, and theoretically we could obtain them one by one with the methods of the proof of Theorem 5.1. However, once we start studying power series for holomorphic functions, we will obtain such a result much more easily; so we save the derivation of integral formulas for higher derivatives of \( f \) for later (Corollary 8.11). #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 105 Context: # VARIATIONS OF A THEME Theorem 5.1 has several important consequences. For starters, it can be used to compute certain integrals. ## Example 5.2 \[ \int_{C(0,1)} \frac{\sin(z)}{z^2} \, dz = \frac{2\pi i}{2\pi} \left. \frac{d}{dx} \sin(z) \right|_{z=0} = 2\pi i \cos(0) = 2\pi i. \] ![Figure 5.1: The integration paths in Example 5.3.](path/to/figure5.1.png) ## Example 5.3 To compute the integral \[ \int_{C(0,2)} \frac{dz}{z^2(z-1)}, \] we could employ a partial fractions expansion similar to the one in Example 4.23, or moving the integration path similar to the one in Exercise 4.29. To exhibit an alternative, we split up the integration path as illustrated in Figure 5.1: we introduce an additional path that separates 0 and 1. If we integrate on these two new closed paths (\(Y_1\) and \(Y_2\)) counterclockwise, the two contributions along the new path will cancel each other. The effect is that we transformed an integral for which two singularities were inside the integration path into a sum of two integrals, each of which has only one singularity inside the integration path; these new integrals will... #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 106 Context: # Consequences of Cauchy’s Theorem We know how to deal with, using Theorems 4.24 and 5.1: \[ \int_{\mathcal{C}(0)} \frac{dz}{z^2(z-1)} = \int_{\mathcal{C}(1)} \frac{dz}{z(z-1)^2} = 2\pi i \left( \frac{1}{z} \bigg|_{z=1} + \frac{1}{(-1)^2} \right) = 2\pi i \left( 1 + \frac{1}{2} \right) = 0. \] ## Example 5.4 \[ \int_{\mathcal{C}(1)} \frac{\cos(z)}{z^3} dz = \int_{\mathcal{C}(1)} \frac{d^3}{dz^3} \cos(z) \bigg|_{z=0} = \pi i(-\cos(0)) = -\pi i. \] Theorem 5.1 has another powerful consequence; just from knowing that \( f \) is holomorphic in \( G \), we know of the existence of \( f'' \), that \( f' \) is also holomorphic in \( G \). Repeating this argument for \( f' \), \( f'' \), etc., shows that all derivatives \( f^{(n)} \) exist and are holomorphic. We can translate this into the language of partial derivatives; since the Cauchy–Riemann equations (Theorem 2.13) show that any sequence of partial differentiations of \( f \) results in a constant times \( f^{(n)} \). ## Corollary 5.5 If \( f \) is differentiable in a region \( G \) then \( f \) is infinitely differentiable in \( G \), and all partials of \( f \) with respect to \( x \) and \( y \) exist and are continuous. ## 5.2 Antiderivatives Again Theorem 4.15 gives us an antiderivative for a function that has zero integrals over closed paths in a given region. Now that we have Corollary 5.5, mediating just a bit more over Theorem 4.15 gives a converse of sorts to Corollary 4.20. ### Corollary 5.6 (Morera's Theorem) Suppose \( f \) is continuous in the region \( G \) and \[ \int_{C} f = 0 \] for all piecewise smooth closed paths \( C \subset G \). Then \( f \) is holomorphic in \( G \). [^1]: Named after Giacinto Morera (1856–1907). #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 109 Context: ```markdown # Taking Cauchy’s Formulas to the Limit ## Proof Suppose (by way of contradiction) that \( p(z) \) does not have any roots, that is, \( p(z) \neq 0 \) for all \( z \in \mathbb{C} \). Then \( \frac{1}{p(z)} \) is entire, and so Cauchy’s Integral Formula (Theorem 4.24) gives \[ \frac{1}{p(0)} = \frac{1}{2\pi i} \int_{\mathcal{C}(0, R)} \frac{p'(z)}{p(z)} \, dz, \quad \text{for any } R > 0. \] Let \( d \) be the degree of \( p(z) \) and \( a_k \) its leading coefficient. Propositions 4.6(d) and 5.10 allow us to estimate, for sufficiently large \( R \): \[ \left| \frac{1}{p(0)} \right| = \frac{1}{2\pi} \int_{\mathcal{C}(0, R)} \frac{1}{|p(z)|} \, |dz| \leq \frac{1}{2\pi} \cdot \frac{1}{\text{ext } |a_k| R^{d}} \cdot |2\pi R| = \frac{2 |a_k|}{R^{d}}. \] The left-hand side is independent of \( R \), while the right-hand side can be made arbitrarily small (by choosing \( R \) sufficiently large), and so we conclude that \( \frac{1}{p(0)} = 0 \), which is impossible. ## Theorem 5.11 Theorem 5.11 implies that any polynomial \( p \) can be factored into linear terms of the form \( z - a \) where \( a \) is a root of \( p \) as we can apply the corollary, after getting a root to \( z^n = 0 \) (which is again a polynomial by the division algorithm), etc. (see also Exercise 5.11). A compact reformulation of the Fundamental Theorem of Algebra (Theorem 5.11) is to say that \( \mathbb{C} \) is algebraically closed in contrast, \( \mathbb{R} \) is not algebraically closed. ### Example 5.12 The polynomial \( p(z) = z^4 + 5z^2 + 3 \) has no roots in \( \mathbb{R} \). The Fundamental Theorem of Algebra (Theorem 5.11) states that \( p \) must have a root (in fact, four roots) in \( \mathbb{C} \): \[ p(z) = (z^2 + 1)(z^2 + 3) = (z + i)(z - i)(\sqrt{3} + z)(\sqrt{3} - z) \quad \Box \] Another powerful consequence of Theorem 5.1 is the following result, which again has no counterpart in real analysis (consider, for example, the ratio test). ``` #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 111 Context: # TAKING CAUCHY'S FORMULAS TO THE LIMIT ## Figure 5.2: The integration paths in Example 5.14 We computed the integral over \( \sigma_R \) already in Example 4.29: \[ \int_{-R}^{R} \frac{dz}{z^2 + 1} = \pi. \] This holds for any \( R > 1 \), and so we can take the limit as \( R \to \infty \). By Proposition 4.6(d) and the reverse triangle inequality (Corollary 1.7(b)): \[ \left| \int_{-R}^{R} \frac{dz}{z^2 + 1} \right| \leq \max_{\sigma_R} \left| \frac{1}{z^2 + 1} \right| \cdot |R| \leq \max_{\sigma_R} \left( \frac{1}{|z|^2 - 1} \right) R = \frac{R}{R^2 - 1} \] which goes to \( 0 \) as \( R \to \infty \). Thus, \[ \pi = \lim_{R \to \infty} \int_{-R}^{R} \frac{dz}{z^2 + 1} = \lim_{R \to \infty} \left( \int_{-\infty}^{-R} \frac{dz}{z^2 + 1} + \int_{R}^{\infty} \frac{dz}{z^2 + 1} \right). \] Of course, this integral can be evaluated almost as easily using standard formulas from calculus. However, just slight modifications of this example lead to improper integrals that are beyond the scope of basic calculus; see Exercises 5.18 and 5.19. #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 114 Context: 5.12. Show that a polynomial of odd degree with real coefficients must have a real zero. *(Hint: Use Exercise 1.24.)* 5.13. Suppose \( f \) is entire and \( |f(z)| \leq \sqrt{|z|} \) for all \( z \in \mathbb{C} \). Prove that \( f \) is identically 0. *(Hint: Show first that \( f \) is constant.)* 5.14. Suppose \( f \) is entire and there exists \( M > 0 \) such that \( |f(z)| \leq M \) for all \( z \in \mathbb{C} \). Prove that \( f \) is constant. 5.15. Suppose \( f \) is entire with bounded real part, i.e., writing \( f(z) = u(z) + iv(z) \), there exists \( M > 0 \) such that \( |u(z)| \leq M \) for all \( z \in \mathbb{C} \). Prove that \( f \) is constant. *(Hint: Consider the function \( \exp(f(z)) \).)* 5.16. Suppose \( f \) is entire and there exist constants \( a \) and \( b \) such that \( |f(z)| \leq |z|^a + b \) for all \( z \in \mathbb{C} \). Prove that \( f \) is polynomial of degree at most 1. *(Hint: Use Theorem 5.1 and Exercise 2.29.)* 5.17. Suppose \( f: D(0, 1) \to D(0, 1) \) is holomorphic. Prove that for \( |z| < 1 \), \[ |f'(z)| \leq \frac{1}{1 - |z|}. \] 5.18. Compute \[ \int_0^{\infty} \frac{dx}{x^4 + 1}. \] 5.19. In this problem \( f(z) = \frac{e^{iz}}{e^{iz} + 1} \) and \( R > 1 \). Modify our computations in Example 5.14 as follows: (a) Show that \( f^* = f \) where \( f^* \) is again (as in Figure 5.2) the counterclockwise semicircle formed by the segment \([-R, R]\) on the real axis, followed by the circular arc \( \gamma_R \) of radius \( R \) in the upper half plane from \( R \) to \(-R\). (b) Show that \( |f(z)| \leq \frac{1}{\sqrt{2}} \) for \( z \) in the upper half plane, and conclude that \( \lim_{|z| \to \infty} f(z) = 0 \) and hence \( \lim_{R \to \infty} f_{R} = 0 \). #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 119 Context: # Definition and Basic Properties Let \( f \). Namely, by Theorem 2.13 such a function \( f = u + iv \) must satisfy \[ f' = u_x + iv_x = u_y - iv_y. \] (The second equation follows from the Cauchy–Riemann equations (2.3).) It is also worth mentioning that our proof of Theorem 6.6 shows that if \( u \) is harmonic in \( G \), then \( v \) is the real part of the function \( g = u - iv \), which is holomorphic in \( G \) regardless of whether \( G \) is simply connected or not. As our proof of Theorem 6.6 is constructive, we can use it to produce harmonic conjugates. ## Example 6.7 Revisiting Example 6.1 for the second time, we can construct a harmonic conjugate of \( u(x, y) = xy \) along the lines of our proof of Theorem 6.6: first let \[ g = u_x - i v = y - i x = -i z, \] which has an antiderivative \[ h(z) = \frac{1}{2} z^2 = xy - \frac{1}{2}(x^2 - y^2) \] whose real part is \( u \) and whose imaginary part \[ v(x, y) = -\frac{1}{2}(x^2 - y^2) \] gives a harmonic conjugate for \( u \). We can give a more practical machinery for computing harmonic conjugates, which only depends on computing certain (calculus) integrals; thus this can be easily applied, e.g., to polynomials. We state it for functions that are harmonic in the whole complex plane, but you can easily adjust it to functions that are harmonic on certain subsets of \( \mathbb{C}^2 \). ## Theorem 6.8 Suppose \( u \) is harmonic on \( C \). Then \[ u(x, y) := \int \frac{\partial u}{\partial x}(x, t) \, dt - \int \frac{\partial u}{\partial y}(y, t) \, dt \] is a harmonic conjugate for \( u \). *Theorem 6.8 is due to Sheldon Axler and the basis for his Mathematica package Harmonic Function Theory.* #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 121 Context: Theorem 6.10. Suppose \( u \) is harmonic in the region \( G \) and \( \overline{D(u, r)} \subset G \). Then \[ u(x) = \frac{1}{2\pi} \int_{D(u,r)} u(y + r e^{i\theta}) d\theta. \] **Proof.** Exercise 6.14 provides \( x \in D(u,r) \subset D[u,r] \subset G \). The open disk \( D[u,r] \) is simply connected, so by Theorem 6.6 there is a function \( f \) holomorphic in \( D[u,r] \) such that \( u = \operatorname{Re} f \) on \( D[u,r] \). Now we apply Corollary 4.25 to \( f \): \[ f(x) = \frac{1}{2\pi} \int_{0}^{2\pi} f(x + r e^{i\theta}) d\theta. \] Theorem 6.10 follows by taking the real part on both sides. **Corollary** 4.25 and Theorem 6.10 say that holomorphic and harmonic functions have the mean-value property. Our next result is an important consequence of this property to extreme values of a function. **Definition.** Let \( G \) be a region. The function \( u : G \to \mathbb{R} \) has a strong relative maximum at \( x_0 \in G \) if there exists a disk \( D[u, r] \subset G \) such that \( u(x) \leq u(x_0) \) for all \( x \in D[u, r] \) and \( u(x_0) < u(x) \) for some \( y \in D[u, r] \). The definition of a strong relative minimum is analogous. Theorem 6.11. If \( u \) is harmonic in the region \( G \), then it does not have a strong relative maximum or minimum in \( G \). **Proof.** Assume, by way of contradiction, that \( u \) has a strong relative maximum. Then there is a disk in \( G \) centered at \( x_0 \) containing a point \( x \) with \( u(x) < u(x_0) \). Let \( r = |x - x_0| \) and apply Theorem 6.10: \[ u(x) = \frac{1}{2\pi} \int_{0}^{2\pi} u(x_0 + r e^{i\theta}) d\theta. \] Intuitively, this cannot hold, because some of the function values we're integrating are smaller than \( u(x_0) \), contradicting the mean-value property. To make this into a thorough argument, suppose that \( u(x) < u(x_0) \) for \( r \leq 5 \) and \( x \in [0, 2r] \). Now we split up the mean-value... #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 126 Context: # Harmonic Functions (a) Recall the Möbius function \[ f(z) = \frac{z - a}{1 - \overline{a}z}, \] for some fixed \( a \in \mathbb{C} \) with \( |a| < 1 \), from Exercise 3.9. Show that \( u(f(z)) \) is harmonic on an open disk \( D[0, R] \) containing \( D(0, 1) \). (b) Apply Theorem 6.10 to the function \( u(f(z)) \) with \( v = 0 \) to deduce \[ u(a) = \frac{1}{2\pi} \int_{C(a)} \frac{u(f(z))}{z} dz. \] (c) Recalling, again from Exercise 3.9, that \( f(z) \) maps the unit circle to itself, apply a change of variables to (6.3) to prove \[ u(a) = \frac{1}{2\pi} \int_{0}^{2\pi} u(e^{i\theta}) \left| 1 - \frac{|a|^2}{|e^{i\theta} - a|^2} \right| d\theta. \] (d) Deduce (6.2) by setting \( a = re^{i\theta} \). ## Exercise 6.14 Suppose \( G \) is open and \( \overline{D(a, r)} \subset G \). Show that there is \( R > r \) so that \( \overline{D(a, r)} \subset D[R] \subset G \). (Hint: If \( G = \mathbb{C} \) just take \( R = r + 1 \). Otherwise choose some \( w \in G \), let \( M = |w - a| \), and let \( K = \overline{D(a, M)} \subset G \). Show that \( K \) is nonempty, closed, and bounded, and apply Theorem A1 to find a point \( x_0 \in K \) that minimizes \( f(z) = |z - a| \) on \( K \). Show that \( R = |x_0 - a| \) works.) #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 127 Context: # Chapter 7 ## Power Series *It is a pain to think about convergence but sometimes you really have to.* **Sinai Robbins** Looking back to what machinery we have established so far for integrating complex functions, there are several useful theorems we developed in Chapters 4 and 5. But there are some simple-looking integrals, such as: \[ \int_{C} \frac{\exp(z)}{\sin(z)} \, dz, \tag{7.1} \] that we cannot compute with this machinery. The problems, naturally, come from the singularities at \(0\) and \(x\) inside the integration path, which in turn stem from the roots of the sine function. We might try to simplify this problem a bit by writing the integral as the sum of integrals over the two "D" shaped paths shown in Figure 5.1. ![Figure 7.1: Modifying the integration path for (7.1).](path/to/figure5.1) #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 129 Context: To prove that a sequence \((a_n)\) is divergent, we have to show the negation of the statement that defines convergence; that is, given any \(L \in \mathbb{C}\), there exists \(\epsilon > 0\) such that, given any integer \(N\), there exists an integer \(n > N\) such that \(|a_n - L| > \epsilon\). (If you have not negated any mathematical statements, this is worth meditating about.) ### Example 2.7: The sequence \((a_n = i^n)\) diverges: Given \(L \in \mathbb{C}\), choose \(L = \frac{1}{2}\). We consider two cases: If \(\text{Re}(L) > 0\), then for any \(N\), choose \(n > N\) such that \(a_n = -1\). (This is always possible since \(a_{2k} = i^{2k} = -1\) for any \(k \geq 0\).) Then \[ |a_n - L| = |1 + L| > \frac{1}{2}. \] If \(\text{Re}(L) < 0\), then for any \(N\), choose \(n > N\) such that \(a_n = 1\). (This is always possible since \(a_{4k} = i^{4k} = 1\) for any \(k \geq 0\).) Then \[ |a_n - L| = |1 - L| > \frac{1}{2}. \] This proves that \((a_n = i^n)\) diverges. The following limit laws are the cousins of the identities in Propositions 2.4 and 2.6, with one little twist. ### Proposition 7.3. Let \((a_k)\) and \((b_k)\) be convergent sequences and \(c \in \mathbb{C}\). Then 1. \(\lim_{k \to \infty} (a_k + b_k) = \lim_{k \to \infty} a_k + \lim_{k \to \infty} b_k.\) 2. \(\lim_{k \to \infty} (a_k - b_k) = \lim_{k \to \infty} a_k - \lim_{k \to \infty} b_k.\) 3. \(\lim_{k \to \infty} (c a_k) = c \cdot \lim_{k \to \infty} a_k.\) 4. \(\lim_{k \to \infty} (a_k b_k) = \lim_{k \to \infty} a_k \cdot \lim_{k \to \infty} b_k.\) 5. \(\lim_{k \to \infty} \frac{a_k}{b_k} = \frac{\lim_{k \to \infty} a_k}{\lim_{k \to \infty} b_k}\) where \(b_k \neq 0\) for all sufficiently large \(k\), assuming \(\lim_{k \to \infty} b_k \neq 0.\) 6. \(\lim_{k \to \infty} k = \infty.\) Again, the proof of this proposition is essentially a repeat from arguments we have given in Chapters 2 and 3, as you should convince yourself in Exercise 7.4. We will assume, as an axiom, that it is complete. To phrase this precisely, we need the following. #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 130 Context: # POWER SERIES ## Definition The sequence \((a_n)\) is monotonic if it is either nondecreasing \((a_n \geq a_{n-1}\) for all \(n)\) or nonincreasing \((a_n \leq a_{n-1}\) for all \(n)\). There are many equivalent ways of formulating the completeness property for the reals. Here is what we'll go by: ## Axiom (Monotone Sequence Property) Any bounded monotonic sequence converges. This axiom (or one of its many equivalent statements) gives arguably the most important property of the real number system: namely, that we can, in many cases, determine that a given sequence converges without knowing the value of the limit. In this sense we can use the sequence to define a real number. ## Example 7.4 Consider the sequence \((a_n)\) defined by \[ a_n = 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n}. \] This sequence is increasing (by definition) and each \(a_n \leq 3\) by Exercise 7.9. By the Monotone Sequence Property, \((a_n)\) converges, which allows us to define one of the most famous numbers in all of mathematics, \[ \epsilon = 1 + \lim_{n \to \infty} a_n. \] ## Example 7.5 Fix \(0 < r < 1\). We claim that \(\lim_{n \to \infty} r^n = 0\). First, the sequence \((a_n = r^n)\) converges because it is decreasing and bounded below by 0. Let \(L = \lim_{n \to \infty} r^n\). By Proposition 7.3, \[ L = \lim_{n \to \infty} r^n = r^1 = \lim_{n \to \infty} r^n = 0. \] Thus \((1 - r)L = 0\) and (since \(1 - r \neq 0\)) we conclude that \(L = 0\). We remark that the Monotone Sequence Property implies the Least Upper Bound Property: every nonempty set of real numbers with an upper bound has a least upper bound. The Least Upper Bound Property, in turn, implies the following theorem, which is often stated as a separate axiom: **Footnote:** Both the Archimedean Property and the Least Upper Bound Property can be used in (different) axiomatic developments of \(\mathbb{R}\). #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 131 Context: # Theorem 7.6. [Archimedean Property] If \( x \) is any real number then there is an integer \( N \) that is greater than \( x \). For a proof see Exercise 7.10. Theorem 7.6 essentially says that infinity is not part of the real numbers. Note that we already used Theorem 7.6 in Example 7.1. The Archimedean Property underlies the construction of an infinite decimal expansion for any real number, while the Monotone Sequence Property shows that such infinite decimal expansion actually converges to a real number. We close this discussion of limits with a pair of standard limits. The first of these can be established by calculus methods (such as L'Hôpital's rule (Theorem A.11), by treating \( x \) as the variable); both of them can be proved by more elementary considerations. Either way, we leave the proof of the following to Exercise 7.11. ## Proposition 7.7. (a) Exponential best polynomials: For any polynomial \( p(x) \) (with complex coefficients) and any \( x \in \mathbb{C} \) with \( |x| > 1 \), \[ \lim_{x \to \infty} \frac{p(x)}{e^x} = 0. \] (b) Factorials best exponentials: For any \( z \in \mathbb{C} \), \[ \lim_{n \to \infty} \frac{z^n}{n!} = 0. \] ## 7.2 Series **Definition.** A series is a sequence \( (a_n) \) whose members are of the form \( a_n = \sum_{k=1}^{n} b_k \) (or \( a_n = \sum_{k=1}^{\infty} b_k \)); we call \( (b_k) \) the sequence of terms of the series. The \( a_n = \sum_{k=1}^{n} b_k \) (or \( a = \sum_{k=1}^{\infty} b_k \)) are the partial sums of the series. If we wanted to be lazy we would define convergence of a series simply by referring to convergence of the partial sums of the series—after all, we just defined series through sums. However, there are some convergence features that hinge on special appearances for series, so we mention them here explicitly. For starters, a *Archimedes of Syracuse (287–212 BCE) attributes this property to Eudoxus of Cnidus (408–355 BCE). #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 134 Context: ```markdown # Power Series ## Proof of Proposition 7.13 Suppose \(\sum_{k=1}^{\infty} b_k\) converges. Then, using Proposition 7.3(a) & (d), \[ 0 = \lim_{n \to \infty} \sum_{k=1}^{n} b_k = \lim_{n \to \infty} \left( \sum_{k=0}^{n} b_k - \sum_{k=1}^{n} b_k \right) = \lim_{n \to \infty} b_n. \] A common mistake is to try to use the converse of Proposition 7.13, but the converse is false: ### Example 7.16 The harmonic series \(\sum_{k=1}^{\infty} \frac{1}{k}\) diverges (even though the terms go to 0). If we assume the series converges to \(L\), then \[ L = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots \] \[ > \frac{1}{2} + \frac{1}{4} + \frac{1}{6} + \cdots = \frac{1}{2} \left( 1 + \frac{1}{2} + \frac{1}{3} + \cdots \right) = \frac{1}{2}L. \] Thus, \[ L > \frac{L}{2} \] which leads to \[ L = 1 + \frac{1}{2} + \frac{1}{3} + \cdots = L, \] a contradiction. ![Figure 7.2: The integral test.](link_to_image) ``` #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 138 Context: Let's digest these two notions of convergence of a function sequence by describing them using quantifiers: as usual, \( \forall \) denotes for all and \( \exists \) means there exists. Pointwise convergence on \( G \) translates into \[ V \to \forall z \in G \, \exists N \in \mathbb{N} \, |f_n(z) - f(z)| < \epsilon, \] whereas uniform convergence on \( G \) translates into \[ V \to \exists N \in \mathbb{N} \, \forall z \in G \, \forall n \in \mathbb{N} \, |f_n(z) - f(z)| < \epsilon. \] No big deal — we only exchanged two of the quantifiers. In the first case, \( N \) may well depend on \( z \); in the second case, we need to find an \( N \) that works for all \( z \in G \). And this can make all the difference ... ### Example 7.23 Let \( f: [0, 1] \to \mathbb{R} \) be defined by \( f_n(x) = x^n \). We claim that this sequence of functions converges pointwise to \( f: [0, 1] \to \mathbb{R} \) given by \( f(x) = 0 \). This is immediate for the point \( z = 0 \). Now given any \( \epsilon > 0 \) and \( 0 < |x| < 1 \), choose \( N > \frac{1}{\epsilon} \). Then for all \( n \geq N \), \[ |f_n(x) - f(x)| = |x^n - 0| = |x|^n \leq |x|^N < \epsilon. \] (You ought to check carefully that all our inequalities work the way we claim they do.) ### Example 7.24 Let \( f: [0, 1] \to \mathbb{R} \) be defined by \( f_n(x) = x^n \). We claim that this sequence of functions converges uniformly to \( f: [0, 1] \to \mathbb{R} \) given by \( f(x) = 0 \). Given any \( \epsilon > 0 \) and \( |x| < \frac{1}{n} \), choose \( N \geq \frac{1}{\epsilon} \). Then for all \( n \geq N \), \[ |f_n(x) - f(x)| = |x^n - 0| \leq |x|^n \left( \frac{1}{n} \right)^N. \] (Again, you should carefully check our inequalities.) The differences between Example 7.23 and Example 7.24 are subtle, and we suggest you meditate over them for a while with a good cup of coffee. You might already suspect that the function sequence in Example 7.23 does not converge uniformly, as we will see in a moment. The first application illustrating the difference between pointwise and uniform convergence says, in essence, that if we have a sequence of functions \( f_n \) that #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 140 Context: # POWER SERIES uniform either, though this needs a separate proof, as the domain of the functions in Example 7.23 is the unit disk (Exercise 7.20(b)). Now that we have established Proposition 7.25 about continuity, we can ask about integration of sequences or series of functions. The next theorem should come as no surprise; however, its consequences (which we will see shortly) are wide ranging. ## Proposition 7.27 Suppose \( f_n : G \to \mathbb{C} \) is continuous, for \( n \geq 1 \). Then \( (f_n) \) converges uniformly to \( f : G \to \mathbb{C} \) and \( G \) is a piecewise smooth path. Then \[ \lim_{n \to \infty} \int_{C} f_n(z) \, dz = \int_{C} f(z) \, dz. \] **Proof:** We may assume that \( z \) is not just a point, in which case the proposition holds trivially. Given \( \epsilon > 0 \), there exists \( N \) such that for all \( z \in G \) and all \( n \in \mathbb{N}, \) \[ |f_n(z) - f(z)| < \frac{\epsilon}{\text{length}(C)}. \] With Proposition 4.6(d) we can thus estimate \[ \left| \int_{C} f_n(z) \, dz - \int_{C} f(z) \, dz \right| \leq \max_{z \in C} |f_n(z) - f(z)| \cdot \text{length}(C) < \epsilon. \] All of these notions for sequences of functions hold verbatim for series of functions. For example, if \( \sum_{n} f_n(z) \) converges uniformly on \( G \) and \( G \) is a piecewise smooth path, then \[ \int_{C} \sum_{n} f_n(z) \, dz = \sum_{n} \int_{C} f_n(z) \, dz. \] In some sense, the above identity is why we care about uniform convergence. There are several criteria for uniform convergence; see, e.g., Exercises 7.19 and 7.20, and the following result, sometimes called the Weierstrass M-test. ## Proposition 7.28 Suppose \( f_n : G \to \mathbb{C} \) for \( n \geq 1 \), and \( |f_n(z)| \leq M_n \) for all \( z \in G \), where \( \sum_{n} M_n \) converges. Then \( \sum_{n} f_n(z) \) converges absolutely and uniformly in \( G \). (We say the series \( \sum_{n} f_n(z) \) converges absolutely and uniformly.) **Proof:** For each fixed \( z \), the series \( \sum_{n} |f_n(z)| \) converges absolutely by Corollary 7.12. To show that the convergence is uniform, let \( \epsilon > 0 \). Then there exists \( N \) such that #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 143 Context: Now if \( |z - z_0| < r - |a| \), then \[ \sum_{k=0}^{\infty} |(z - z_0)^k|^p < \sum_{k=0}^{\infty} \frac{|(z - z_0)|^k}{r^k} \leq \frac{1}{1 - \frac{|z - z_0|}{r}}. \] The sum on the right-hand side is a convergent geometric sequence, since \( |z - z_0| < r \), and so \( \sum_{k=0}^{\infty} (z - z_0)^k \) converges absolutely by Corollary 7.12. **Proof of Theorem 7.31.** Consider the set \[ S = \left\{ x \in \mathbb{R}^2 : \sum_{k \neq 0} c_k (x - z_0)^k \text{ converges} \right\}. \] (This set is nonempty since \( 0 \in S \).) If \( R \) is unbounded, then \( \sum_{k \neq 0} c_k (z - z_0)^{-k} \) converges absolutely and uniformly for \( |z - z_0| < r \), for any \( r \) and so this gives the \( R = 0 \) case of Theorem 7.31: choose \( S \) with \( x \) such that \( R < R \). Then Proposition 7.32 says that \( \sum_{k} c_k (z - z_0)^k \) converges absolutely. Since \( |(z - z_0)|^k \leq |z - z_0|^k \), we can now use Proposition 7.28. If \( R \) is bounded, let \( R \) be its least upper bound. If \( R < \infty \), then \( \sum_{k \neq 0} c_k (z - z_0)^{-k} \) converges only for \( z \in R \), which establishes Theorem 7.31 in this case. Now assume \( R > 0 \). If \( |z - z_0| < R \) for some \( r < R \), again we can find \( z \) such that \( r < x < R \). Then \( \sum_{k} c_k |(z - z_0)|^k \) converges absolutely and uniformly for \( |z - z_0| < |z| \) by Proposition 7.13. This proves (b). Finally, if \( |z - z_0| > R \) then there exists \( r < R \) such that \[ |z - z_0| < R - |a|. \] But \( \sum_{k} c_k (z - z_0)^{-k} \) diverges, so by the contrapositive of Theorem 7.20, \( \sum_{k} |c_k| |(z - z_0)|^k \) diverges, and so by the contrapositive of Proposition 7.32, \( \sum_{k} c_k (z - z_0)^{-k} \) diverges, which finishes the proof. #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 147 Context: # REGIONS OF CONVERGENCE ## 7.9 (a) Show that \( \frac{1}{3} \sum_{k=1}^{\infty} \frac{1}{k} \) for any positive integer \( k \). (b) Conclude with Example 7.9 that for any positive integer \( n \), \[ 1 + \frac{1}{2} + \frac{1}{6} + \ldots + \frac{1}{n!} \geq 3. \] ## 7.10 Derive the Archimedean Property from the Monotone Sequence Property. ## 7.11 Prove Proposition 7.7. ## 7.12 Prove that \( (c_n) \) converges if and only if \( (\text{Re} c_n) \) and \( (\text{Im} c_n) \) converge. ## 7.13 Prove that \( Z \) is complete and that \( Q \) is not complete. ## 7.14 Prove that, if \( a_n \leq b_n \leq c_n \) for all \( n \) and \( \lim_{n \to \infty} a_n = L, \, b_n = \lim_{n \to \infty} b_n = L, \, c_n = L \), then \( \lim_{n \to \infty} b_n = L \). This is called the Squeeze Theorem, and is useful in testing a sequence for convergence. ## 7.15 Find the least upper bound of the set \( \{ \text{Re}(e^{(2\pi i t)}) : t \in Q \} \). ## 7.16 (a) Suppose that the sequence \( (c_n) \) converges to zero. Show that \( \sum_{n=0}^{\infty} c_n \) converges if and only if \( \sum_{n=0}^{\infty}(c_n + c_{n+1}) \) converges. Moreover, if the two series converge then they have the same limit. (b) Give an example where \( (c_n) \) does not converge to 0 and one of the series in (a) diverges while the other converges. ## 7.17 Prove that the series \( \sum b_n \) converges if and only if \( \lim_{n \to \infty} \sum_{k=0}^{n} b_k = 0 \). #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 148 Context: # Power Series ![Functions](https://example.com/path/to/image) **Figure 7.3:** The functions \( f_n(x) = \sin\left(\frac{x}{n}\right) \) in Exercise 7.21. ## 7.18 1. Show that \[ \sum_{k=1}^{\infty} \frac{k}{k+1} \text{ diverges.} \] 2. Show that \[ \sum_{k=1}^{\infty} \frac{k}{k^2 + 1} \text{ converges.} \] ## 7.19 1. Suppose \( G \subset \mathbb{C} \) and \( f_n : G \to \mathbb{C} \) for \( n \geq 1 \). Suppose \( (a_n) \) is a sequence in \( \mathbb{R} \) with \( \lim_{n \to \infty} a_n = 0 \) and, for each \( n \geq 1 \), \[ |f_n(a_n)| \leq a_n \quad \text{for all } a \in G. \] Show that \( f_n \) converges uniformly to the zero function in \( G \). 2. Re-prove the statement of Example 7.24 using part (a). ## 7.20 1. Suppose \( G \subset \mathbb{C}, f_n : G \to \mathbb{C} \) for \( n \geq 1 \), and \( f_n \) converges uniformly to the zero function in \( G \). Show that if \( (a_n) \) is any sequence in \( G \), then \[ \lim_{n \to \infty} f_n(a_n) = 0. \] 2. Apply (a) to the function sequence given in Example 7.23, together with the sequence \( (b_n = \frac{1}{n}) \), to prove that the convergence given in Example 7.23 is not uniform. Image Analysis: ### Comprehensive Examination #### 1. Localization and Attribution - **Image Positioning and Numbering**: - **Image 1 (Top)**: Positioned at the upper part of the page. - **Image 2 (Middle)**: Text content starting from "7.18" to "7.20" located beneath the Image 1. #### 2. Object Detection and Classification - **Image 1**: - **Objects**: Graph/Chart depicting a series of curves. - **Classification**: Mathematical function graphs. #### 3. Scene and Activity Analysis - **Image 1**: - **Scene Description**: Graphical representation of functions \( f_n(x) = \sin^n(x) \) for different values of \( n \). - **Activity**: Visualization of function behavior as \( n \) increases, showing how the functions change shape. #### 4. Text Analysis - **Image 2**: - **Text Content**: - **Figure Caption**: "Figure 7.3: The functions \( f_n(x) = \sin^n(x) \) in Exercise 7.21." - **7.18**: Two parts (a) and (b) discussing convergence and divergence of series. - **7.19**: Two parts (a) and (b), with (a) detailing properties of a function sequence and convergence criteria; (b) asks to re-prove part (a). - **7.20**: Two parts (a) and (b), with (a) expanding on function convergence criteria and (b) applying these criteria to a given example. - **Significance**: The text provides mathematical exercises and explanations related to the visual content (graph of sin functions). #### 5. Diagram and Chart Analysis - **Image 1**: - **Diagram**: Graph of \( f_n(x) = \sin^n(x) \) - **Axes**: - **X-Axis**: Represents the variable \( x \). - **Y-Axis**: Represents the value of \( f_n(x) \). - **Scales**: Standard numerical scales for both x and y axes. - **Key Insights**: Demonstrates how the function \( f_n(x) \) narrows and approaches a delta-like function centered at \( \pi \) and other periodic points as \( n \) increases. #### 8. Color Analysis - **Image 1**: - **Dominant Colors**: Blue curves on a white background. - **Impact on Perception**: The color choice helps emphasize the variation among different curves, making it easy to distinguish between different values of \( n \). #### 9. Perspective and Composition - **Image 1**: - **Perspective**: Straight-on view of the graph. - **Composition**: Centralized graph with labeled axes. The curves are symmetric around the origin, following the properties of sine functions. #### 10. Contextual Significance - **Image 1**: - **Context**: Related to exercises and explanations in the study of power series and function convergence in mathematics. - **Contribution**: Provides visual understanding and concrete examples to complement the theoretical exercises below. #### 12. Graph Numbers - **Image 1**: - **Data Points**: Not explicitly provided, but the curves can be understood as sampled points from the functions \( \sin^n(x) \). - **Trends**: As \( n \) increases, the curve sharply peaks at specific points (multiples of \( \pi \)), demonstrating convergence behavior. #### 13. Graph and Trend Analysis - **Image 1**: - **Trends**: The sequence of functions \( f_n(x) = \sin^n(x) \) becomes more peaked at multiples of \( \pi \) as \( n \) increases, indicating how the function evolves with higher powers. - **Interpretation**: This trend is significant in understanding pointwise convergence and properties of Fourier series in mathematical analysis. #### Additional Aspects - **Prozessbeschreibungen (Process Descriptions)**: - **Series Convergence** (7.18): Analysis of divergence and convergence criteria. - **Function Sequence Convergence** (7.19, 7.20): Conditions and proof of uniform convergence of function sequences on given domains. #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 150 Context: ``` # Power Series ## 7.27 (a) Suppose that the sequence \( (c_n) \) is bounded. Show that the radius of convergence of \( \sum_{n=0}^\infty c_n (z - a)^n \) is at least 1. (b) Suppose that the sequence \( (c_n) \) does not converge to 0. Show that the radius of convergence of \( \sum_{n=0}^\infty c_n (z - a)^n \) is at most 1. ## 7.28 Find the power series centered at 1 and compute its radius of convergence for each of the following functions: (a) \( f(z) = \frac{1}{z} \) (b) \( f(z) = \log(z) \) ## 7.29 Use the Weierstrass M-test to show that each of the following series converges uniformly on the given domain: (a) \( \sum_{k=0}^\infty \frac{z^k}{k!} \) on \( [0, 1] \) (b) \( \sum_{k=0}^\infty z^k \) on \( \{ z \in \mathbb{C} : |z| < 2 \} \) (c) \( \sum_{k=0}^\infty \frac{z^k}{z^k + 1} \) on \( D(0, r) \) where \( 0 < r < 1 \) ## 7.30 Fix \( c \in \mathbb{C} \) and \( r > |z| \). Prove that \( \sum_{k=0}^\infty \left( \frac{z}{w} \right)^k \) converges uniformly in the variable \( z \) for \( |z| > r \). ## 7.31 Complete our proof of Corollary 7.33 by considering the case \( R = \infty \). ## 7.32 Prove that, if \( \lim_{n \to \infty} \left| c_n \right|^{\frac{1}{n}} \) exists then the radius of convergence of the series \( \sum_{n=0}^\infty c_n (z - a)^n \) equals \[ R = \begin{cases} \infty & \text{if } \lim_{n \to \infty} \left| c_n \right|^{\frac{1}{n}} = 0, \\ 0 & \text{otherwise}. \end{cases} \] ## 7.33 Find the radius of convergence for each of the following series: (a) \( \sum_{n=0}^\infty a^n z^n \) where \( a \in \mathbb{C} \) (b) \( \sum_{k=0}^\infty k! z^k \) where \( n \in \mathbb{Z} \) ``` #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 155 Context: # POWER SERIES AND HOLOMORPHIC FUNCTIONS ## Proof For starters, \( f(0) = c_0 \). Theorem 8.2 gives \( f'(0) = c_1 \). Applying the same theorem to \( f' \) gives \[ f''(z) = \sum_{k=2}^{\infty} k(k-1)c_kz^{k-2} \] and so \( f''(c_0) = 2c_2 \). A quick induction game establishes \( f^{(n)}(0) = n!c_n \) for all \( k \geq 0 \). Taylor's formula shows that the coefficients of any power series converging to \( f \) on some open disk \( D \) can be determined from the function \( f \) restricted to \( D \). It follows immediately that the coefficients of a power series are unique: ### Corollary 8.6 If \( \sum_{k=0}^{\infty} a_k(z - z_0)^k \) and \( \sum_{k=0}^{\infty} b_k(z - z_0)^k \) are two power series that both converge to the same function on an open disk centered at \( z_0 \), then \( a_k = b_k \) for all \( k \geq 0 \). ### Example 8.7 We'd like to compute a power series expansion for \( f(z) = \exp(z) \) centered at \( z_0 = 0 \): \[ f^{(k)}(0) = \frac{1}{k!} \exp(0) = 1. \] Corollary 8.5 suggests that this power series is \[ \sum_{k=0}^{\infty} \frac{z^k}{k!} \] which converges for all \( z \in \mathbb{C} \) (essentially by Example 7.35). We now turn to the second cornerstone result of this section, that a holomorphic function can be locally represented by a power series. ### Theorem 8.8 Suppose \( f \) is a function holomorphic in \( D\left(z_0, r\right) \). Then \( f \) can be represented as a power series centered at \( z_0 \) with a radius of convergence \( r \): \[ f(z) = \sum_{k=0}^{\infty} c_k(z - z_0)^k \] with \[ c_k = \frac{1}{2\pi i} \oint_{\gamma} \frac{f(z)}{(z - z_0)^{k+1}} \, dz \] where \( \gamma \) is any positively oriented, simple, closed, piecewise smooth path in \( D\left(z_0, r\right) \) for which \( z_0 \) is inside \( \gamma \). #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 159 Context: # Classification of Zeros and the Identity Principle Continuing in this way, we see that we can factor \( p(z) \) as \( p(z) = (z-a)^m g(z) \) where \( m \) is a positive integer and \( g(z) \) is a polynomial that does not have a zero at \( a \). The integer \( m \) is called the multiplicity of the zero of \( p(z) \). Almost exactly the same thing happens for holomorphic functions. ## Theorem 8.14 (Classification of Zeros) Suppose \( f : G \to \mathbb{C} \) is holomorphic and \( f \) has a zero at \( a \in G \). Then either: 1. \( f \) is identically zero on some disk \( D \) centered at \( a \) (that is, \( f(z) = 0 \) for all \( z \in D \)); or 2. There exists a positive integer \( m \) and a holomorphic function \( g : G \to \mathbb{C} \), such that \( f(z) = (z-a)^m g(z) \) for all \( z \in G \). In this case, the zero is isolated: there is a disk \( D[a,r] \) which contains no other zero of \( f \). The integer \( m \) in the second case is uniquely determined by \( f \) and is called the multiplicity of the zero at \( a \). ### Proof By Theorem 8.8, there exists \( R > 0 \) such that we can expand \[ f(z) = \sum_{k=0}^{\infty} c_k (z-a)^k \text{ for } z \in D[a,R]. \] and \( c_0 = f(a) = 0 \). There are now exactly two possibilities: either 1. \( c_0 = 0 \) for all \( k \geq 0 \); or 2. There is some positive integer \( m \) so that \( c_k = 0 \) for all \( k < m \) but \( c_m \neq 0 \). The first case gives \( f(z) = 0 \) for all \( z \in D[a,R] \). So now consider the second case. We note that for \( z \in D[a,R] \): \[ f(z) = (z-a)^m \left( c_m + c_{m+1} (z-a) + \ldots \right) = (z-a)^m \sum_{n=0}^{\infty} c_{m+n} (z-a)^n. \] #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 160 Context: # TAYLOR AND LAURENT SERIES Thus we can define a function \( g : G \rightarrow \mathbb{C} \) through \( g(z) = \begin{cases} \sum_{n=0}^{\infty} a_n (z - a)^n & \text{if } z \in D(a, R), \\ f(z) & \text{if } z \in G \setminus \{ a \}. \end{cases} \) (According to our calculations above, the two definitions give the same value when \( z \in D(a, R) \). The function \( g \) is holomorphic in \( D(a, R) \) by the first definition, and \( g \) is holomorphic in \( G \setminus \{ a \} \) by the second definition. Note that \( g(a) = c \neq 0 \) and, by construction, \( f(z) = (z - a)^n s(z) \quad \text{for all } z \in G. \) Since \( s(a) \neq 0 \) by continuity, \( r > 0 \) so that \( s(z) \neq 0 \) for all \( z \in D(a, r) \), so \( D(a, r) \) contains no other zero of \( f \). The integer \( r \) is unique, since it is defined in terms of the power series expansion of \( f \) at \( a \), which is unique by Corollary 8.6. Theorem 8.14 gives rise to the following result, which is sometimes called the identity principle or the uniqueness theorem. ## Theorem 8.15 Suppose \( G \) is a region, \( f : G \rightarrow \mathbb{C} \) is holomorphic, and \( f(a) = 0 \) where \( (a_k) \) is a sequence of distinct numbers that converges in \( G \). Then \( f \) is the zero function on \( G \). Applying this theorem to the difference of two functions immediately gives the following variant. ## Corollary 8.16 Suppose \( f \) and \( g \) are holomorphic in a region \( G \) and \( f(a) = g(a) = 0 \) at a sequence that converges to \( a \in G \) with \( a_k \neq a \) for all \( k \). Then \( f(z) = g(z) \) for all \( z \in G \). ### Proof of Theorem 8.15 Consider the following two subsets of \( G \): - \( X := \{ z \in G : \text{ there exists } z \text{ such that } F(z) = 0 \text{ for all } z \in D(a, r) \} \) - \( Y := \{ z \in G : \text{ there exists } z \text{ such that } F(z) \neq 0 \text{ for all } z \in D(a, r) \setminus \{ a \} \} \) If \( f(a) \neq 0 \) then, by continuity of \( f \), there exists a disk centered at \( a \) in which \( f \) is nonzero, and so \( a \in Y \). #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 163 Context: Here \( a_k \in \mathbb{C} \) are terms indexed by the integers. The double series above converges if and only if the two series on the right-hand side do. Absolute and uniform convergence are defined analogously. Equipped with this, we can now introduce the following central concept. ## Definition A Laurent series centered at \( z_0 \) is a double series of the form \[ \sum_{n \in \mathbb{Z}} c_n (z - z_0)^{n}. \] ### Example 8.21 The series that started this section is the Laurent series of \(\exp\left(\frac{1}{z}\right)\) centered at 0. ### Example 8.22 Any power series is a Laurent series (with \( a_k = 0 \) for \( k < 0 \)). We should pause for a minute and ask for which \( z \) a general Laurent series can possibly converge. By definition, \[ \sum_{n \in \mathbb{Z}} c_n (z - z_0)^n = \sum_{n \geq 0} c_n (z - z_0)^n + \sum_{n < 0} c_n (z - z_0)^n. \] The first series on the right-hand side is a power series with some radius of convergence \( R_z \), that is, with Theorem 7.31, it converges in \( \{ z \in \mathbb{C} : |z - z_0| < R_z \} \), and the convergence is uniform in \( \{ z \in \mathbb{C} : |z - z_0| \leq r \} \) for any fixed \( r < R_z \). For the second series, we invite you (in Exercise 8.30) to revise our proof of Theorem 7.31 to show that this series converges for \[ \frac{1}{|z - z_0|} < \frac{1}{R_1} \] for some \( R_1 \), and that the convergence is uniform in \( \{ z \in \mathbb{C} : |z - z_0| \leq r \} \) for any \( r > R_1 \). Thus the Laurent series converges in the annulus \[ A = \{ z \in \mathbb{C} : R_1 < |z - z_0| < R_2 \} \] (assuming this set is not empty, i.e., \( R_1 < R_2 \)), and the convergence is uniform on any set of the form \[ \{ z \in \mathbb{C} : R_1 < |z - z_0| < r \} \] for \( R_1 < r < R_2 \). *Jean-Pierre Antoine Laurent (1813–1894).* #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 165 Context: # LAURENT SERIES ## Theorem 8.24 Suppose \( f \) is a function that is holomorphic in \( A = \{ z \in \mathbb{C} : R_1 < |z - z_0| < R_2 \} \). Then \( f \) can be represented in \( A \) as a Laurent series centered at \( z_0 \): \[ f(z) = \sum_{n=-\infty}^{\infty} c_n (z - z_0)^n \] with \[ c_n = \frac{1}{2 \pi i} \int_{C_{r_1}} \frac{f(w)}{(w - z_0)^{n+1}} \, dw, \] where \( R_1 < r < R_2 \). By Cauchy's Theorem 4.18, we can replace the circle \( C_{r_1} \) in the formula for the Laurent coefficients by any path \( \gamma \subset A [C_{r_1}, r_2] \). ![Figure 8.1: The path γ in our proof of Theorem 8.24.](path_to_figure) ## Proof Let \( g(z) = f(z + z_0) \), so \( g \) is a function holomorphic in \( \{ z \in \mathbb{C} : R_1 < |z| < R_2 \} \). Fix \( R_1 < r_1 < r_2 < R_2 \), and let \( \gamma \) be the path in Figure 8.1, where \( \gamma_1 = C[r_1, r_2] \) and \( \gamma_2 = C[0, r_2] \). By Cauchy's Integral Formula (Theorem 4.27), \[ g(z) = \frac{1}{2 \pi i} \int_{\gamma} \frac{g(w)}{w - z} \, dw. \tag{8.1} \] Complete any missing details from the original image as necessary while ensuring the accuracy of the markup. #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 170 Context: 8.11. Suppose \( G \) is a region and \( f : G \to \mathbb{C} \) is holomorphic. Prove that the sets - \( X = \{ a \in G : \text{there exists } r \text{ such that } f(a) = 0 \text{ for all } z \in D(a, r) \} \) - \( Y = \{ a \in G : \text{there exists } r \text{ such that } f(z) \neq 0 \text{ for all } z \in D(a, r) \} \) in our proof of Theorem 8.15 are open. 8.12. Prove the Minimum-Modulus Theorem (Corollary 8.19): Suppose \( f \) is holomorphic and nonconstant in a region \( G \). Then \( |f| \) does not attain a weak relative minimum at a point \( a \in G \) unless \( f(a) = 0 \). 8.13. Prove Corollary 8.20: Assume that \( u \) is harmonic in a region \( G \) and has a weak local maximum at \( a \in G \). (a) If \( G \) is simply connected then apply Theorem 8.17 to \( u = \operatorname{Re}(f) \) where \( f \) is a harmonic conjugate of \( u \). Conclude that \( u \) is constant on \( G \). (b) If \( G \) is not simply connected, then the above argument applies to \( u \) on any disk \( D(a, r) \cap G \). Conclude that the partials \( u_x \) and \( u_y \) are zero on \( G \), and adapt the argument of Theorem 2.17 to show that \( u \) is constant. 8.14. Let \( C : \mathbb{C} \to \mathbb{C} \) be given by \( C(z) = z^2 - 2 \). Find the maximum and minimum of \( |f(z)| \) on the closed unit disk. 8.15. Give another proof of the Fundamental Theorem of Algebra (Theorem 5.11), using the Minimum-Modulus Theorem (Corollary 8.19). (Hint: Use Proposition 5.10 to show that a polynomial does not achieve its minimum modulus on a large circle; then use the Minimum-Modulus Theorem to deduce that the polynomial has a zero.) 8.16. Give another proof of (a variant of) the Maximum-Modulus Theorem 8.17 via Corollary 8.11, as follows: Suppose \( f \) is holomorphic in a region containing \( \overline{D(a, r)} \) and \( |f(z)| \leq M \) for each point \( z \in D(a, r) \). Show (e.g., by Corollary 8.11) that there is a constant \( c \in \mathbb{C} \ such that \[ |f(z)| \leq c < M. \] Conclude that \( |f(a)| \leq M \). #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 174 Context: # TAYLOR AND LAURENT SERIES ### 8.36. (a) Find a Laurent series for \[ \frac{1}{(z^2 - 4)(z - 2)} \] centered at \( z = 2 \) and specify the region in which it converges. (b) Compute \[ \int_{C_{[1]}} \frac{dz}{(z^2 - 4)(z - 2)}. \] ### 8.37. (a) Find the power series of \( \exp(z) \) centered at \( z = -1 \). (b) Compute \[ \int_{C_{[-2]}} \frac{\exp(z)}{(z + 1)^3} dz. \] ### 8.38. Compute \[ \int \frac{\exp(z)}{\sin(z)} dz \] where \( \gamma \) is a closed curve not passing through integer multiples of \( \pi \). #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 175 Context: # Chapter 9 ## Isolated Singularities and the Residue Theorem \(\frac{1}{z}\) has a nasty singularity at \(z = 0\), but it did not bother Newton—the reason is far enough. Edward Witten We return one last time to the starting point of Chapters 7 and 8: the quest for \[ \int_{C} \frac{\exp(z)}{\sin(z)} \, dz. \] We computed this integral in Example 8.28 crawling on hands and knees (but we finally computed it), by considering various Taylor and Laurent expansions of \(\exp(z)\) and \(\frac{1}{\sin(z)}\). In this chapter, we develop a calculus for similar integral computations. ## 9.1 Classification of Singularities What are the differences among the functions \(\frac{\exp(z)}{z}\), \(\frac{1}{z}\), and \(\exp(z)\) at \(z = 0\)? None of them are defined at \(0\), but each singularity is of a different nature. We will frequently consider functions in this chapter that are holomorphic in a disk except at its center (usually because that’s where a singularity lies), and it will be handy to define the punctured disk with center \(z_0\) and radius \(R\), \[ D(z_0, R) = \{ z \in \mathbb{C} : 0 < |z - z_0| < R \}. \] We extend this definition naturally with \(D[z_0, R] = \{ z \in \mathbb{C} : |z - z_0| \leq R \}\). For complex functions there are three types of singularities, which are classified as follows: #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 177 Context: # Classification of Singularities show that \( f \) has neither a removable singularity nor a pole. To get a feel for the different types of singularities, we start with the following criteria. **Proposition 9.5.** Suppose \( z_0 \) is an isolated singularity of \( f \). Then 1. \( z_0 \) is removable if and only if \( \lim_{z \to z_0} (z - z_0)f(z) = 0; \) 2. \( z_0 \) is a pole if and only if it is not removable and \( \lim_{z \to z_0} (z - z_0)^n f(z) = 0 \) for some positive integer \( n \). **Proof.** (a) Suppose that \( z_0 \) is a removable singularity of \( f \); so there exists a holomorphic function \( h \) on \( D(z_0, R) \) such that \( f(z) = h(z) \) for all \( z \in D(z_0, R) \). But then \( h \) is continuous at \( z_0 \) and \[ \lim_{z \to z_0} f(z) = \lim_{z \to z_0} h(z) = h(z_0). \] Conversely, suppose that \( \lim_{z \to z_0} (z - z_0)f(z) = 0 \) and \( f \) is holomorphic in \( D(z_0, R) \). We can define the function \( g: D(z_0, R) \to \mathbb{C} \) by \[ g(z) = \begin{cases} (z - z_0)^n f(z) & \text{if } z \neq z_0 \\ 0 & \text{if } z = z_0 \end{cases} \] Then \( g \) is holomorphic in \( D(z_0, R) \) and \[ g'(z_0) = \lim_{z \to z_0} \frac{g(z) - g(z_0)}{z - z_0} = \lim_{z \to z_0} \frac{(z - z_0)^{n-1}f(z)}{z - z_0} = \lim_{z \to z_0} (z - z_0)^{n-1}f(z) = 0, \] so \( g \) is holomorphic in \( D(z_0, R) \). We can then expand \( g \) into a power series: \[ g(z) = \sum_{k \geq 0} a_k(z - z_0)^k \] whose first two terms are zero: \( a_0 = g(z_0) = 0 \) and \( a_1 = g'(z_0) = 0 \). But then we can write \[ f(z) = \frac{g(z)}{(z - z_0)^n} = \sum_{k \geq 0} a_k(z - z_0)^{k - n} \] and so \[ f(z) = \sum_{k \geq 0} a_{k + n}(z - z_0)^k \] for all \( z \in D(z_0, R) \). #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 179 Context: # CLASSIFICATION OF SINGULARITIES on \( D[\alpha, R] \) and non-zero at \( \alpha_n > 0 \), and \( f(z) = \frac{g(z)}{z - \alpha} \) for all \( z \in D[\alpha, R] \). But then \( \alpha \) is a pole of order \( m \) since \[ \lim_{z \to \alpha} |f(z)| = \lim_{z \to \alpha} \left| \frac{g(z)}{(z - \alpha)^m} \right| = \lim_{z \to \alpha} \frac{|g(z)|}{|z - \alpha|^m} = \infty. \] We underline one feature of the last part of our proof: ## Corollary 9.6 Suppose \( f \) is holomorphic in \( D[\alpha, R] \). Then \( f \) has a pole at \( \alpha \) if and only if there exists a positive integer \( m \) and a holomorphic function \( g : D[\alpha, R] \to \mathbb{C} \) such that \( g(\alpha) \neq 0 \) and \[ f(z) = \frac{g(z)}{(z - \alpha)^m} \] for all \( z \in D[\alpha, R] \). If \( \alpha \) is a pole then \( m \) is unique. ### Proof The only part not covered in the proof of Theorem 9.5 is uniqueness of \( m \). Suppose \( f(z) = \frac{(z - \alpha)^{m}}{g(z)} \) and \( f(z) = \frac{(z - \alpha)^{n}}{h(z)} \) both work, with \( m > n \). Then \( g(z) = (z - \alpha)^{m - n} h(z) \), and plugging in \( z = \alpha \) yields \( g(\alpha) = 0 \), violating \( g(\alpha) \neq 0 \). ### Definition The integer \( m \) in Corollary 9.6 is the order of the pole \( \alpha \). This definition, naturally coming out of Corollary 9.6, parallels that of the multiplicity of a zero, which naturally came out of Theorem 8.14. The two results also show that if \( f \) has a zero at \( \alpha \) of multiplicity \( m \) and only if \( j \) has a pole of order \( m \), we will make use of the notions of zeros and poles quite extensively in this chapter. You might have noticed that the Proposition 9.5 did not include any results on essential singularities. Not only does the next theorem make up for this but it also nicely illustrates the strangeness of essential singularities. To appreciate the following result, we suggest meditating about its statement over a good cup of coffee. #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 180 Context: ```markdown # Theorem 9.7 (Casorati–Weierstrass) If \( f \) is an essential singularity of \( r \) and \( r \) is any positive real number, then every \( u \in C \) is arbitrarily close to a point in \( f(B(z_0, r)) \). That is, for any \( x \in C \) and \( \epsilon > 0 \) there exists \( z \in B(z_0, r) \) such that \( |u - f(z)| < \epsilon \). In the language of topology, Theorem 9.7 says that the image of any punctured disk centered at an essential singularity is dense in \( C \). There is a stronger theorem, beyond the scope of this book, which implies the Casorati–Weierstrass Theorem 9.7. It is due to Charles Émile Picard (1856–1941) and says that the image of any punctured disk centered at an essential singularity misses at most one point of \( C \). (It is worth coming up with examples of functions that do not miss any point in \( C \) and functions that miss exactly one point. Try it!) ## Proof Suppose (by way of contradiction) that there exists \( w \in C \) and \( \epsilon > 0 \) such that for all \( z \in B(z_0, r) \) \[ |u - f(z)| \geq \epsilon. \] Then the function \( g(z) = \frac{f(z) - w}{z - z_0} \) stays bounded as \( z \to z_0 \) and so \[ \lim_{z \to z_0} (z - z_0) g(z) = \lim_{z \to z_0} \frac{f(z) - w}{z - z_0} = 0. \] (Proposition 9.5(a) tells us that \( f \) has a removable singularity at \( z_0 \).) Hence \[ \lim_{z \to z_0} \left| \frac{f(z) - w}{z - z_0} \right| = \infty \] and so the function \( f(z) - w \) has a pole at \( z_0 \). By Proposition 9.5(b), there is a positive integer \( n \) so that \[ \lim_{z \to z_0} (z - z_0)^{n+1} f(z) = \lim_{z \to z_0} (z - z_0)^n g(f(z)) = 0. \] Invoking Proposition 9.5 again, we conclude that the function \( f(z) \) has a pole or removable singularity at \( z_0 \), which implies the same holds for \( f(z) \), a contradiction. ``` #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 185 Context: ```markdown # RESIDUES **Proof:** The functions \( f \) and \( g \) have power series centered at \( z_0 \); the one for \( g \) has by assumption no constant term: \[ f(z) = \sum_{n=0}^{\infty} a_n (z - z_0)^n \] \[ g(z) = \sum_{n=1}^{\infty} b_n (z - z_0)^n \] Let \( h(z) = \sum_{n=1}^{\infty} b_n (z - z_0)^{-n} \) and note that \( h(z_0) = b_1 \neq 0 \). Hence, \[ \frac{f(z)}{g(z)} = \frac{f(z)}{(z - z_0) h(z)} \] and the function \( \frac{f(z)}{g(z)} \) is holomorphic at \( z_0 \). By Prop 9.11 and Taylor's formula (Corollary 8.5), \[ \text{Res}_{z \to z_0} \left( \frac{f(z)}{g(z)} \right) = \lim_{z \to z_0} \left( (z - z_0) \frac{f(z)}{g(z)} \right) = \frac{f(z_0)}{h(z_0)} \frac{f(z)}{h(z)} \cdot \frac{d}{dz} \frac{f(z)}{g(z)} \Big|_{z=z_0} \] Example 9.15. Revisiting once more Example 8.28, we note that \( f(z) = e^{z} \) and \( g(z) = \sin(z) \) fill the bill. Thus, \[ \text{Res}_{z \to 0} \left( \frac{e^{z}}{\sin(z)} \right) = \frac{e^0}{\cos(0)} = 1 \] and confirming once more our computations in Examples 8.28 and 9.12. Example 9.16. We compute the residue of \( \frac{e^{z} + z^2}{e^{2z} - z^2} \) at \( z_0 = 2\pi i \), by applying Proposition 9.14 with \( f(z) = e^{z} + z^2 \) and \( g(z) = e^{2z} - z^2 \). Thus, \[ \text{Res}_{z \to 2\pi i} \left( \frac{e^{z} + z^2}{e^{2z} - z^2} \right) = \frac{-4\pi^2 + 2}{\cosh(2\pi)} \] An extension of Proposition 9.14 of sorts is given in Exercise 9.12. ``` #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 188 Context: ```markdown 182 ISOLATED SINGULARITIES AND THE RESIDUE THEOREM zeros and poles in G. By (9.1) and (9.2), $$ f\left(z\right) = \frac{n_1}{z - z_1} + \frac{n_2}{z - z_2} - \frac{m_1}{z - p_1} + \frac{s'(z)}{g(z)} $$ where \(s\) is a function that is holomorphic in \(G\) (in particular, without poles) and never zero. Thanks to Cauchy’s Theorem 4.18 and Exercise 4.4, the integral is equal to $$ \int_{\gamma} \frac{f'(z)}{g(z)} dz = 2\pi i \left(n_1 + \ldots + n_j - m_1 - \ldots - m_k\right) \int_{\gamma} \frac{s'(z)}{g(z)}. $$ Finally, \( \frac{s'}{g} \) is holomorphic in \(G\) (because \(g\) is never zero in \(G\)), so Corollary 4.20 gives $$ \int_{\gamma} \frac{s'}{g} = 0. $$ As mentioned above, this beautiful theorem helps to locate poles and zeros of a function \(f\). The idea is simple: we can first numerically integrate \(f\) over a big circle \(\gamma\) that includes all possible paths over which we potentially will be integrating \(f\). Then the numerical value of \(\int_{\gamma} f\) will be close to an integer that, according to the Argument Principle, equals \(Z(f) - P(f)\). Then we can integrate \(\frac{f}{g}\) over a smaller closed path \(\gamma'\) that encompasses half of the interior of \(\gamma\) and \(Z(f) - P(f)\). Continuing this process for smaller and smaller regions will (after certain verification) produce small regions where \(f\) has exactly one zero or exactly one pole. Integrating \(f\) over the boundaries of those small regions that contain poles and dividing by \(Z(f)\) gives all residues of \(f\). Another nice relevant application of the Argument Principle is a famous theorem due to Rouche (1832–1910). **Theorem 9.18 (Rouche's Theorem):** Suppose \(f\) and \(g\) are holomorphic in a region \(G\) and \(r\) is a positively oriented, simple, closed, piecewise smooth path, such that \( |g(z)| < |f(z)| \) for all \( z \in \gamma \). Then $$ Z(f + g, \gamma) = Z(f, \gamma). $$ ``` #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 189 Context: # ARGUMENT PRINCIPLE AND ROUCHÉ'S THEOREM This theorem is of surprising practicality. It allows us to locate the zeros of a function fairly precisely. Here is an illustration. ## Example 2.19 All the roots of the polynomial \( p(z) = z^5 + z^4 + z^3 + z^2 + z + 1 \) have modulus less than two. To see this, let \( f(z) = z^5 + z^4 + z^3 + z^2 + z + 1 \). For \( z \in [0, 2] \): \[ |f(z)| \leq |z^5| + |z^4| + |z^3| + |z^2| + |z| + 1 = 16 + 8 + 4 + 2 + 1 = 31 < 32 = |z^5| \text{ for } z \in [\sqrt{2}, 2]. \] So \( g \) and \( f \) satisfy the condition of Theorem 9.18. But \( f \) has just one root, of multiplicity 5 at the origin, whence \[ Z(f) \cap (C[0, 2]) = Z(f + g, C[0, 2]) = Z(f, C[0, 2]) = 5. \] ### Proof of Theorem 9.18 By (9.1) and the Argument Principle (Theorem 9.17): \[ Z(f + g, r) = \frac{1}{2\pi i} \int_{\gamma} \frac{f + g}{f + g} \, \frac{(f + g)'}{f + g} \, dz = \frac{1}{2\pi i} \int_{\gamma} \frac{(f + g)'}{f + g} \, dz, \] where \( \gamma \) is a contour around \( r \). We are assuming that \( |g| < |f| \) on \( \gamma \), which means that the function \( \frac{1}{f + g} \) evaluated on \( \gamma \) stays away from \( R_+ \). But then \( \log\left(\frac{1}{f + g}\right) \) is a well-defined holomorphic function on \( \gamma \). Its derivative is \[ \frac{(f + g)'}{f + g}. \] --- > *The Fundamental Theorem of Algebra (Theorem 5.1) asserts that \( p \) has \( n \) roots in \( C \). What's special about the theorem of Example 2.19 is that they all have modulus < 2. Note also that there is no general formula for composing roots of a polynomial of degree \( n \). Although for this \( n \) it’s not hard to find one root—and therefore all of them.* #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 193 Context: # Argument Principle and Rouche's Theorem ## 9.21. Find the number of zeros of 1. \( 3z^2 - z \) in \( D[0, 1] \) 2. \( \{ z \} \in D[0, 1] \) 3. \( e^{-z} - 5 + 1 \) in \( \{ z \in \mathbb{C} : |z| \leq 2 \} \) ## 9.22. Give another proof of the Fundamental Theorem of Algebra (Theorem 5.11), using Rouche's Theorem 9.18. *(Hint: If \( p(z) = z^n + a_{n-1}z^{n-1} + \ldots + a_0 \), let \( f(z) = a_n z^n \) and \( g(z) = a_{n-1}z^{n-1} + a_{n-2}z^{n-2} + \ldots + a_0 \), and choose \( r \) a circle that is large enough to make the condition of Rouche's theorem work. You might want to first apply Proposition 5.10 to \( g(z) \).)* ## 9.23. Suppose \( S \subset C \) is closed and bounded and all points of \( S \) are isolated points of \( S \). Show that \( S \) is finite, as follows: (a) For each \( x \in S \), we can choose \( \varepsilon > 0 \) so that \( D[x, \varepsilon] \) contains no points of \( S \) except \( x \). Show that \( f \) is continuous. *(Hint: This is really easy if you use the first definition of continuity in Section 2.1.)* (b) Assume \( S \) is non-empty. By the Extreme Value Theorem A.1, \( S \) has a minimum value, \( r_0 > 0 \). Let \( r = r_0/2 \). Since \( S \) is bounded, it lies in a disk \( D[0, M] \) for some \( M > 0 \). Show that the small disks \( D[x, r] \) for \( x \in S \) are disjoint and lie in \( D[0, M+r] \). (c) Find a bound on the number of such small disks. *(Hint: Compare the areas of \( D[x, r] \) and \( D[0, M + r] \).)* #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 195 Context: # Binomial Coefficients (4) Evaluate \(\sum_{n=1}^{\infty} \frac{1}{n^2}\). (5) Repeat the exercise with the function \(f(z) = \frac{z^{-1}}{e^{z} - 1}\) to arrive at an evaluation of \[ \sum_{n=1}^{\infty} \frac{(-1)^{n}}{n^2}. \] (Hint: To bound this function, you may use the fact that \(\frac{1}{e^z - 1} = 1 + \cos(z_n)\).) (6) Evaluate \(\sum_{n=1}^{\infty} n\) and \(\sum_{n=1}^{\infty} n^{-1}\). We remark that, in the language of Example 7.21, you have computed the evaluations \(\zeta(2)\) and \(\zeta(4)\) of the Riemann zeta function. The function \(\zeta'(s) := \sum_{n=1}^{\infty} \frac{1}{n^s}\) is called the alternating zeta function. ## 10.2 Binomial Coefficients The binomial coefficient \(\binom{n}{k}\) is a natural candidate for being explored analytically, as the binomial theorem states: \[ (x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{k} y^{n-k} \] (for \(x, y \in \mathbb{C}\) and \(n \in \mathbb{Z}_{\geq 0}\)) tells us that \(\binom{n}{k}\) is the coefficient of \(x^{k}\) in \((x + y)^{n}\). You will derive two simple identities in the course of the exercises below. 1. Convince yourself that \[ \binom{n}{k} = \frac{1}{2\pi i} \oint_{C} \frac{(x + 1)^{n}}{z^{k + 1}} dz, \] where \(C\) is any simple closed piecewise smooth path such that \(0\) is inside \(r\). 2. Derive a recurrence relation for binomial coefficients from the fact that \[ \frac{1}{z^{n}} = \frac{1}{2^{n}} \cdot \left( \text{Multiply both sides by } \frac{1}{z^{n}}\right). \] 3. Now suppose \(z\) is such that \(|z| < 1/4\). Find a simple path \(P\) surrounding the origin such that \[ \sum_{n=0}^{\infty} \left( \frac{(z + 1)^{k}}{k} \right) \] converges uniformly on \(y\) as a function of \(\epsilon\). Evaluate this sum. #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 196 Context: ``` Keeping \( x \) and \( y \) from (3), convince yourself that \[ \sum_{k=0}^{\infty} \left( \frac{2k}{k} \right) k^k = \frac{1}{2\pi i} \int \left( \frac{x^{n+1}}{z^{n+1}} \right) dz, \] use (3) to interchange summation and integral, and use the Residue Theorem 9.10 to evaluate the integral, giving an identity for \( \sum_{k=0}^{\infty} \left( \frac{1}{2} \right)^k \). ### 10.3 Fibonacci Numbers The Fibonacci\(^1\) numbers are a sequence of integers defined recursively through \[ f_0 = 0, \] \[ f_1 = 1, \] \[ f_n = f_{n-1} + f_{n-2} \quad \text{for } n \geq 2. \] Let \( F(z) = \sum_{n=0}^{\infty} f_n z^n \). 1. Show that \( F \) has a positive radius of convergence. 2. Show that the recurrence relation among the \( f_n \) implies that \( F(z) = \frac{z}{1 - z - z^2} \). (Hint: Write down the power series for \( F(z) \) and \( z^2 F(z) \) and rearrange both so that you can easily add.) 3. Verify that \[ \text{Res}_{z=0} \left( \frac{1}{z(1 - z - z^2)} \right) = f_0. \] 4. Use the Residue Theorem 9.10 to derive an identity for \( f_n \). (Hint: Integrate \[ \frac{1}{z(1 - z - z^2)} \] around \( C[0, R] \) and show that this integral vanishes as \( R \to \infty \). \(^1\) Named after Leonardo Pisano Fibonacci (1170–1250). ``` #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 197 Context: # The Coin-Exchange Problem (5) Generalize to other sequences defined by recurrence relations, e.g., the Fibonacci numbers - \( t_0 = 0 \) - \( t_1 = 0 \) - \( t_2 = 1 \) - \( t_n = t_{n-1} + t_{n-2} + t_{n-3} \) for \( n \geq 3 \). ## 10.4 The Coin-Exchange Problem In this exercise, we will solve and extend a classical problem of Ferdinand Georg Frobenius (1849–1917). Suppose \( a \) and \( b \) are relatively prime positive integers, and suppose \( r \) is a positive integer. Consider the function \[ f(c) = \frac{1}{(1 - z)(1 - z^2)z^r} \] 1. Compute the residues at all nonzero poles of \( f \). 2. Verify that \( Res_{z = 0}(f) = N(r) \), where \[ N(r) = \{(m, n) \in \mathbb{Z}^2 : m \geq 0, n \geq 0, m + nb = r\} \]. 3. Use the Residue Theorem, Theorem 9.10, to derive an identity for \( N(r) \). (Hint: Integrate \( f \) around \( [0, R] \) and show that this integral vanishes as \( R \to \infty \).) 4. Use the following three steps to simplify this identity to \[ N(t) = \frac{1}{ab} \left( \frac{b-1}{a} - \left( \frac{a-1}{b} \right) \right) + 1. \] *This means that the integers do not have any common factor.* #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 198 Context: # Discrete Applications of the Residue Theorem Here, \(\{x_n\}\) denotes the fractional part of \(x\): \(a_n \equiv 1 \,(\text{mod } b)\), and \(b \cdot 6 \equiv 1 \,(\text{mod } d)\). ## (a) Verify that for \(b = 1\), \[ N(n) = \left| \{(m,n) \in \mathbb{Z}^2 : m \geq 0, n + n \leq s \} \right| = \left| \{ m \in \mathbb{Z} : m \geq 0, m \cdot s \leq 1 \} \right| = \left[ \frac{0}{d} \right] n^2 = \left\lfloor \frac{1}{d} \right\rfloor. \] ## (b) Use this together with the identity found in (3) to obtain: \[ -\frac{1}{2} \sum_{k=1}^{\infty} \frac{1}{(1 - 2e^{2\pi i k})\cdot 2k} = \left( -\frac{1}{2} + \frac{1}{2} \right) + \frac{1}{z} \, \] ## (c) Verify that \[ \sum_{k=1}^{\infty} \frac{1}{(1 - 2e^{2\pi i k})} = \sum_{k=1}^{\infty} \frac{1}{(1 - 2e^{2\pi i k})} \] ## (5) Prove that \(N(a - b) = 0\), and \(N(n) > 0\) for all \(x > a\) and \(x < b\). ### Historical Remark Given relatively prime positive integers \(a_1, a_2, \ldots\), let’s call an integer \(t\) representable if there exist nonnegative integers \(m_1, m_2, \ldots, m_k\) such that \[ t = m_1 a_1 + m_2 a_2 + \cdots + m_k a_k. \] (There are many scenarios in which you may ask whether or not \(t\) is representable, given fixed \(a_1, a_2, \ldots, a_k\); for example, if the \(a_i\) are co-prime, this question asks whether you can give exact change for \(t\). In the late 19th century, Frobenius raised the problem of finding the largest integer that is not representable. We will call this largest integer the Frobenius number \(g(a_1, a_2, \ldots, a_k)\). It is well known (probably at least since the 1880s, when James Joseph Sylvester (1814–1897) studied the Frobenius problem) that \(g(a_1, a_2) = a_1 a_2 - a_1 - a_2\). You verified this result in (5). For \(x > 2\), there is no closed formula for \(g(a_1, a_2, \ldots, a_k)\). The formula in (4) states that The fractional part of a real number \(x\) is loosely speaking the part after the decimal point. More thoroughly, the principal integer function of \(x\), denoted by \(\lfloor x \rfloor\), is the largest integer not exceeding \(x\). The fractional part is defined as \( \{x\} = x - \lfloor x \rfloor \). **"This means that \(c\) is an integer such that \(c = a + 1 + b + k\) for some \(k \in \mathbb{Z}\)."** #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 199 Context: # Dedekind Sums This exercise outlines one more nontraditional application of the Residue Theorem 9.10. Given two positive, relatively prime integers \( a \) and \( b \), let \[ f(z) = \cot(\pi z) \cot(b \pi z). \] 1. Choose an \( R > 0 \) such that the rectangular path \( \gamma_R \) from \( -1 - iR \) to \( -1 + iR \) to \( 1 + iR \) to \( 1 - iR \) back to \( -1 - iR \) does not pass through any of the poles of \( f \). (a) Compute the residues for the poles of \( f \) inside \( \gamma \). Hint: Use the periodicity of the cotangent and the fact that \[ \cot z = \frac{1}{z} - \frac{1}{3} z + \text{higher-order terms}. \] (b) Prove that \( \lim_{R \to \infty} \int_{\gamma_R} f \, dz = -2i \) and deduce that for any \( R > 0 \) \[ \int_{-1}^{1} f \, dz = -2i. \] 2. Define \[ t(a, b) = \frac{1}{48} \sum_{k=1}^{b-1} \cot \left( \frac{\pi k a}{b} \right) \left( \frac{k}{b} \right). \] Use the Residue Theorem 9.10 to show that \[ t(a, b) + t(b, a) = -\frac{1}{4} + \frac{1}{12} \left( \frac{a}{b} + \frac{b}{a} \right). \] 3. Generalize (10.1) and (10.2). #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 200 Context: # Discrete Applications of the Residue Theorem ## Historical Remark The sum in (10.1) is called a Dedekind* sum. It first appeared in the study of the Dedekind \( \eta \)-function \[ \eta(z) = \exp\left(\frac{\pi i}{12} z\right) \prod_{n=1}^{\infty} \left(1 - \exp(2 \pi i n z)\right) \] in the 1870s and has since intrigued mathematicians from such different areas as topology, number theory, and discrete geometry. The reciprocity law (10.2) is the most important and famous identity of the Dedekind sum. The proof that is outlined here is due to Hans Rademacher (1892–1969). --- *Named after Julius Wilhelm Richard Dedekind (1831–1916). #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 201 Context: # Appendix: Theorems From Calculus First, it is necessary to study the facts, to multiply the number of observations, and then later to search for formulas that connect them so as to discern the particular law governing a certain class of phenomena. In general, it is not until after these particular laws have been established that one can expect to discover and articulate the more general law that connects them by bringing a multitude of apparently very diverse phenomena together under a single governing principle. _Augustin Louis Cauchy (1789–1857)_ Here we collect a few theorems from real calculus that we make use of in the course of the text. ## Theorem A.1 (Extreme-Value Theorem) Suppose \( K \subset \mathbb{R}^n \) is closed and bounded and \( f: K \to \mathbb{R} \) is continuous. Then \( f \) has a minimum and maximum value, i.e., \[ \min_{x \in K} f(x) \quad \text{and} \quad \max_{x \in K} f(x) \] exist in \( \mathbb{R} \). ## Theorem A.2 (Mean-Value Theorem) Suppose \( I \subset \mathbb{R} \) is an interval. \( f: I \to \mathbb{R} \) is differentiable, and \( x, x + h \in I \). Then there exists \( 0 < c < h \) such that \[ f(x + h) - f(x) = f'(c)(x + h - x). \] Many of the most important results of analysis concern combinations of limit operations. The most important of all calculus theorems combines differentiation and integration (in two ways): #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 206 Context: 200 # Solutions to Selected Exercises ## 4.1 (a) \(6\) (b) \(\pi\) (c) \(4\) (d) \(\sqrt{7} + \frac{1}{2} \sinh^{-1}(4)\) ## 4.5 (a) \(8i\) (b) \(0\) (c) \(0\) ## 4.6 (a) \(1-i\) (b) \(i\pi, -\pi, 0, 2\pi\) (c) \(i^2, -2, 0, 2r^2\) ## 4.7 (a) \(\frac{1}{3} (e^{3z} - e^{-3z})\) (b) \(0\) (c) \(\{g(\exp(3 + 3i)) - 1\}\) ## 4.12 \(\sqrt{3} i\) ## 4.18 (a) \(-4 + i\left(4 + \frac{1}{3}\right)\) (b) \((\ln(17) - 1) + i \left(\frac{\pi}{4} + 1\right)\) (c) \(2\sqrt{2} - 1 + 2i\sqrt{2}\) (d) \(\frac{1}{4} \sin(8) - 2 + i \left(2 - \frac{1}{4} \sinh(8)\right)\) ## 4.27 For \(|r| < |z| \Rightarrow 2\pi for \ |r| < |z|\) ## 4.30 \(\frac{24}{30}\) ## 4.34 \(0\) ## 4.35 For \(r = 1\): \( \frac{-1}{3} \); for \(r = 3\): \(0\); for \(r = 5\) ## 4.37 (a) \(2\pi\) (b) \(0\) (c) \(-\frac{3}{2}\) (d) \(\frac{2}{2r^2 - 1}\) ## 5.1 (a) \(2\pi\) (b) \(-6i\) (c) \(\pi\) (d) \(0\) ## 5.3 (a) \(0\) (b) \(2\pi i\) (c) \(0\) (d) \(\pi\) (e) \(f(0)\) ## 5.4 \(2\pi \exp{(iw)}\) ## 5.18 \(\frac{\sqrt{3}}{2}\) ## 5.20 \(\frac{\sqrt{z}}{\sin{\left(\frac{1}{\sqrt{2}} z\right)}}\) ## 7.1 (a) Divergent (b) Convergent (limit \(0\)) (c) Divergent (d) Convergent (limit \(2 -\frac{1}{2}\)) (e) Convergent (limit \(0\)) ## 7.3 \(\sum_{n=2}^{\infty} (-4)^n z^n = \sum_{n=3}^{\infty} \frac{1}{6} z^n = \frac{k + 1}{2} z^{n+2}\) #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 209 Context: # Table of Contents 1. **Convergence** - Sequence - Series - Cosine - Cotangent - Cross Ratio - Curve - Cycloid 2. **Numerical Concepts** - Dedekind Sum - Dense - Derivative - Partial - Difference Quotient - Differentiable - Dilation - Discriminant 3. **Geometric Concepts** - Disk - Closed - Open - Punctured - Unit - Distance of Complex Numbers - Divergent - Domain - Double Series 4. **Mathematical Constants** - e - Embedding of R into C - Empty Set - Entire Functions - Essential Singularity - Euclidean Plane - Euler’s Formula 5. **Functions and Theorems** - Even - Exponential Function - Exponential Rules - Extended Complex Plane - Fibonacci Numbers - Field - Fixed Point - Frobenius Problem - Functions - Conformal - Even - Exponential - Logarithmic - Odd - Trigonometric - Fundamental Theorem - Of Algebra - Of Calculus 6. **Applications** - Geogebra - Geometric Interpretation of Multiplication - Geometric Series - Group 7. **Harmonic Analysis** - Harmonic - Harmonic Conjugate - Holomorphic - Homotopy - Hyperbolic Trig Functions #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 210 Context: ``` # Index - i, 4 - identity, 3 - identity map, 23 - identity principle, 154 - image - of a function, 27 - of a point, 23 - imaginary axis, 5 - imaginary part, 4 - improper integral, 104, 186 - infinity, 46 - inside, 88 - integral, 71 - path independent, 102 - integral test, 129 - integration by parts, 92 - interior point, 13 - inverse function, 31 - of a Möbius transformation, 43 - inverse parametrization, 74 - inversion, 45 - isolated point, 13 - isolated singularity, 170 - Jacobian, 65 - Jordan curve theorem, 88 - L'Hôpital's rule, 197 - Laplace, 110 - Laurent series, 157 - least upper bound, 124, 137 - Leibniz's rule, 83, 196 - length, 73 - limit - infinity, 46 - of a function, 24 - of a sequence, 122 - of a series, 126 - linear fractional transformation, 43 - Liouville's theorem, 104 - Log, 59 - log, 61 - logarithm, 59 - logarithmic derivative, 180 - max/min property for harmonic functions, 115, 155 - maximum - strong relative, 115 - weak relative, 116, 155 - mean-value theorem - for harmonic functions, 114 - for holomorphic functions, 86 - for real functions, 195 - meromorphic, 181 - minimum - strong relative, 115 - weak relative, 155 - Möbius transformation, 43 - modulus, 5 - monotone, 124 - monotone sequence property, 124 - Morera's theorem, 100 - multiplication, 2 - north pole, 51 ``` #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 211 Context: ``` obvious, 23 odd, 167 one-to-one, 31 onto, 31 open disk, 12 set, 13 order of a pole, 173 orientation, 15 partial derivative, 32 path, 14 closed, 16 inside of, 88 interior of, 88 polygonal, 79 positively oriented, 88 path independent, 102 periodic, 55, 193 Picard's theorem, 174 piecewise smooth, 72 plane, 12 pointwise convergence, 131 Poisson integral formula, 119 Poisson kernel, 95, 118 polar form, 9 pole, 170 polynomial, 0, 20, 40, 102 positive orientation, 88 power series, 135 differentiation of, 147 integration of, 139 primitive, 76 primitive root of unity, 9 principal argument, 59 principal logarithm, 59 principal value of z, 61 punctured disk, 169 real axis, 5 real number, 2 real part, 4 rectangular form, 9 region, 14 of convergence, 136 simply-connected, 101, 111 removable singularity, 170 reparameterization, 73 residue theorem, 176 reverse triangle inequality, 11, 20 Riemann hypothesis, 130 Riemann sphere, 47 Riemann zeta function, 130 root, 4 root of unity, 9 root test, 138 Rouché's theorem, 182 separated, 14 sequence, 122 convergent, 122 divergent, 122 limit, 122 monotone, 124 series, 125 ``` ########## """QUERY: in 4 sentences please summarize your context""" Consider the chat history for relevant information. 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Page 13, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 14, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 15, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 16, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 19, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 21, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 22, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - 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Page 160, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 163, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 165, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 170, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 174, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 175, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 177, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 179, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 180, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 185, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 188, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 189, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 193, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 195, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 196, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 197, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 198, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 199, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 200, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 201, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 206, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 209, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 210, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 211 ================================================== **Elapsed Time: 5.74 seconds** ================================================== FINAL ANSWER Answer: *A First Course in Complex Analysis* is a textbook designed for undergraduate students, focusing on the foundational concepts of complex analysis, including complex numbers, differentiation, integration, and the properties of holomorphic functions. The authors, Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka, aim to provide a rigorous yet accessible introduction, minimizing reliance on prior real analysis knowledge and treating series and sequences from scratch. The book emphasizes the importance of the Residue Theorem and includes numerous examples, exercises, and proofs to reinforce understanding. Additionally, it highlights the historical context and significance of complex analysis within the broader field of mathematics, making it suitable for students encountering rigorous analysis for the first time. ================================================== **Elapsed Time: 3.13 seconds** ==================================================