{ "query": "in 6 sentences can you explain e equals mc squared including history", "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": "*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." }, { "role": "user", "content": "in 5 sentences please summarize your context" }, { "role": "assistant", "content": "The document titled *A First Course in Complex Analysis* is a comprehensive textbook aimed at undergraduate students, focusing on the fundamental concepts of complex analysis. It covers a wide 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, emphasize a rigorous yet accessible approach, incorporating numerous examples, exercises, and proofs to reinforce understanding. The book also explores significant results such as the Residue Theorem and its applications, providing insights into the historical context and importance of complex analysis in mathematics. Overall, it serves as a foundational resource for students encountering rigorous analysis for the first time, preparing them for further studies in mathematics and related fields." }, { "role": "user", "content": "in 5 sentences what is complex numbers?" }, { "role": "assistant", "content": "Complex numbers are an extension of the real numbers, defined as ordered pairs of real numbers, typically expressed in the form \\( z = x + iy \\), where \\( x \\) is the real part, \\( y \\) is the imaginary part, and \\( i \\) is the imaginary unit satisfying \\( i^2 = -1 \\). This formulation allows for the representation of numbers that cannot be expressed as real numbers, such as the square root of negative numbers, thereby providing solutions to equations like \\( x^2 + 1 = 0 \\). Complex numbers can be added, subtracted, multiplied, and divided (except by zero), following algebraic rules similar to those of real numbers, which include properties like commutativity and associativity. Geometrically, complex numbers can be represented as points or vectors in a two-dimensional plane, known as the complex plane, where the x-axis represents the real part and the y-axis represents the imaginary part. The introduction of complex numbers not only enriches the number system but also plays a crucial role in various fields of mathematics, physics, and engineering, particularly in solving polynomial equations and analyzing waveforms." }, { "role": "user", "content": "in 6 sentences what is complex numbers?" }, { "role": "assistant", "content": "Complex numbers are an extension of the real numbers, represented in the form \\( z = x + iy \\), where \\( x \\) is the real part, \\( y \\) is the imaginary part, and \\( i \\) is the imaginary unit defined by \\( i^2 = -1 \\). This formulation allows for the inclusion of solutions to equations that do not have real solutions, such as \\( x^2 + 1 = 0 \\). Complex numbers can be added, subtracted, multiplied, and divided (except by zero), following algebraic rules similar to those of real numbers, which include properties like commutativity and associativity. Geometrically, complex numbers can be visualized as points or vectors in a two-dimensional plane known as the complex plane, where the x-axis represents the real part and the y-axis represents the imaginary part. The introduction of complex numbers not only enriches the number system but also plays a crucial role in various fields of mathematics, physics, and engineering, particularly in solving polynomial equations and analyzing waveforms. Furthermore, every nonconstant polynomial has roots in the complex numbers, as stated by the Fundamental Theorem of Algebra, which asserts that the complex numbers are algebraically closed." }, { "role": "user", "content": "in 5 sentences what is complex numbers?" }, { "role": "assistant", "content": "Complex numbers are an extension of the real numbers, represented in the form \\( z = x + iy \\), where \\( x \\) is the real part, \\( y \\) is the imaginary part, and \\( i \\) is the imaginary unit defined by \\( i^2 = -1 \\). This formulation allows for the inclusion of solutions to equations that do not have real solutions, such as \\( x^2 + 1 = 0 \\), thus providing a complete number system. Complex numbers can be added, subtracted, multiplied, and divided (except by zero), following algebraic rules similar to those of real numbers, which include properties like commutativity and associativity. Geometrically, complex numbers can be visualized as points or vectors in a two-dimensional plane known as the complex plane, where the x-axis represents the real part and the y-axis represents the imaginary part. The introduction of complex numbers not only enriches the number system but also plays a crucial role in various fields of mathematics, physics, and engineering, particularly in solving polynomial equations and analyzing waveforms." }, { "role": "user", "content": "in 6 sentences can you explain e equals mc squared" }, { "role": "assistant", "content": "Einstein's equation \\(E = mc^2\\) expresses the equivalence of mass (\\(m\\)) and energy (\\(E\\)), indicating that mass can be converted into energy and vice versa. In this equation, \\(c\\) represents the speed of light in a vacuum, approximately \\(3 \\times 10^8\\) meters per second, and it is squared, which emphasizes the vast amount of energy contained in even a small amount of mass. This relationship implies that a small quantity of mass can yield a tremendous amount of energy, which is the principle behind nuclear reactions, where mass is converted into energy. The equation fundamentally changed our understanding of physics, showing that mass and energy are interchangeable and that they are two forms of the same physical entity. It also laid the groundwork for modern physics, influencing theories in both quantum mechanics and cosmology. Overall, \\(E = mc^2\\) encapsulates a profound insight into the nature of the universe, illustrating the deep connection between mass and energy." }, { "role": "user", "content": "in 6 sentences can you explain e equals mc squared including history" }, { "role": "assistant", "content": "" } ], "stream": true, "language_level": "", "chat_channel": ":3990:1726564953700", "language": "German", "tone": "neutral", "writing_style": "standard", "model": "gpt-4o-mini-2024-07-18", "knowledgebase": "ki-dev-large", "seed": 1055, "client_id": 1726564953700, "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 6 sentences can you explain e equals mc squared including history Model: gpt-4o-mini-2024-07-18 Use Curl?: None ================================================== **Elapsed Time: 0.00 seconds** ================================================== ROUTING Query type: summary ================================================== **Elapsed Time: 1.56 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: 0.11 seconds** ================================================== VECTOR SEARCH DONE ================================================== **Elapsed Time: 6.09 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: 0.17 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: 6 Context: ``` # Contents 6. Harmonic Functions ................................. 110 6.1 Definitions and Basic Properties ................. 114 6.2 Mean-Value Principle ............................ 114 7. Power Series ....................................... 121 7.1 Sequences and Completeness ...................... 122 7.2 Series .......................................... 125 7.3 Sequences and Series of Functions ............... 131 7.4 Regions of Convergence .......................... 135 8. Taylor and Laurent Series .......................... 146 8.1 Power Series and Holomorphic Functions .......... 146 8.2 Classification of Zeros and the Identity Principle 152 8.3 Laurent Series .................................. 156 9. Isolated Singularities and the Residue Theorem..... 169 9.1 Classification of Singularities ................... 169 9.2 Residues ........................................ 176 9.3 Argument Principle and Rouché’s Theorem ........ 180 10. Discrete Applications of the Residue Theorem ...... 188 10.1 Infinite Sums ................................... 188 10.2 Binomial Coefficients ........................... 189 10.3 Fibonacci Numbers ............................... 190 10.4 The Coin-Exchange Problem ...................... 193 10.5 Dedekind Sums .................................. 193 ## Theorems From Calculus ## Solutions to Selected Exercises ## Index ``` #################### 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: 8 Context: ``` # Complex Numbers It is not hard to come up with examples for \( p \) and \( q \) for which the argument of this square root becomes negative and thus not computable within the real numbers. On the other hand (e.g., by arguing through the graph of a cubic polynomial), every cubic polynomial has at least one real root. This seeming contradiction can be solved using complex numbers, as was probably first exemplified by Rafael Bombelli (1526–1572). In the next section we show exactly how the complex numbers are set up, and in the rest of this chapter we will explore the properties of the complex numbers. These properties will be of both algebraic (such as the commutative and distributive properties mentioned already) and geometric nature. You will see, for example, that multiplication can be described geometrically. In the rest of the book, the calculus of complex numbers will be built on the properties that we develop in this chapter. ## 1.1 Definitions and Algebraic Properties There are many equivalent ways to think about a complex number, each of which is useful in its own right. In this section, we begin with a formal definition of a complex number. We then interpret this formal definition in more useful and easier-to-work-with algebraic language. Later we will see several more ways of thinking about complex numbers. **Definition.** The complex numbers are pairs of real numbers: \[ C = \{(x, y) : x, y \in \mathbb{R} \} \] equipped with the addition \[ (x, y) + (a, b) = (x + a, y + b) \quad (1.2) \] and the multiplication \[ (x, y) \cdot (a, b) = (x a - y b, x b + y a) \quad (1.3) \] One reason to believe that the definitions of these binary operations are acceptable is that \( C \) is an extension of \( \mathbb{R} \), in the sense that the complex numbers of the form \((x, 0)\) behave just like real numbers: \[ (x, 0) + (y, 0) = (x + y, 0) \quad \text{and} \quad (x, 0) \cdot (y, 0) = (xy, 0). \] ``` #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 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: 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: 17 Context: # GEOMETRIC PROPERTIES We note that \( f(z) \) yields a neat formula for the inverse of a nonzero complex number, which is implicit already in (1.14): \[ z^{-1} = \frac{1}{|z|^2} \overline{z}. \] A famous geometric inequality (which holds, more generally, for vectors in \( \mathbb{R}^n \)) goes as follows. ## Proposition 1.6 (Triangle Inequality) For any \( z_1, z_2 \in \mathbb{C} \) we have \( |z_1 + z_2| \leq |z_1| + |z_2| \). By drawing a picture in the complex plane, you should be able to come up with a geometric proof of the triangle inequality. Here we proceed algebraically: ### Proof We make extensive use of Proposition 1.5: \[ |z_1 + z_2|^2 = |(z_1 + z_2)(\overline{z_1 + z_2})| = z_1\overline{z_1} + z_1\overline{z_2} + z_2\overline{z_1} + z_2\overline{z_2} = |z_1|^2 + z_1\overline{z_2} + z_2\overline{z_1} + |z_2|^2 \] \[ \leq |z_1|^2 + 2\Re(z_1\overline{z_2}) + |z_2|^2 \] \[ = |z_1|^2 + |z_2|^2 + 2|z_1||z_2| \cdot \frac{\Re(z_1\overline{z_2})}{|z_1||z_2|} \leq (|z_1| + |z_2|)^2, \] where the inequality follows from (1.16). Taking square roots on the left- and right-hand sides proves our claim. For future reference we list several useful variants of the triangle inequality: ## Corollary 1.7 For \( z_1, z_2, \ldots, z_n \in \mathbb{C} \), we have the following relations: (a) The triangle inequality: \( |z_1 + z_2| \leq |z_1| + |z_2|. \) (b) The reverse triangle inequality: \( |z_1 + z_2| \geq ||z_1| - |z_2||. \) (c) The triangle inequality for sums: \[ \left| \sum_{k=1}^{n} z_k \right| \leq \sum_{k=1}^{n} |z_k|. \] Inequality (a) is just a rewrite of the original triangle inequality, using the fact that \( |z| = |z| \) and follows by induction. The proof of the reverse triangle inequality (b) is left as Exercise 1.25. #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 18 Context: # 1.4 Elementary Topology of the Plane In Section 1.2 we saw that the complex numbers \( C \), which were initially defined algebraically, can be identified with the points in the Euclidean plane \( \mathbb{R}^2 \). In this section we collect some definitions and results concerning the topology of the plane. \[ \begin{array}{c} y \\ \text{C}(2+i,2) \\ \hspace{-30pt} \text{D}(-2, 1) \\ \hspace{80pt} 1 \\ \hspace{-30pt} x \end{array} \] **Figure 1.6:** Sample circle and disk. In Proposition 1.2, we interpreted \( |z - w| \) as the distance between the complex numbers \( z \) and \( w \), viewed as points in the plane. So if we fix a complex number \( a \) and a positive real number \( r \), then all \( z \in C \) satisfying \( |z - a| = r \) form the set of points at distance \( r \) from \( a \); this set is the circle with center \( a \) and radius \( r \), which we denote by \[ C(a, r) = \{ z \in C : |z - a| = r \} \] The inside of this circle is called the open disk with center \( a \) and radius \( r \); we use the notation \[ D(a, r) = \{ z \in C : |z - a| < r \} \] Note that \( D(a, r) \) does not include the points on \( C(a, r) \). Figure 1.6 illustrates these definitions. Next we need some terminology for talking about subsets of \( C \). **Definition.** Suppose \( G \) is a subset of \( C \). #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 20 Context: # Complex Numbers ![Figure 1.7: The intervals [0,1) and (1,2) are separated.](image_link) ## Definition Two sets \( X, Y \subset \mathbb{C} \) are separated if there are disjoint open sets \( A, B \subset C \) so that \( X \subset A \) and \( Y \subset B \). A set \( G \subset \mathbb{C} \) is connected if it is impossible to find two separated nonempty sets whose union is \( G \). A region is a connected open set. The idea of separation is that the two open sets \( A \) and \( B \) ensure that \( X \) and \( Y \) cannot just "stick together." It is usually easy to check that a set is not connected. On the other hand, it is hard to use the above definition to show that a set is connected, since we have to rule out any possible separation. ## Example 1.10 The intervals \( X = [0,1) \) and \( Y = (1,2) \) on the real axis are separated: There are infinitely many choices for \( A \) and \( B \) that work; one choice is \( A = D[0,1) \) and \( B = D(1,2) \), depicted in Figure 1.7. Hence \( X \cup Y = [0,1) \cup (1,2) \) is not connected. One type of connected set that we will use frequently is a path. ## Definition A path (or curve) in \( \mathbb{C} \) is a continuous function \( \gamma : [a,b] \to \mathbb{C} \), where \([a,b]\) is a closed interval in \( \mathbb{R} \). We may think of \( \gamma \) as a parametrization of the image that is painted by the path and will often write this parametrization as \( \gamma(t) \), for \( a \leq t \leq b \). The path is smooth if it is differentiable and the derivative \( \gamma' \) is continuous and nonzero. #################### 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: 23 Context: # Exercises ## 1.1. Let \( z = 1 + 2i \) and \( w = 2 - i \). Compute the following: (a) \( z + 3w \) (b) \( w - z \) (c) \( z^2 \) (d) \( \text{Re}(wz + w) \) (e) \( z^2 + z + i \) ## 1.2. Find the real and imaginary parts of each of the following: (a) \( \frac{e^{ia}}{1 + i} \) for any \( a \in \mathbb{R} \) (b) \( 1 - i \) (c) \( \left( \frac{1}{\sqrt{2}} \right)^3 \) (d) \( i^n \) for any \( n \in \mathbb{Z} \) ## 1.3. Find the absolute value and conjugate of each of the following: (a) \( -2 + i \) (b) \( (2 + i\sqrt{3}) \) (c) \( \frac{3}{\sqrt{5}} - \sqrt{5}i \) (d) \( (1 + i)^{6} \) ## 1.4. Write in polar form: (a) \( 2i \) (b) \( 1 + i \) (c) \( 3 + \sqrt{3}i \) (d) \( -i \) (e) \( (2 - i)^2 \) (f) \( [3 - 4i] \) (g) \( \sqrt{5} - i \) (h) \( \left( \frac{1}{4} \right)^{4} \) ## 1.5. Write in rectangular form: (a) \( \sqrt{2}e^{i\frac{\pi}{4}} \) (b) \( 3e^{i\frac{\pi}{6}} \) (c) \( -r^{2}e^{i2\theta} \) (d) \( 2e^{i\frac{\pi}{3}} \) ## 1.6. Write in both polar and rectangular form: (a) \( e^{i\frac{\pi}{6}} \) (b) \( \frac{1}{2} e^{i\theta} \) #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 24 Context: # COMPLEX NUMBERS ## 1.7 Show that the quadratic formula works. That is, for \( a, b, c \in \mathbb{R} \) with \( a \neq 0 \), prove that the roots of the equation \( ax^2 + bx + c = 0 \) are \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here we define \( \sqrt{b^2 - 4ac} = i \sqrt{-(b^2 - 4ac)} \) if the discriminant \( b^2 - 4ac \) is negative. ## 1.8 Use the quadratic formula to solve the following equations: - (a) \( x^2 + 25 = 0 \) - (b) \( 2x^2 + 2x + 5 = 0 \) - (c) \( 5x^2 + 4x + 1 = 0 \) - (d) \( x^2 - 2x = 0 \) - (e) \( x^2 = 2x \) ## 1.9 Find all solutions of the equation \( x^2 + 2x + (1-i) = 0 \). ## 1.10 Fix \( a \in \mathbb{C} \) and \( b \in \mathbb{R} \). Show that the equation \( |z|^2 + \text{Re}(z) + b = 0 \) has a solution if and only if \( |a|^2 \geq 4b \). When solutions exist, show the solution set is a circle. ## 1.11 Find all solutions to the following equations: - (a) \( z^6 = 1 \) - (b) \( z^6 = -16 \) - (c) \( z^6 = e^{-3} \) - (d) \( z^6 - z^2 = 0 \) ## 1.12 Show that \( |z| = 1 \) if and only if \( z = \bar{z} \). ## 1.13 Show that: - (a) \( z \) is a real number if and only if \( z = \bar{z} \). - (b) \( z \) is either real or purely imaginary if and only if \( |z|^2 = z^2 \). ## 1.14 Review Proposition 1.1. ## 1.15 Show that if \( z_1 z_2 = 0 \) then \( z_1 = 0 \) or \( z_2 = 0 \). #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 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: 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: 37 Context: # Differentiability and Holomorphicity ## Proof Let \( \gamma_1(t) \) and \( \gamma_2(t) \) be parametrizations of the two paths such that \( \gamma_1(0) = \gamma_2(0) = \gamma(0) \). Then \( \gamma_2(t) \) (considered as a vector) is the tangent vector of \( \gamma_1 \) at the point \( a \), and \( \gamma_2'(0) \) is the tangent vector of \( \gamma_2 \) at the point \( f(a) \) given by: \[ \frac{d}{dt} \bigg|_{t=0} \gamma_1(t) = f'(\gamma_1(0)) \cdot \gamma_1'(0) \] and similarly, the tangent vector of \( f(\gamma_2) \) at the point \( f(a) \) is \( f'(\gamma_2'(0)) = f'[\gamma_2'(0)] \). This means that the action of \( f \) multiplies the two tangent vectors \( \gamma_1'(0) \) and \( \gamma_2'(0) \) by the same nonzero complex number \( f'(a) \), and so the two tangent vectors get dilated by \( |f'(a)| \) (which does not affect their direction) and rotated by the same angle (an argument of \( f(a) \)). We end this section with yet another differentiation rule, that for inverse functions. As in the real case, this rule is only defined for functions that are bijections. ## Definition A function \( f: G \to H \) is one-to-one if for every image \( w \in H \) there is a unique \( z \in G \) such that \( f(z) = w \). The function is onto if every \( w \in H \) has a preimage \( z \in G \) that gives \( f(z) = w \). A bijection is a function that is both one-to-one and onto. If \( f: G \to H \) is a bijection then \( g: H \to G \) is the inverse of \( f \) if \( f(g(z)) = z \) for all \( z \in H \); in other words, the composition \( g \circ f \) is the identity function on \( H \). ## Proposition 12 Suppose \( G, H \subset \mathbb{C} \) are open sets, \( f: G \to H \) is a bijection, \( s: H \to G \) is the inverse function of \( f \), and \( z_0 \in H \). If \( f \) is differentiable at \( s(z_0) \) with \( f'(s(z_0)) \neq 0 \) and \( g \) is continuous at \( z_0 \), then \( g \) is differentiable at \( z_0 \), with \[ g'(z_0) = \frac{1}{f'(s(z_0))} \] ## Proof Since \( f(z_0) = z \) for all \( z \in H \), \[ g'(z_0) = \lim_{z \to f(z_0)} \frac{g(z) - g(f(z_0))}{z - f(z_0)} = \lim_{z \to f(z_0)} \frac{1}{f'(s(z_0))} \cdot \frac{g(f(z_0)) - g(z)}{g'(z_0)} \] \[ = \lim_{z \to f(z_0)} \frac{g(f(z_0)) - g(z)}{g'(z_0)(z - f(z_0))} \] #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 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: 40 Context: # Differentiation (and an analogous identity for \( x \)). Functions with continuous second partials satisfying this partial differential equation on a region \( G \subset \mathbb{C} \) (though not necessarily (2.3)) are called harmonic on \( G \); we will study such functions in Chapter 6. Again, as we will see later, if \( f \) is holomorphic in an open set then the partials of any order of \( u \) and \( v \) exist; hence we will show that the real and imaginary parts of a function that is holomorphic in an open set are harmonic on that set. ## Example 2.14 We revisit Example 2.7 and again consider \( f: \mathbb{C} \to \mathbb{C} \) given by \[ f(z) = z^2 = (x + iy)^2 = (x^2 - 3y^2) + i(3x^2 - y^2). \] Thus, \[ f_x(z) = 3x^2 - 3y^2 + 6ixy \] and \[ f_y(z) = -6xy - 3y^2 - 3ix^2 \] are continuous on \( C \) and satisfy \( f_x = \overline{f_y} \). Thus by Theorem 2.13(b), \( f(z) = z^2 \) is entire. ## Example 2.15 Revisiting Example 2.8 (you saw that coming, didn't you?), we consider \( f: \mathbb{C} \to \mathbb{C} \) given by \[ f(z) = (z^2) = (x - iy)^2 = x^2 - y^2 - 2iy. \] Now, \[ f_x(z) = 2x - 2iy \] and \[ f_y(z) = -2y - 2ix, \] which satisfy \( f_y = \overline{f_x} \) only when \( z = 0 \). (The contrapositive of Theorem 2.13(a)) thus implies that \( f(z) = (z^2) \) is not differentiable on \( C \setminus \{0\} \). ### Proof of Theorem 2.13 (a) If \( f \) is differentiable at \( c = (x_0, y_0) \) then \[ f'(x_0) = \lim_{h \to 0} \frac{f(x_0 + h) - f(x_0)}{h}. \] As we know by now, we must get the same result if we restrict \( z \) to be on the real axis and if we restrict it to be on the imaginary axis. In the first case, let \( h = \Delta x \) and \[ f'(x_0) = \lim_{\Delta x \to 0} \frac{f(x_0 + \Delta x) - f(x_0)}{\Delta x} = \lim_{\Delta x \to 0} \frac{f(x_0 + \Delta x, y_0)}{\Delta x} \frac{\partial f}{\partial x}(x_0, y_0). \] #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 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: 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: 61 Context: # Exponential and Trigonometric Functions ## 3.4 Exponential and Trigonometric Functions To define the complex exponential function, we once more borrow concepts from calculus, namely the real exponential \( e^x \) and the real sine and cosine, and we finally make sense of the notation \( e^{x + iy} = e^x (\cos y + i \sin y) \). ### Definition The **complex exponential function** is defined for \( z = x + iy \) as: \[ \exp(z) = e^x (\cos y + i \sin y) = e^z. \] This definition seems a bit arbitrary. Our first justification is that all exponential rules that are used to from real numbers carry over to the complex case. They mainly follow from Proposition 1.3 and are collected in the following. ### Proposition 3.16 For all \( z_1, z_2 \in \mathbb{C} \): (a) \(\exp(z_1) \exp(z_2) = \exp(z_1 + z_2)\) (b) \(\exp(-z) = \frac{1}{\exp(z)}\) (c) \(\exp(iz) \neq 0\) (d) \(\exp(z) = \exp(\text{Re}(z))\) (e) \(\exp(z + 2\pi i) = \exp(z)\) (f) \(\exp(0) = \exp(z)\). Identity (f) is very special and has no counterpart for the real exponential function. It says that the complex exponential function is periodic with period \( 2\pi i \). This has many interesting consequences; one that may not seem too pleasant at first sight is the fact that the complex exponential function is not one-to-one. Identity (f) is not only remarkable, but we invite you to meditate on its proof below; it gives a strong indication that our definition of exp is reasonable. We also note that (f) implies that \( \exp \) is infinite. We leave the proof of Proposition 3.16 as Exercise 3.34 but give one example. ### Proof of (f) The partial derivatives of \( f(z) = \exp(z) \) are: \[ \frac{\partial f}{\partial x} = e^{x + iy} \cos y \quad \text{and} \quad \frac{\partial f}{\partial y} = e^{x + iy} (-\sin y + i \cos y). \] #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 64 Context: # Examples of Functions A similar calculation holds for the cosine. Not too surprisingly, the following properties follow mostly from Proposition 3.16. ## Proposition 3.17. For all \( z_1, z_2 \in \mathbb{C} \): \[ \begin{align*} \sin(-z) &= -\sin z & \cos(-z) &= \cos z \\ \sin(z + 2\pi) &= \sin z & \cos(z + 2\pi) &= \cos z \\ \tan(z + \pi) &= \tan z & \cot(z + \pi) &= \cot z \\ \sin(z + \frac{\pi}{2}) &= \cos z & \cos(z + \frac{\pi}{2}) &= -\sin z \\ \cos(z_1 + z_2) &= \cos z_1 \cos z_2 - \sin z_1 \sin z_2 \\ \sin(z_1 + z_2) &= \sin z_1 \cos z_2 + \cos z_1 \sin z_2 \\ \cos^2 z + \sin^2 z &= 1 & \frac{d}{dz} \sin z &= \cos z \\ \frac{d}{dz} \cos z &= -\sin z \end{align*} \] Finally, we end with caution: unlike in the real case, the complex sine and cosine are not bounded—consider, for example, \(\sin(y)\) as \(y \to \infty\). We end this section with a remark on hyperbolic trigonometric functions. The hyperbolic sine, cosine, tangent, and cotangent are defined as in the real case: ## Definition: \[ \begin{align*} \sinh z &= \frac{1}{2} \left( e^{z} - e^{-z} \right) & \cosh z &= \frac{1}{2} \left( e^{z} + e^{-z} \right) \\ \tanh z &= \frac{\sinh z}{\cosh z} = \frac{e^{z} - e^{-z}}{e^{z} + e^{-z}} & \coth z &= \frac{1}{\tanh z} = \frac{\cosh z}{\sinh z} - 1 \end{align*} \] As such, they are yet more special combinations of the exponential function. They still satisfy the identities you already know, e.g., \[ \frac{d}{dz} \sinh z = \cosh z \quad \text{and} \quad \frac{d}{dz} \cosh z = \sinh z. \] Moreover, they are related to the trigonometric functions via: \[ \sinh(iz) = i \sin z \quad \text{and} \quad \cosh(iz) = \cos z. \] #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 65 Context: # Logarithms and Complex Exponentials The complex logarithm is the first function we'll encounter that is of somewhat tricky nature. It is motivated as an inverse to the exponential function, that is, we're looking for a function \( \Log \) such that \[ \exp(\Log(z)) = z = \Log(\exp z). \] But because \( \exp \) is not one-to-one, this is too much to hope for. In fact, given a function \( \Log \) that satisfies the first equation, the function \( f(z) = \Log(z) + 2\pi i k \) does as well, and so there cannot be an inverse of \( \exp \) (which would have to be unique). On the other hand, \( \exp \) becomes one-to-one if we restrict its domain, so there is hope for a logarithm if we're careful about its construction and about its domain. ## Definition Given a region \( G \), any continuous function \( \Log: G \rightarrow \mathbb{C} \) that satisfies \( \exp(\Log(z)) = z \) is a branch of the logarithm (on \( G \)). To make sure this definition is not vacuous, let's write, as usual, \( z = re^{i \theta} \), and suppose that \( \Log(z) = \log |z| + i \arg(z) \). Then for the first equation to hold, we need \[ \exp(\Log(z)) = e^{\log |z| + i \arg(z)} = z, \] that is, \( e^{\log r} = r \). The latter equation is equivalent to \( r = e^{\log r} \) for some \( k \in \mathbb{Z} \), and denoting the natural logarithm of the positive real number \( r \) by \( \ln(r) \), the former equation is equivalent to \( u = \ln |z| \). A reasonable definition of a logarithm function \( \Log \) would hence be \( \Log(z) = \ln |z| + i \arg(z) \) where \( \arg(z) \) gives the argument for the complex number \( z \) according to some convention—here is an example: ## Definition Let \( \Arg \) denote the unique argument \( \theta \) of \( z \) that lies in \( (-\pi, \pi) \) (the principal argument of \( z \)). Then the principal logarithm is the function \( \Log: \mathbb{C} \setminus \{0\} \rightarrow \mathbb{C} \) defined through \[ \Log(z) = \ln |z| + i \Arg(z). \] #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 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: 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: 87 Context: # Cauchy's Theorem Exercise 4.23 shows that \( \sim_0 \) is an equivalence relation on the set of closed paths in \( G \). The definition of homotopy applies to parameterization of curves; but Exercise 4.24, together with transitivity of \( \sim_0 \), shows that homotopy is invariant under reparameterizations. ## Theorem 4.18 (Cauchy’s Theorem) Suppose \( G \subset \mathbb{C} \) is a region, \( f \) is holomorphic in \( G \), \( \gamma_0 \) and \( \gamma_1 \) are piecewise smooth paths in \( G \), and \( \gamma_0 \sim \gamma_1 \). Then \[ \int_{\gamma_0} f = \int_{\gamma_1} f. \] As a historical aside, it is assumed that Johann Carl Friedrich Gauss (1777–1855) knew a version of this theorem in 1811 but published it only in 1831. Cauchy (of course) was the first to publish a complete 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: 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: 91 Context: Example 4.23. We'd like to compute \[ \int_{C} \frac{dz}{z^2 - 2z} \] where \(C\) is the unit circle, oriented counter-clockwise. (Try computing it from first principles.) We use a partial fractions expansion to write \[ \int_{C} \frac{dz}{z^2 - 2z} = \frac{1}{2} \int_{C} \frac{dz}{z - 2} - \frac{1}{2} \int_{C} \frac{dz}{z}. \] The first integral on the right-hand side is zero by Corollary 4.20 applied to the function \(f(z) = \frac{1}{z}\) (note that \(f\) is holomorphic in \(C \setminus \{2\}\) and \(C\) is \((C \setminus \{2\})\)-contractible). The second integral is \(2\pi i\) by Exercise 4.4, and so \[ \int_{C} \frac{dz}{z^2 - 2z} = -\pi i. \] Sometimes Corollary 4.20 itself is known as Cauchy's Theorem. See Exercise 4.26 for a related formulation of Corollary 4.20, with a proof based on Green's Theorem. ### 4.4 Cauchy's Integral Formula We recall our notations: - \(C(r) = \{ z \in \mathbb{C} : |z - a| = r \}\) - \(D(r) = \{ z \in \mathbb{C} : |z - a| < r \}\) - \(D[-r] = \{ z \in \mathbb{C} : |z - a| \leq r \}\) Theorem 4.24. If \(f\) is holomorphic in an open set containing \(D[r]\) then \[ f(w) = \frac{1}{2\pi i} \int_{C(r)} f(z) \frac{dz}{z - w}. \] This is Cauchy's Integral Formula for the case that the integration path is a circle; we will prove the general statement at the end of this chapter. However, already this special case is worth meditating over: the data on the right-hand side of Theorem 4.24 #################### 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: 110 Context: # Corollary 5.13 (Liouville's Theorem) Any bounded entire function is constant. ## Proof: Suppose \( |f(z)| \leq M \) for all \( z \in \mathbb{C} \). Given any \( \epsilon > 0 \), we apply Theorem 5.1 with the circle \( C(R) \); note that we can choose any \( R > 0 \) because \( f \) is entire. By Proposition 4.16(1), \[ \left| \frac{f(w)}{2\pi i \int_{C(R)} (z - w)^{-1} dz} \right| = \frac{1}{2\pi i} \int_{C(R)} \frac{f(z)}{(z - w)^2} dz \leq \frac{1}{2\pi} \max_{z \in C(R)} |f(z)| \cdot \frac{1}{2\pi R}. \] The right-hand side can be made arbitrarily small, as we are allowed to choose \( R \) as large as we want. This implies that \( f' = 0 \), and hence, by Theorem 2.17, \( f \) is constant. As an example of the usefulness of Liouville's theorem (Corollary 5.13), we give another proof of the Fundamental Theorem of Algebra, close to Gauss's original proof. ## Second proof of Theorem 5.11 (Fundamental Theorem of Algebra): Suppose (by way of contradiction) that \( f \) does not have any roots, that is, \( f(z) \neq 0 \) for all \( z \in \mathbb{C} \). Then the function \( f(z) = \frac{1}{f(z)} \) is entire. But \( f \to 0 \) as \( |z| \to \infty \), by Proposition 5.10; consequently, by Exercise 5.10, \( f \) is bounded. Now we apply Corollary 5.13 to deduce that \( f \) is constant. Hence \( f \) is constant, which contradicts our assumptions. As one more example of the theme of getting results from Cauchy's Integral Formulas by taking the limit as a path “goes to infinity,” we compute an improper integral. ## Example 5.14: We will compute the (real) integral \[ \int_0^{\infty} \frac{dx}{x^2 + 1} = \pi. \] Let \( C_R \) be the counterclockwise semicircle formed by the segment \([-R, R]\) of the real axis from \(-R\) to \(R\), followed by the circular arc \( \gamma_R \) of radius \( R \) in the upper half plane from \( R \to -R \), see Figure 5.2. > *This theorem (the historical reason commonly attributed to Joseph Liouville (1809–1883). It was published earlier by Cauchy in fact, Gauss may well have known about it before Cauchy.)* #################### 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: 112 Context: # Exercises ## 5.1 Compute the following integrals, where \(\square\) is the boundary of the square with vertices at \(\pm 4i\), positively oriented: (a) \(\int_{\square} \frac{e^{z^2}}{z^2} \, dz\) (b) \(\int_{\square} \frac{z^3}{(x - \pi^2)} \, dz\) (c) \(\int_{\square} \frac{\sin(2z)}{(z - \pi)^2} \, dz\) (d) \(\int_{\square} \frac{e^{z} \cos(z)}{(x - \pi)} \, dz\) ## 5.2 Prove the formula for \(f^{(n)}\) in Theorem 5.1. *Hint: Modify the proof of the integral formula for \(f^{(n)}(w)\) as follows:* (a) Write a difference quotient for \(f^{(n)}(w)\), and use the formula for \(f^{(n)}(w)\) in Theorem 5.1 to convert this difference quotient into an integral of \(f(z)\) divided by some polynomial. (b) Subtract the desired integral formula for \(f^{(n)}\) from your integral for the difference quotient, and simplify to get the analogue of (5.1). (c) Find a bound as in the proof of Theorem 5.1 for the integrand, and conclude that the limit of the difference quotient is the desired integral formula. ## 5.3 Integrate the following functions over the circle \(C(0,3)\): (a) \(\log(z - 4i)\) (b) \(\frac{1}{z - \frac{1}{2}}\) (c) \(\frac{1}{z^2}\) (d) \(\frac{e^{z}}{z^3}\) (e) \(\cos(z) \, z\) (f) \(z^{-3}\) (g) \(\frac{\sin(z)}{(z^2 + 1)}\) (h) \(\frac{1}{(4 + z^2 + 1)}\) (i) \(\frac{e^{2z}}{(z - 1)(\sqrt{z - 2})}\) ## 5.4 Compute \(\int_{C(2)} \frac{e^{z}}{(z - w)^2}\, dz\) where \(w\) is any fixed complex number with \(|w| \neq 2\). #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 113 Context: 5.5. Define \( f : [0, 1] \to \mathbb{C} \) through \[ f(x) := \int_0^1 \frac{dw}{1 - wx} \] (the integration path is from 0 to 1 along the real line). Prove that \( f \) is holomorphic in the unit disk \( D(0, 1) \). 5.6. To appreciate Corollary 5.5, show that the function \( f : \mathbb{R} \to \mathbb{R} \) given by \[ f(x) = \begin{cases} x^2 \sin\left(\frac{1}{x}\right) & \text{if } x \neq 0 \\ 0 & \text{if } x = 0 \end{cases} \] is differentiable in \( \mathbb{R} \), yet \( f' \) is not even continuous (much less differentiable) at 0. 5.7. Prove that \( f(2) = e^2 \) does not have an antiderivative in any nonempty region. 5.8. Show that \( \exp(\sin z) \) has an antiderivative on \( \mathbb{C} \). (What is it?) 5.9. Find a region on which \( f(z) = \exp(z^2) \) has an antiderivative. (Your region should be as large as you can make it. How does this compare with the real function \( f(x) = e^x \)?) 5.10. Suppose \( f \) is continuous on \( \mathbb{C} \) and \( \lim_{z \to \infty} f(z) = L \). Show that \( f \) is bounded. (Hint: If \( \lim_{z \to \infty} f(z) = L \), use the definition of the limit at infinity to show that there is \( R > 0 \) so that \( |f(z) - L| < \epsilon \) if \( |z| > R \). Now argue that \( |f(z)| < |L| + 1 \) for \( |z| \geq R \). Use an argument from calculus to show that \( |f(z)| \) is bounded for \( |z| \leq R \).) 5.11. Let \( p(z) \) be a polynomial of degree \( n > 0 \). Prove that there exist complex numbers \( c_1, c_2, \ldots, c_n \), and positive integers \( j_1, \ldots, j_n \) such that \[ p(z) = c e^{-j_1} (z - z_1)^{j_1} \cdots (z - z_n)^{j_n} \] where \( j_1 + \ldots + j_n = n \). #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 116 Context: # Chapter 6 ## Harmonic Functions The shortest route between two truths in the real domain passes through the complex domain. Jacques Hadamard (1865–1963) We will now spend a short while on certain functions defined on subsets of the complex plane that are real valued, namely those functions that are harmonic in some region. The main motivation for studying harmonic functions is that the partial differential equation they satisfy is very common in the physical sciences. Their definition briefly showed its face in Chapter 2, but we study them only now in more detail, since we have more machinery at our disposal. This machinery comes from complex-valued functions, which are, nevertheless, intimately connected to harmonic functions. ### 6.1 Definition and Basic Properties Recall from Section 2.3 the definition of a harmonic function: **Definition.** Let \( G \subset \mathbb{C} \) be a region. A function \( u : G \rightarrow \mathbb{R} \) is harmonic in \( G \) if it has continuous second partials in \( G \) and satisfies the Laplace* equation \[ u_{xx} + u_{yy} = 0. \] **Example 6.1.** The function \( u(x, y) = xy \) is harmonic in \( C \) since \( u_{xx} + u_{yy} = 0 + 0 = 0. \) **Example 6.2.** The function \( u(x, y) = e^x \cos(y) \) is harmonic in \( C \) because \[ u_{xx} + u_{yy} = e^x \cos(y) - e^x \cos(y) = 0. \] 1. Named after Pierre-Simon Laplace (1749–1827). #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 118 Context: # HARMONIC FUNCTIONS ## Theorem 6.6. Suppose \( u \) is harmonic on a simply-connected region \( G \). Then there exists a harmonic function \( v \) in \( G \) such that \( f = u + iv \) is holomorphic in \( G \). The function \( v \) is called a **harmonic conjugate of** \( u \). **Proof:** We will explicitly construct a holomorphic function \( f \) (and thus \( v = \text{Im } f \)). First, let \[ g = u_x - i u_y. \] The plan is to prove that \( g \) is holomorphic, and then to construct an antiderivative of \( g \), which will be almost the function \( f \) that we are after. To prove that \( g \) is holomorphic, we use Theorem 2.13: first because \( u \) is harmonic, \( \text{Re } g = u_x \) and \( \text{Im } g = -u_y \) have continuous partials. Moreover, again because \( u \) is harmonic, \( \text{Re } g \) and \( \text{Im } g \) satisfy the Cauchy-Riemann equations (2.3): \[ \begin{align*} \text{(Re } g)_y & = u_{xy} = -u_{yx} = -(\text{Im } g)_x, \\ \text{(Re } g)_x & = u_{xx} = -u_{yy} = -(\text{Im } g)_y. \end{align*} \] Theorem 2.13 implies that \( g \) is holomorphic in \( G \), and so we can use Corollary 5.8 to obtain an antiderivative \( f \) of \( g \) (here is where we use the fact that \( G \) is simply connected). Now we decompose \( g \) into its real and imaginary parts as \( h = a + ib \). Then, again using Theorem 2.13, \[ g = b' - a' i, \] (The second equation follows from the Cauchy-Riemann equations (2.3)). But the real part of \( g \) is \( g_x = a \), and thus \( u_x = a = \text{Re } f \) for some function \( f \) that depends only on \( y \). On the other hand, comparing the imaginary parts of \( g' \) yields \( -u_y = -a' = b_y = \text{Im } f \) where \( c \) depends only on \( y \). Hence \( f \) has the form \[ f(x) = h(x) + c \] is a function holomorphic in \( G \) whose real part is \( u \), as promised. > As a side remark, with hindsight it should not be surprising that the function \( g \) that we first constructed in our proof is the derivative of the sought-after function. #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 120 Context: ``` 114 # HARMONIC FUNCTIONS ## Proof: We will prove that \( u + iv \) satisfies the Cauchy–Riemann equations (2.3). The first follows from \[ \frac{\partial u}{\partial y}(x, y) = \frac{\partial v}{\partial x}(x, y) \] by the Fundamental Theorem of Calculus (Theorem A.3). Second, by Leibniz's Rule (Theorem A.9), the Fundamental Theorem of Calculus (Theorem A.3), and the fact that \( v \) is harmonic, \[ \frac{\partial v}{\partial x}(x, y) = \int_{0}^{y} \frac{\partial^2 v}{\partial x^2}(x, t) \, dt = -\int_{0}^{y} \frac{\partial^2 u}{\partial y^2}(x, t) \, dt = \frac{\partial u}{\partial y}(x, 0) - \frac{\partial^2 u}{\partial y \partial x}(x, 0). \] As you might imagine, Proposition 6.3 and Theorem 6.6 allow for a powerful interplay between harmonic and holomorphic functions. In that spirit, the following theorem appears not so surprising. You might appreciate its depth better when looking back at the simple definition of a harmonic function. ## Corollary 6.9 A harmonic function is infinitely differentiable. ## Proof: Suppose \( u \) is harmonic in \( G \) and \( z_0 \in G \). We wish to show that \( u \in C^\infty \) exists for all positive integers \( n \). Let \( r > 0 \) such that the disk \( D(z_0, r) \) is contained in \( G \). Since \( D(z_0, r) \) is simply connected, Theorem 6.6 asserts the existence of a holomorphic function \( f \) in \( D(z_0, r) \) such that \( \text{Re} \, f = u \) by Corollary 5.5. \( f \) is infinitely differentiable on \( D(z_0, r) \) and hence so is its real part \( u \). This proof is the first in a series of proofs that use the fact that the property of being harmonic is local—if it is a property at each point of a certain region. Note that in our proof of Corollary 6.9 we did not construct a function \( f \) that is holomorphic in \( G \); we only constructed such a function on the disk \( D(z_0, r) \). This might very well differ from one disk to the next. ## 6.2 Mean-Value and Maximum/Minimum Principle We have established an intimate connection between harmonic and holomorphic functions, and so it should come as no surprise that some of the theorems we proved for holomorphic functions have analogues in the world of harmonic functions. Here is such a harmonic analogue of Cauchy's Integral Formula (Theorems 6.24 and 6.27). ``` #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 122 Context: ``` # Harmonic Functions ## Integral \[ u(w) = \frac{1}{2\pi} \int_{0}^{2\pi} \frac{|u(w+r e^{i\theta})|}{r} d\theta \] \[ = \frac{1}{2\pi} \left( \int_{0}^{\pi} |u(w+r e^{i\theta})| d\theta + \int_{\pi}^{2\pi} |u(w+r e^{i\theta})| d\theta \right) \] All the integrands can be bounded by \(u(w)\); for the middle integral we get a strict inequality. Hence \[ a(w) < \frac{1}{2\pi} \left( \int_{0}^{\pi} u(x) dx + \int_{\pi}^{2\pi} u(x) dx \right) = u(w). \] a contradiction. The same argument works if we assume that \(u\) has a relative minimum. But in this case there’s a shortcut argument: if \(u\) has a strong relative minimum then the harmonic function \(-u\) has a strong relative maximum, which we just showed cannot exist. So far, harmonic functions have benefited from our knowledge of holomorphic functions. Here is a result where the benefit goes in the opposite direction. ## Corollary 6.12 If \(f\) is holomorphic and nonzero in the region \(G\), then \(|f|\) does not have a strong relative maximum or minimum in \(G\). ### Proof By Exercise 6.6, the function \(\ln |f(z)|\) is harmonic on \(G\) and so, by Theorem 6.11, does not have a strong relative maximum or minimum in \(G\). But then neither does \(|f(z)|\), because it is monotonic. We finish our excursion about harmonic functions with a preview and its consequences. We say a real valued function on a region \(G\) has a weak relative maximum at \(x_0\) if there exists a disk \(D(x_0, r) \subset G\) such that \[ u(x_0) \geq u(x) \quad \text{for all } x \in D(x_0, r). \] We define weak relative minimum similarly. In Chapter 8 we will strengthen Theorem 6.11 and Corollary 6.12 to Theorem 8.17 and Corollary 8.20 by replacing strong relative extremum in the hypotheses with weak relative extremum. A special but important case is when \(u\) does not assume that \(f\) is nonzero in a region \(G\) to have a strong relative maximum 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: 123 Context: # Mean-Value Principle functions, Corollary 8.20, concerns bounded regions. In Chapter 8 we will establish that, if \( u \) is harmonic in a bounded region \( G \) and continuous on its closure, then \[ \sup_{x \in G} u(x) = \max_{x \in \partial G} u(x) \quad \text{and} \quad \inf_{x \in G} u(x) = \min_{x \in \partial G} u(x) \tag{6.1} \] where, as usual, \( \partial G \) denotes the boundary of \( G \). We'll exploit this in the next two corollaries. ## Corollary 6.13 Suppose \( u \) is harmonic in the bounded region \( G \) and continuous on its closure. If \( u \) is zero on \( \partial G \) then it is zero in \( G \). **Proof.** By (6.1), \[ u(z) \leq \sup_{x \in G} u(x) = \max_{x \in \partial G} u(x) \] and \[ u(z) \geq \inf_{x \in G} u(x) = \min_{x \in \partial G} u(x), \] so \( u \) must be zero in \( G \. \) ## Corollary 6.14 Suppose \( u \) and \( v \) are harmonic in the bounded region \( G \) and continuous on its closure. If \( u(x) = v(x) \) for all \( x \in \partial G \) then \( u \equiv v \) for all \( x \in G \). **Proof.** \( u - v \) is harmonic in \( G \) (Exercise 6.2) and is continuous on the closure \( \overline{G} \), and \( u - v \) is zero on \( \partial G \). Now apply Corollary 6.13. ## Corollary 6.15 Corollary 6.14 says that if a function \( u \) is harmonic in a bounded region \( G \) and is continuous on the closure \( \overline{G} \) then the values of \( u \) at points in \( G \) are completely determined by the values of \( u \) on the boundary of \( G \). We should remark, however, that this result is of a completely theoretical nature: it says nothing about how to extend a continuous function \( f \) given on the boundary of a region to be harmonic in the full region. This problem is called the Dirichlet problem, and it has a solution for all bounded simply-connected regions. If the region \( G \) is the unit disk and \( f \) is a continuous function on the unit circle, define \[ \hat{u}(r, \theta) = r^{\ell} \cdot f(\theta) \quad \text{and} \quad \hat{u}(r, \theta) = \frac{1}{2\pi} \int_0^{2\pi} r^{\ell} f(\phi) \, d\phi \quad \text{for } r < 1, \] where \(\ell\) is a suitable integer. #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 124 Context: where \( P_\nu(r) \) is the Poisson kernel which we introduced in Exercise 4.31. Then \( u \) is the desired extension: it is continuous on the closed unit disk, harmonic in the open unit disk, and agrees with \( u \) on the unit circle. In simple cases this solution can be converted to solutions in other regions, using a conformal map to the unit disk. All of this is beyond the scope of this book, though Exercise 6.13 gives some indication why the above formula does the trick. At any rate, we remark that Corollary 6.14 says that the solution to the Dirichlet problem is unique. ## Exercises 6.1. Show that all partial derivatives of a harmonic function are harmonic. 6.2. Suppose \( u(x, y) \) and \( v(x, y) \) are harmonic in \( G \), and \( \varepsilon \in \mathbb{R} \). Prove that \( u(x, y) + \varepsilon v(x, y) \) is also harmonic in \( G \). 6.3. Give an example that shows that the product of two harmonic functions is not necessarily harmonic. 6.4. Let \( u(x, y) = e^y \sin y \). (a) Show that \( u \) is harmonic on \( \mathbb{C} \). (b) Find an entire function \( f \) such that \( \text{Re}(f) = u \). 6.5. Consider \( u(x, y) = \ln(x^2 + y^2) \). (a) Show that \( u \) is harmonic on \( \mathbb{C} \setminus \{(0, 0)\} \). (b) Prove that \( u \) is not the real part of a function that is holomorphic in \( \mathbb{C} \setminus \{(0, 0)\} \). 6.6. Show that if \( f \) is holomorphic and nonzero in \( G \), then \( \ln|f(x, y)| \) is harmonic in \( G \). 6.7. Suppose \( u(x, y) \) is a function \( \mathbb{R}^2 \rightarrow \mathbb{R} \) that depends only on \( x \). When is \( u \) harmonic? #################### 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: 135 Context: Proposition 7.17 (Integral Test). If \( f: [1, \infty) \to \mathbb{R} \) is continuous and nonincreasing, then \[ \int_{1}^{\infty} f(t) \, dt \leq \sum_{k=1}^{\infty} f(k) \leq \int_{1}^{\infty} f(t) \, dt + f(1). \] The Integral Test literally comes with a proof by picture—see Figure 7.2: the integral of \( f \) on the interval \([k, k + 1]\) is bounded between \( f(k) \) and \( f(k + 1) \). Adding the pieces gives the inequalities above for the partial sum versus the integrals from \( 1 \) to \( n \) and from \( 1 \) to \( n + 1 \), and the inequality persists in the limit. Corollary 7.18. If \( f: [1, \infty) \to \mathbb{R} \) is continuous and nonincreasing, then \( \sum_{k=1}^{\infty} f(k) \) converges if and only if \( \int_{1}^{\infty} f(t) \, dt \) is finite. Proof. Suppose \( \int_{1}^{\infty} f(t) \, dt = \infty \). Then the first inequality in Proposition 7.17 implies that the partial sums \( \sum_{k=1}^{n} f(k) \) are unbounded, and so Corollary 7.10 says that \( \sum_{k=1}^{\infty} f(k) \) cannot converge. Conversely, if \( \int_{1}^{\infty} f(t) \, dt \) is finite, then the second inequality in Proposition 7.17 says that the partial sums \( \sum_{k=1}^{n} f(k) \) are bounded; thus, again with Corollary 7.10, we conclude that \( \sum_{k=1}^{\infty} f(k) \) converges. Example 7.19. The series \( \sum_{k=1}^{\infty} \frac{1}{k^p} \) converges for \( p > 1 \) and diverges for \( p \leq 1 \) (and the case \( p = 1 \) was the subject of Example 7.16) because \[ \int_{1}^{\infty} \frac{dx}{x^p} = \lim_{t \to \infty} \left( \frac{x^{1-p}}{1-p} \Big|_1^t + 1 \right) = \frac{1}{1 - p} \text{ if and only if } p > 1. \] By now you might be amused that we have collected several results on series whose terms are nonnegative real numbers. One reason is that such series are a bit easier to handle; another one is that there is a notion of convergence special to series that relates any series to one with only nonnegative terms. Definition. The series \( \sum_{k=1}^{\infty} a_k \) converges absolutely if \( \sum_{k=1}^{\infty} |a_k| \) converges. Theorem 7.20. If a series converges absolutely then it converges. This seems like an obvious statement, but its proof is, nevertheless, nontrivial. #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 136 Context: ``` # 130 Power Series ## Proof Suppose \( \sum_{k=1}^{\infty} |\phi_k| \) converges. We first consider the case that each \( b_k \) is real. Let \[ b_k^* = \begin{cases} b_k & \text{if } b_k \geq 0, \\ 0 & \text{otherwise.} \end{cases} \] and \[ b_k^{**} = \begin{cases} 0 & \text{if } b_k < 0, \\ b_k & \text{otherwise.} \end{cases} \] Then \( 0 \leq b_k^* \leq |b_k| \) for all \( k \geq 1 \), and so by Corollary 7.12, both \[ \sum_{k=1}^{\infty} b_k^* \] and \[ -\sum_{k=1}^{\infty} b_k^{**} \] converge. But then so does \[ \sum_{k=1}^{\infty} b_k = \sum_{k=1}^{\infty} b_k^* + \sum_{k=1}^{\infty} b_k^{**}. \] For the general case \( b_k \in \mathbb{C} \), we write each term as \( b_k = c_k + i d_k \). Since \( 0 \leq |c_k| \leq |b_k| \) for all \( k \geq 1 \), Corollary 7.12 implies that \( \sum_{k=1}^{\infty} |c_k| \) converges absolutely, and by an analogous argument, so does \( \sum_{k=1}^{\infty} |d_k| \). But now we can use the first case to deduce that both \( \sum_{k=1}^{\infty} c_k \) and \( \sum_{k=1}^{\infty} d_k \) converge. ### Example 7.21. Continuing Example 7.19, we find: \[ \zeta(s) = \sum_{k=1}^{\infty} \frac{1}{k^s} \] converges for \( \text{Re}(s) > 1 \), because then (using Exercise 3.49) \[ \sum_{k=1}^{\infty} k^{-s} = \sum_{n=1}^{\infty} n^{-\text{Re}(s)} \] converges. Viewed as a function in \( s \), the series \( \zeta(s) \) is the Riemann zeta function, an indispensable tool in number theory and many other areas in mathematics and physics. > **Note:** The Riemann zeta function is the subject of the subtly most famous open problem in mathematics, the **Riemann Hypothesis**. It turns out that \( \zeta(s) \) can be extended to a function that is holomorphic on \( \mathbb{C} \setminus \{1\} \), and the Riemann hypothesis asserts that the roots of this extended function in the strip \( 0 < \text{Re}(s) < 1 \) are all on the critical line \( \text{Re}(s) = \frac{1}{2} \). ``` #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 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: 141 Context: for all \( n \in \mathbb{N} \). \[ \sum_{n=1}^N M_n - \sum_{n=1}^N M_n < \epsilon. \] Thus for all \( z \in G \) and \( n \in \mathbb{N} \): \[ \sum_{k=1}^{\infty} \left( f_k(z) - \sum_{k=1}^N f_k(z) \right) = \left| \sum_{k=1}^{\infty} f_k(z) \right| \leq \sum_{k=1}^N \left| f_k(z) \right| < \epsilon, \] which proves uniform convergence. Replace \( f_k \) with \( |f_k| \) in this argument to see that \( \sum_{k=1}^{\infty} |f_k| \) also converges uniformly. Example 7.29. We revisit Example 7.8 and consider the geometric series \( \sum_{k=1}^{\infty} z^k \) as a series of functions in \( z \). We know from Example 7.8 that this function series converges pointwise for \( |z| < 1 \): \[ \sum_{k=0}^{\infty} z^k = \frac{1}{1-z}. \] To study uniform convergence, we apply Proposition 7.28 with \( f(k) = z^k \). We need a series of upper bounds that converges, so fix a real number \( 0 < r < 1 \) and let \( M_k = M_r^k \). Then \[ |f_k(z)| = |z|^k < r^k \quad \text{for } |z| < r. \] And \( \sum_{k=1}^{\infty} r^k \) converges by Example 7.8. Thus, Proposition 7.28 says that \( \sum_{k=1}^{\infty} z^k \) converges uniformly for \( |z| < r \). We note the subtle distinction of domains for pointwise/uniform convergence: \[ \sum_{k=1}^{\infty} z^k \text{ converges (absolutely) for } |z| < 1, \text{ but to force uniform convergence, we need to shrink the domain to } |z| < r \text{ (arbitrary but fixed)} < 1. \] ### 7.4 Regions of Convergence For the remainder of this chapter (indeed, this book) we concentrate on some very special series of functions. **Definition.** A power series centered at \( z_0 \) is a series of the form \[ \sum_{k=0}^{\infty} c_k (z-z_0)^k \] where \( c_0, c_1, \ldots \in \mathbb{C} \). #################### 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: 144 Context: Corollary 7.33. If \( \lim_{n \to \infty} \sqrt{|z|} \) exists, then the radius of convergence of the series \[ \sum_{n=0}^{\infty} c_n (z-a)^n \] equals \[ R = \begin{cases} \frac{\infty}{\lim_{n \to \infty} \sqrt{|c_n|}} & \text{if } \lim_{n \to \infty} \sqrt{|c_n|} = 0, \\ \text{otherwise}. \end{cases} \] **Proof:** We treat the case that \( R \) is finite and leave the case \( R = \infty \) to Exercise 7.31. Given \( R \) as in the statement of the corollary, it suffices (by Theorem 7.31) to show that \( \sum_{n=0}^{\infty} c_n (z-a)^n \) converges for \( |z-a| < R \) and diverges for \( |z-a| > R \). Suppose \( r = |z-a| < R \). Since \( \lim_{n \to \infty} \sqrt{|c_n|} = \frac{1}{R} \) and \( \sqrt{R} < r \), there exists \( N \) such that \( \sqrt{|c_n|} < \frac{R}{2} \) for those \( n \geq N \). For those \( k \): \[ |c_n (z-a)^n| < |c_n| \left( \sqrt{|z-a|} \right)^n < \left( \sqrt{|c_n|} \right)^n \left( \frac{2R}{R+r} \right)^k \] and so \( \sum_{n=0}^{\infty} (c_n (z-a)^n) \) converges (absolutely) by Proposition 7.28, because \( |z-a| < 1 \) and thus \( \sum \frac{2R}{R+r} \) converges as a geometric series. Thus \( \sum c_n (z-a)^n \) converges. Now suppose \( r = |z-a| > R \). Again, using \( \sqrt{|c_n|} = \frac{1}{R} \) and now \( \sqrt{R} < r \), there exists \( N \) such that \( \sqrt{|c_n|} < \frac{R}{k} \) for \( n \geq N \). For those \( k \): \[ |c_n (z-a)^n| > \left( \sqrt{|c_n|} \right)^n \left( \frac{2R}{R+r} \right)^k > 1, \] and so the sequence \( \sum c_n (z-a)^n \) cannot converge to 0. Subsequently (by Corollary 7.14), \( \sum_{n=0}^{\infty} c_n (z-a)^n \) diverges. You might remember this corollary from calculus, where it goes by the name root test. Its twin sister, the ratio test, is the subject of Exercise 7.32. **Example 7.34.** For the power series \( \sum_{k=0}^{\infty} k z^k \) we compute \[ \lim_{|z| \to 1} \sqrt[k]{k} = \lim_{k \to \infty} \sqrt{k} = e^{\lim_{k \to \infty} \frac{1}{k} \ln(k)} = 1, \] and Corollary 7.33 gives the radius of convergence \( R = 1 \). (Alternatively, we can argue by differentiating the geometric series.) #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 146 Context: # Exercises ## 7.1 For each of the sequences, prove convergence or divergence. If the sequence converges, find the limit. (a) \( a_n = e^{n} \) (b) \( a_n = \frac{1}{n} \) (c) \( a_n = \cos(n) \) (d) \( a_n = \sin\left(\frac{1}{n}\right) \) (e) \( a_n = 2 - \frac{n^2}{2n^2+1} \) ## 7.2 Determine whether each of the following series converges or diverges. (a) \( \sum_{n=2}^{\infty} \left(\frac{1}{n}\right)^{\frac{3}{2}} \) (b) \( \sum_{n=2}^{\infty} \left(\frac{1}{n}\right)^{1} \) (c) \( \sum_{n=2}^{\infty} \left( \frac{1 + 2i}{\sqrt{5}} \right) \) (d) \( \sum_{n=2}^{\infty} \frac{1}{n^2 + 2n} \) ## 7.3 Compute \( \sum_{n=2}^{\infty} \frac{1}{n + 2n} \). ## 7.4 Prove Proposition 7.3. ## 7.5 Prove the following: (a) \( \lim_{n \to \infty} a_n = L \iff \lim_{n \to \infty} |a_n| = |L| \). (b) \( \lim_{n \to \infty} a_n = 0 \implies \lim_{n \to \infty} |a_n| = 0 \). ## 7.6 Show that a convergent sequence is bounded, i.e.: if \( \lim_{n \to \infty} a_n = L \) exists, then there is an \( M \) such that \( |a_n| \leq M \) for all \( n \geq 1 \). ## 7.7 Show that the limit of a convergent sequence is unique. ## 7.8 Let \( (a_n) \) be a sequence. A point \( a \) is an accumulation point of the sequence if for every \( \epsilon > 0 \) and every \( N \in \mathbb{Z}^+ \) there exists some \( n > N \) such that \( |a_n - a| < \epsilon \). Prove that if a sequence has more than one accumulation point then the sequence diverges. #################### 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: 153 Context: # POWER SERIES AND HOLOMORPHIC FUNCTIONS A special case of this result concerns power series with infinite radius of convergence: those represent entire functions. Now that we know that power series are differentiable in their regions of convergence, we can also know how to find their derivatives. The next result says that we can similarly differentiate the series term by term. ## Theorem 8.2 Suppose \( f(z) = \sum_{k=0}^{\infty} c_k (z - z_0)^k \) has radius of convergence \( R > 0 \). Then \[ f'(z) = \sum_{k=1}^{\infty} k c_k (z - z_0)^{k-1} \quad \text{for any } z \in D_{[z_0,R]}. \] The radius of convergence of this power series is also \( R \). **Proof.** If \( z \in D_{[z_0,R]} \) then \( |z - z_0| < R \), so we can choose \( R' \) such that \( |z - z_0| < R' < R \). Then the circle \( \gamma = C(z_0,R') \) lies in \( D_{[z_0,R]} \) and is inside \( S \). Since \( f \) is holomorphic in \( D_{[z_0,R]} \), we can use Cauchy’s Integral Formula for \( f \) (Theorem 5.1), as well as Corollary 3.7: \[ f'(z) = \frac{1}{2 \pi i} \int_{\gamma} \frac{f(w)}{(w - z)^2} \, dw = \frac{1}{2 \pi i} \int_{\gamma} \frac{(w - z)^k}{(w - z)^2} \, dw = \sum_{k=0}^{\infty} k c_k \int_{\gamma} (w - z)^{k-1} \, dw \] Note that we used Theorem 5.1 again in the penultimate step, but now applied to the function \( f(w) \). The last statement of the theorem is easy to show: the radius of convergence of \( f'(z) \) is at least \( R \) since we have shown that the series for \( f' \) converges whenever \( |z - z_0| < R \), and it cannot be larger than \( R \) by comparison to the series for \( f(z) \), since the coefficients for \( (z - z_0)^{k-1} \) are larger than the corresponding ones for \( f(z) \). ## Example 8.3 Let \[ f(z) = \sum_{k=0}^{\infty} \frac{z^k}{k!}. \] #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 156 Context: Proof: Let \( g(z) = f(z + z_0) \), so \( g \) is a function holomorphic in \( D[0, R] \). Given \( z \in D[0, R] \), let \( r = |z| \leq R \). By Cauchy's Integral Formula (Theorem 4.27), \[ g(z) = \frac{1}{2\pi i} \int_{C(z_0, r)} \frac{g(w)}{w - z} \, dw. \] The factor \( \frac{1}{w - z} \) in this integral can be expanded into a geometric series (note that \( w \in C[0, R] \) and \( |z| < 1 \)): \[ \frac{1}{w - z} = \frac{1}{w} \cdot \frac{1}{1 - \frac{z}{w}} = \frac{1}{w} \sum_{k=0}^{\infty} \left(\frac{z}{w}\right)^k \] which converges uniformly in the variable \( w \in C[0, R] \) by Exercise 7.30. Hence Proposition 7.27 applies: \[ g(z) = \frac{1}{2\pi i} \int_{C(z_0, r)} g(w) \left( \sum_{k=0}^{\infty} \left(\frac{z}{w}\right)^k \right) \, dw. \] Now, since \( f(z) = g(z - z_0) \), we apply a change of variables to obtain \[ f(z) = \sum_{k=0}^{\infty} \frac{1}{2\pi i} \int_{C(z_0, r)} \frac{g(w)}{(w - z_0)^{k+1}} \, dw \cdot z^k. \] The only differences of this right-hand side to the statement of the theorem are the paths we’re integrating over. However, by Cauchy's Theorem 4.18, \[ \int_{C[z_0, r]} \frac{f(w)}{(w - z_0)^{k+1}} \, dw = \int_{D[w - z_0, R]} \frac{f(w)}{(w - z_0)^{k+1}} \, dw. \] We note a remarkable feature of our proof: namely, if we have a holomorphic function \( f: G \to \mathbb{C} \) and are interested in expanding \( f \) into a power series centered at \( z_0 \in G \), then we maximize the radius of convergence \( R \) of the power series, in the sense that its region of convergence reaches to the boundary of \( G \). Let’s make this precise. **Definition:** For a region \( G \subset \mathbb{C} \) and a point \( z_0 \in G \), we define the distance of \( z_0 \) to \( \partial G \), as the greatest lower bound of \( |z - z_0| : z \in \partial G \); if this set is empty, we define the distance of \( z_0 \) to \( \partial G \) to be \( \infty \). #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 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: 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: 164 Context: Example 8.23. We’d like to compute the start of a Laurent series for \(\frac{1}{\sin z}\) centered at \(z_0 = 0\). We start by considering the function \(g : D(0, \epsilon) \setminus \{0\} \to \mathbb{C}\) defined by \[ g(z) = \begin{cases} \frac{1}{\sin z} - \frac{2}{z} & \text{if } z \neq 0 \\ \text{undefined} & \text{if } z = 0 \end{cases} \] A quick application of l'Hôpital's Rule (A.11) shows that \(g\) is continuous (see Exercise 8.31). Even better, another round of l'Hôpital's Rule proves that \[ \lim_{z \to 0} \frac{\frac{1}{\sin z} - \frac{1}{z}}{z} = \frac{1}{6}. \] But this means that \[ g'(z) = \begin{cases} \frac{(g(z) - \frac{1}{z})' z + g(z) - \frac{1}{z}}{z^2} & \text{if } z \neq 0 \\ \text{undefined} & \text{if } z = 0 \end{cases} \] In particular, \(g\) is holomorphic in \(D(0, \epsilon)\). By Theorem 8.8, \(g\) has a power series expansion at \(0\), which we may compute using Corollary 8.5. It starts with \[ g(z) = \frac{1}{6}z + \frac{1}{z} + \frac{z^2}{360} + \frac{31}{15120} z^3 + \cdots \] and it converges, by Corollary 8.9, for \(|z| < \pi\). But this gives our sought-after Laurent series \[ \frac{1}{\sin z} = z^{-1} + \frac{1}{6} z + \frac{z^2}{360} + \frac{31}{15120} z^3 + \cdots \] which converges for \(0 < |z| < \pi\. Theorem 8.1 implies that a Laurent series represents a function that is holomorphic in its annulus of convergence. The fact that we can conversely represent any function holomorphic in such an annulus by a Laurent series is the substance of the next result. #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 166 Context: # TAYLOR AND LAURENT SERIES For the integral over \( \gamma_r \), we play exactly the same game as in our proof of Theorem 8.8. The factor \[ \frac{1}{w-z} \] in this integral can be expanded into a geometric series (note that \( w \in \gamma_r \) and \( |z| < 1 \)): \[ \frac{1}{w - z} = \frac{1}{w} \sum_{k=0}^{\infty} \left( \frac{z}{w} \right)^k, \] which converges uniformly in the variable \( z \in \gamma_r \) by Exercise 7.30. Hence Proposition 7.27 applies: \[ \int_{\gamma_r} \frac{g(w)}{w - z} \, dw = \int_{\gamma_r} g(w) \sum_{k=0}^{\infty} \left( \frac{z}{w} \right)^k \, dw = \sum_{k=0}^{\infty} \left( \int_{\gamma_r} \frac{g(w)}{w^{k+1}} \, dw \right) z^k. \] The integral over \( \gamma_r \) is computed in a similar fashion; now we expand the factor \( \frac{1}{z-w} \) into the following geometric series (note that \( w \in \gamma_r \) and so \( |z| < 1 \)): \[ \frac{1}{z - w} = -\frac{1}{1 - \frac{w}{z}} = -\sum_{k=0}^{\infty} \left( \frac{w}{z} \right)^k, \] which converges uniformly in the variable \( w \in \gamma_r \). Again, Proposition 7.27 applies: \[ \int_{\gamma_r} \frac{g(w)}{w^{k+1}} \, dw = - \int_{\gamma_r} g(w) \sum_{k=0}^{\infty} \left( \frac{w}{z} \right)^k \, dw = -\sum_{k=0}^{\infty} \left( \int_{\gamma_r} g(w) \, dw \right) z^{-k-1}. \] Putting everything back into (8.1) gives \[ g(z) = \frac{1}{2 \pi i} \left( \sum_{k=0}^{\infty} \int_{\gamma_r} \frac{g(w)}{w^{k+1}} \, dw \cdot z^k + \sum_{k=0}^{\infty} \int_{\gamma_r} g(w) \, dw \cdot z^{-k-1} \right). \] By Cauchy’s Theorem 4.18, we can now change both \( \gamma_r \) and \( z \) to \( C(0, r) \), as long as \( R_1 < r < R_2 \), which finally gives \[ g(z) = \frac{1}{2 \pi i} \sum_{k=0}^{\infty} \int_{C(0, r)} g(w) w^{-k-1} \, dw \cdot z^k. \] The statement follows now with \( f(z) = g(z - z_0) \) and a change of variables in the integral. #################### 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: 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: 184 Context: ```markdown ## ISOLATED SINGULARITIES AND THE RESIDUE THEOREM (b) We know by Proposition 9.8(b) that the Laurent series at \( z_0 \) looks like \[ f(z) = \sum_{k \geq 0} c_k (z - z_0)^k. \] But then \[ (z - z_0)^j f(z) = \sum_{k \geq j} c_k (z - z_0)^{k - j} \] is a power series, and we can use Taylor's formula (Corollary 8.5) to compute \( c_{-l} \). It is worth noting that we are really coming full circle here; compare this proposition to Cauchy's Integral Formulas (Theorems 4.27 & 4.1 and Corollary 8.11). ### Example 9.12 The integral \[ \frac{\operatorname{exp}(z)}{\sin(z)} \] in Example 8.28 has poles of order 1 at 0 and \( \pi \). We thus compute \[ \operatorname{Res}_{z = 0} \left( \frac{\operatorname{exp}(z)}{\sin(z)} \right) = \lim_{z \to 0} \left( \frac{\operatorname{exp}(z)}{\sin(z)} \right) = \operatorname{exp}(0) \lim_{z \to 0} \frac{z}{\sin(z)} = 1 \] and \[ \operatorname{Res}_{z = \pi} \left( \frac{\operatorname{exp}(z)}{\sin(z)} \right) = \lim_{z \to \pi} \left( (z - \pi) \frac{\operatorname{exp}(z)}{\sin(z)} \right) = \operatorname{exp}(\pi) \lim_{z \to \pi} \frac{z - \pi}{\sin(z)} = -e^{-\pi}, \] confirming our computations in Example 8.28. ### Example 9.13 Revisiting Example 9.9, the function \( f(z) = \frac{1}{z^2} \) has a double pole at \( 0 \) with \[ \operatorname{Res}_{z=0} \left( \frac{\sin(z)}{z^2} \right) = \lim_{z \to 0} \frac{d}{dz} \left( \frac{2 \sin(z)}{z^2} \right) = \lim_{z \to 0} \left( \frac{2 \cos(z) - 2 \sin(z)}{z} \right) = 0, \] after a few iterations of 1'Hôpital's Rule. (In this case, it is simpler to read the residue off the Laurent series in Example 9.9.) ### Proposition 9.14 Suppose \( f \) and \( g \) are holomorphic at \( z_0 \), which is a simple zero of \( g \) (i.e., a zero of multiplicity 1). Then \[ \operatorname{Res}_{z \to z_0} \left( \frac{f(z)}{g(z)} \right) = \frac{f(z_0)}{g'(z_0)}. \] ``` #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 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: 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: 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: 203 Context: # Theorem A.9 (Leibniz's Rule) Suppose \( f \) is continuous on \([a, b] \times [c, d] \subset \mathbb{R}^2\) and the partial derivative \(\frac{\partial f}{\partial y}\) exists and is continuous on \([a, b] \times [c, d]\). Then \[ \frac{d}{dx} \int_c^d f(x, y) \, dy = \int_c^d \frac{\partial f}{\partial x}(x, y) \, dy. \] Leibniz's Rule follows from the Fundamental Theorem of Calculus (Theorem A.3). You can try to prove it, e.g., as follows: Define \( F(x) = \int_c^d f(x, y) \, dy \), get an expression for \( F(x) - F(a) \) as an iterated integral by writing \( F(x) - F(a) \) as the integral of \(\frac{\partial f}{\partial y}\), interchange the order of integrations, and then differentiate using Theorem A.3. # Theorem A.10 (Green's Theorem) Let \( C \) be a positively oriented, piecewise smooth, simple, closed path in \(\mathbb{R}^2\) and let \( D \) be the set bounded by \( C \). If \( f(x, y) \) and \( g(x, y) \) have continuous partial derivatives on an open region containing \( D \) then \[ \int_C f \, dx + g \, dy = \iint_D \left( \frac{\partial g}{\partial x} - \frac{\partial f}{\partial y} \right) \, dx \, dy. \] # Theorem A.11 (L'Hôpital's Rule) Suppose \( I \subset \mathbb{R} \) is an open interval and either \( c \) is in \( I \) or \( c \) is an endpoint of \( I \). Suppose \( f \) and \( g \) are differentiable functions on \( I \setminus \{c\} \) with \( g(x) \) never zero. Suppose \[ \lim_{x \to c} f(x) = 0, \quad \lim_{x \to c} g(x) = 0, \quad \lim_{x \to c} \frac{f(x)}{g(x)} = L. \] Then \[ \lim_{x \to c} f(x) = L. \] There are many extensions of L'Hôpital's rule. In particular, the rule remains true if any of the following changes are made: - \( I \) is infinite. - \( I \) is unbounded and \( c \) is an infinite endpoint of \( I \). #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 204 Context: # Solutions to Selected Exercises Well here's another clue for you all. John Lennon & Paul McCartney ("Glass Onion," *The White Album*) ## 1.1 (a) \( 7 - i \) (b) \( 1 - i \) (c) \( -11 - 2i \) (d) \( -2 + 3i \) ## 1.2 (b) \( \frac{1}{3} - \frac{1}{2} i \) ## 1.3 (a) \( \sqrt{5}, -2 \) (b) \( 5\sqrt{5}, 5 - 10i \) (c) \( \frac{\sqrt{2 - 1} + \sqrt{(2 + \sqrt{9})}}{5} \) (d) \( 8 \) ## 1.4 (a) \( 2^{1/2} i \) (b) \( 2\sqrt{5}i^2 e^{2} \) (c) \( e^{5} \) ## 1.5 (a) \( -1 + i \) (b) \( 34i \) (c) \( -1 \) (d) \( 2 \) ## 1.9 \( e^{i \pi} = -1 \) ## 1.11 (a) \( x = e^{i k}, k = 0, 1, \ldots, 5 \) (b) \( x = 2 e^{i t^{5/4}}, k = 0, 1, 2, 3 \) ## 1.18 \( \cos \frac{\pi}{5} \) and \( \sin \frac{3\pi}{5} \) ## 1.33 For instance: (a) \( r(t) = (1 + i)t + e^{it}, 0 \leq t < 2\pi \). (b) \( r(t) = (2 - i)(1 - i)(2i) + (2)(1 + i + 3i), 0 \leq t \leq 1 \). (c) \( r(t) = 34e^{-i \pi} = 34e^{-i(t)}, 0 \leq t \leq \pi \). ## 2.2 (a) \( 1 + i \) ## 2.18 (a) differentiable and holomorphic in \( C \) with derivative \( e^{-z} e^{z} \) (b) nowhere differentiable or holomorphic (c) differentiable only on \( x + iy \in C : x = y \) with derivative \( 2x \), nowhere #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 205 Context: # Holomorphic Functions (a) nowhere differentiable or holomorphic (b) differentiable and holomorphic in \( \mathbb{C} \) with derivative \(-\sin(x) + i\cos(y)\) (c) nowhere differentiable or holomorphic (d) differentiable only at \(0\) with derivative \(2y - 2xi = -2iz\) (e) differentiable only at \(0\) with derivative \(0\), nowhere holomorphic (f) differentiable only at \(y\) with derivative \(0\), nowhere holomorphic (g) differentiable only at \(0\) with derivative \(0\), nowhere holomorphic (h) differentiable only at \(0\) with derivative \(h\), nowhere holomorphic (i) differentiable only at \(y\) with derivative \(0\), nowhere holomorphic (j) differentiable and holomorphic in \( \mathbb{C} \) with derivative \(2y - 2xi = -2iz\) (k) differentiable only at \(0\) with derivative \(0\), nowhere holomorphic (l) differentiable only at \(0\) with derivative \(0\), nowhere holomorphic ## 2.25 (a) \(2xy\) (b) \(\cos(x)\sinh(y)\) ## 2.26 Neither is defined at the origin, nor continuously extendable to the origin, so neither is harmonic on all of \( \mathbb{C} \). Away from the origin, the first is harmonic, while the second is not. ## 3.14 (a) \(\frac{2x+1}{x-3}\) (b) \(\frac{(i-1)z+1}{iz-2}\) (c) \(\frac{z-1}{z+i}\) ## 3.40 (a) \(\ln(2) + \frac{1}{z}\) (b) \(e^z \cdot \left( c \ln(2) + \frac{1}{z^2} \right)\) ## 3.44 (a) differentiable at \(0\), nowhere holomorphic (b) differentiable and holomorphic on \(C \backslash \{-1, -e^{-1}, e^{-i}\}\) (c) differentiable and holomorphic on \(C \backslash \{x + iy \in \mathbb{C} : x \leq -1, y = 2\}\) (d) nowhere differentiable or holomorphic (e) differentiable and holomorphic on \(C \{x + iy \in \mathbb{C} : x \leq 3, y = 0\}\) (f) differentiable and holomorphic in \( \mathbb{C} \) (i.e. entire) ## 3.45 (a) \(z = e^{z}\) (there is no solution) (b) \(z = \ln z + k\pi i\) for \(k \in \mathbb{Z}\) (c) \(z = \frac{1}{2}z + k i\), where \(k \in \mathbb{Z}\) (d) \(z = \frac{j}{2} + k i\), where \(j \in \mathbb{Z}\) (e) \(z = nk\), where \(n, k \in \mathbb{Z}\) (i.e. \(z = 2i\)) ## 3.50 \(f'(z) = e^{z}\) #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 207 Context: ``` 7.26 (a) \(\sum_{k=0}^{\infty} (-1)^k \frac{z^k}{(2k)!}\) (b) \(\sum_{k=0}^{\infty} (-1)^k \frac{z^{2k}}{(2k)!}\) (c) \(\sum_{k=0}^{\infty} (-1)^k \frac{2^{k-3}}{2k^2 + 1} z^{2k}\) (d) \(\sum_{k=0}^{\infty} (-1)^k \frac{(2^k)(k!)}{(2k)!} z^k\) 7.28 \(\sum_{k=0}^{\infty} (-1)^k (z - 1)^k \) \(\sum_{k=0}^{\infty} \frac{(-1)^k}{(k+1)} \) 7.33 \(\forall \, r < 1, \, |f| = 1\) if \(|z| = 1\), and \(|f| \leq 1\) if \(|z| < 1\) \(\text{(b)} \, |f| \leq 1 \, (d) \, 1\) 7.34 (a) \(e^{z^2} \quad (b) \quad \frac{1}{(z-2)} \quad (c) \quad \frac{2z}{(2-z)}\) 8.1 (a) \(C, \, z \in \mathbb{C} : \, |z| < r\) for any \(r\) (b) \(\{z \in \mathbb{C}: |z-3| > 1\}\), \(\{z \in \mathbb{C} : |z-3| < R\}\) for any \(1 < r < R\) 8.2 \(\sin(z)\) \(z = -3\) \(\implies k = 4\) 8.14 The maximum is \(3\) (attained at \(z = \pm 1\)), and the minimum is \(1\) (attained at \(z = 1\)). 8.17 One Laurent series is \(\sum_{n=0}^{\infty} (-2)^{n}(z-1)^{-2-n}\), converging for \(|z| > 1\). 8.18 The Laurent series is \(\sum_{n=0}^{\infty} (-2)^{n}(z-2)^{-3-n}\), converging for \(|z| > 2\). 8.19 One Laurent series is \(3-(z-1)^n\), converging for \(z \neq 1\). 8.25 \(\sum_{k=0}^{\infty} (-1)^k \frac{z^k}{(2k)!}\) 8.36 (a) One Laurent series is \(\sum_{n=0}^{\infty} \frac{(-1)^k}{(1 + (-2)^{n})}\), converging for \(0 < |z| < 4\). 9.5 \(\frac{2\pi}{3}\) \quad \(\frac{27\pi}{4} - \frac{2\pi}{17} - \frac{\pi}{3} \) \quad \(6\) \quad \(2\pi i\) \quad \(f(0)\) 9.21 (a) \(0\) \quad (b) \(1\) \quad (c) \(4\) ``` #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 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: 212 Context: ``` # Index - **simple**, 16 - **simply connected**, 101 - **sine**, 56 - **singularity**, 169 - **smooth**, 14 - **piecewise**, 72 - **south pole**, 51 - **stereographic projection**, 51 - **tangent**, 56 - **Taylor series expansion**, 148 - **topology**, 12, 88 - **translation**, 45 - **triangle inequality**, 11 - **reverse**, 11 - **Tribonacci numbers**, 191 - **trigonometric functions**, 56 - **trigonometric identities**, 7 - **trivial**, 26 - **uniform convergence**, 131 - **uniqueness theorem**, 154 - **unit circle**, 16 - **unit disk**, 16 - **unit sphere**, 51 - **vector**, 5 - **Weierstrass M-test**, 134 - **Weierstrass convergence theorem**, 163 ``` ########## """QUERY: in 6 sentences can you explain e equals mc squared including history""" Consider the chat history for relevant information. 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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 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 203, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 204, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 205, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 207, 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 212 ================================================== **Elapsed Time: 6.12 seconds** ================================================== FINAL ANSWER Answer: Einstein's equation \(E = mc^2\) is a groundbreaking formula that expresses the equivalence of mass (\(m\)) and energy (\(E\)), indicating that mass can be converted into energy and vice versa. The equation was first introduced in Einstein's 1905 paper on special relativity, where he proposed that the laws of physics are the same for all observers, regardless of their relative motion. In this equation, \(c\) represents the speed of light in a vacuum, approximately \(3 \times 10^8\) meters per second, and it is squared, highlighting the immense amount of energy contained in even a small amount of mass. This relationship implies that a small quantity of mass can yield a tremendous amount of energy, which is the principle behind nuclear reactions, where mass is converted into energy. The equation fundamentally altered our understanding of physics, demonstrating that mass and energy are interchangeable and that they are two forms of the same physical entity. Furthermore, the equation laid the groundwork for modern physics, influencing theories in both quantum mechanics and cosmology, and it remains one of the most famous equations in science, symbolizing the profound connection between mass and energy in the universe. ================================================== **Elapsed Time: 6.02 seconds** ==================================================