{ "query": "in 5 sentences can you explain the life of albert einstein", "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": "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." }, { "role": "user", "content": "in 5 sentences can you explain the life of albert einstein" }, { "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 5 sentences can you explain the life of albert einstein Model: gpt-4o-mini-2024-07-18 Use Curl?: None ================================================== **Elapsed Time: 0.00 seconds** ================================================== ROUTING Query type: summary ================================================== **Elapsed Time: 1.61 seconds** ================================================== RAG PARAMETERS Max Context To Include: 100 Lowest Score to Consider: 0.1 ================================================== **Elapsed Time: 0.14 seconds** ================================================== VECTOR SEARCH ALGORITHM TO USE Use MMR search?: True Use Similarity search?: False ================================================== **Elapsed Time: 0.10 seconds** ================================================== VECTOR SEARCH DONE ================================================== **Elapsed Time: 7.01 seconds** ================================================== PRIMER Primer: You are Simon, a highly intelligent personal assistant in a system called KIOS. You are a chatbot that can read knowledgebases through the "CONTEXT" that is included in the user's chat message. Your role is to act as an expert at summarization and analysis. In your responses to enterprise users, prioritize clarity, trustworthiness, and appropriate formality. Be honest by admitting when a topic falls outside your scope of knowledge, and suggest alternative avenues for obtaining information when necessary. Make effective use of chat history to avoid redundancy and enhance response relevance, continuously adapting to integrate all necessary details in your interactions. Use as much tokens as possible to provide a detailed response. ================================================== **Elapsed Time: 0.18 seconds** ================================================== FINAL QUERY Final Query: CONTEXT: ########## File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 3 Context: # "And what is the use of a book," thought Alice, "without pictures or conversations?" Lewis Carroll (Alice in Wonderland) ## About this book *A First Course in Complex Analysis* was written for a one-semester undergraduate course developed at Binghamton University (SUNY) and San Francisco State University, and has been adopted at several other institutions. For many of our students, Complex Analysis is their first rigorous analysis (if not mathematics) class they take, and this book reflects this very much. We tried to rely on as few concepts from real analysis as possible. In particular, series and sequences are treated from scratch, which has the consequence that power series are introduced late in the course. The goal our book works toward is the Residue Theorem, including some nontraditional applications from both continuous and discrete mathematics. A printed paperback version of this open textbook is available from Orthogonal Publishing (www.orthogonalpublishing.com) or your favorite online bookseller. ## About the authors **Matthias Beck** is a professor in the Mathematics Department at San Francisco State University. His research interests are in geometric combinatorics and analytic number theory. He is the author of three other books, *Computing the Continuous Discretion: Integer-point Enumeration in Polyhedra* (with Sinal Robins, Springer 2007), *The Art of Proof: Basic Training for Mathematics* (with Ross Geoghegan, Springer 2010), and *Combinatorial Reciprocity Theorem: An Invitation to Enumerative Geometric Combinatorics* (with Raman Sanyal, AMS 2018). **Gerald M. Green** is a lecturer in the Department of Mathematical Sciences at Binghamton University (SUNY). **Dennis P. Stinson** is a professor emeritus in the Department of Mathematical Sciences at Binghamton University (SUNY). His research interests are in dynamical systems and formal languages. **Lucas Sabalka** is an applied mathematician at a technology company in Lincoln, Nebraska. He works on 3-dimensional computer vision applications. He was formerly a professor of mathematics at St. Louis University, after postdoctoral positions at UC Davis and Binghamton University (SUNY). His mathematical research interests are in geometric group theory, low dimensional topology, and computational algebra. #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 5 Context: # Contents 1. **Complex Numbers** 1.1. Definitions and Algebraic Properties . . . . . . . . 2 1.2. From Algebra to Geometry and Back . . . . . . . . 5 1.3. Geometric Properties . . . . . . . . . . . . . . . . 9 1.4. Elementary Topology of the Plane . . . . . . . . . 12 Optional Lab . . . . . . . . . . . . . . . . . . . . . . 21 2. **Differentiation** 2.1. Limits and Continuity . . . . . . . . . . . . . . . . 23 2.2. Differentiability and Holomorphicity . . . . . . . 28 2.3. The Cauchy–Riemann Equations . . . . . . . . . . . 32 2.4. Constant Functions . . . . . . . . . . . . . . . . . 36 3. **Examples of Functions** 3.1. Möbius Transformations . . . . . . . . . . . . . . . 43 3.2. Infinity and the Cross Ratio . . . . . . . . . . . 46 3.3. Stereographic Projection . . . . . . . . . . . . . . 48 3.4. Exponential and Trigonometric Functions . . . . . 55 3.5. Logarithms and Complex Exponentials . . . . . . . 59 4. **Integration** 4.1. Definition and Basic Properties . . . . . . . . . . 71 4.2. Antiderivatives . . . . . . . . . . . . . . . . . . 76 4.3. Cauchy’s Theorem . . . . . . . . . . . . . . . . . 80 4.4. Cauchy’s Integral Formula . . . . . . . . . . . . 85 5. **Consequences of Cauchy’s Theorem** 5.1. Variations of a Theme . . . . . . . . . . . . . . . 97 5.2. Antiderivatives Again . . . . . . . . . . . . . . 100 5.3. Taking Cauchy's Formulas to the Limit . . . . . . 102 #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 7 Context: # Chapter 1 ## Complex Numbers Die ganzen Zahlen hat der liebe Gott geschaffen, alles andere ist Menschenwerk. (God created the integers, everything else is made by humans.) Leopold Kronecker (1823–1891) The real numbers have many useful properties. There are operations such as addition, subtraction, and multiplication, as well as division by any nonzero number. There are useful laws that govern these operations, such as the commutative and distributive laws. We can take limits and do calculus, differentiating and integrating functions. But you cannot take a square root of \(-1\); that is, you cannot find a real root of the equation $$ x^2 + 1 = 0. \tag{1.1} $$ Most of you have heard that there is a "new" number that is a root of (1.1); that is, \(x^2 + 1 = 0\) or \(x^2 = -1\). We will show that when the real numbers are enlarged to a new system called the **complex numbers**, which includes \(i\), not only do we gain numbers with interesting properties, but we do not lose many of the nice properties that we had before. The complex numbers, like the real numbers, will have the operations of addition, subtraction, multiplication, as well as division by any complex number except zero. These operations will follow all the laws that we are used to, such as the commutative and distributive laws. We will also be able to take limits and do calculus. And there will be a root of (1.1). As a brief historical aside, complex numbers did not originate with the search for a square root of \(-1\); rather, they were introduced in the context of cubic equations. Scipione del Ferro (1465–1526) and Niccolò Tartaglia (1499–1557) discovered a way to find a root of any cubic polynomial, which was publicized by Gerolamo Cardano (1501–1576) and is often referred to as **Cardano's formula**. For the cubic polynomial \(x^3 + px + q\), Cardano's formula involves the quantity $$ \sqrt[3]{-\frac{q}{2} + \sqrt{\left(\frac{q}{2}\right)^2 + \left(\frac{p}{3}\right)^3}} + \sqrt[3]{-\frac{q}{2} - \sqrt{\left(\frac{q}{2}\right)^2 + \left(\frac{p}{3}\right)^3}}. $$ #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 9 Context: So we can think of the real numbers being embedded in \( \mathbb{C} \) as those complex numbers whose second coordinate is zero. The following result states the algebraic structure that we established with our definitions. ## Proposition 1.1 Let \( (x, y) \) be a field, that is, for all \( (a, b) \in \mathbb{C} \): 1. \( (x, y) + (a, b) \in \mathbb{C} \) (1.4) 2. \( (x, y) + (a, d) + (b, c) = (x, y) + ((a + b), c) \) (1.5) 3. \( (x, y) + (0, 0) = (x, y) \) (1.6) 4. \( (x, y) + (-x, -y) = (0, 0) \) (1.7) 5. \( (a, b) + (c, d) = (x, y) \cdot (a, b) + (c, d) \) (1.8) 6. \( (x, y) \in \mathbb{C} \) (1.9) 7. \( (x, y) \cdot (a, b) = (x \cdot a, y \cdot b) \) (1.10) 8. \( (x, y) \cdot (b, d) = (x \cdot b, (a \cdot b)(c, d)) \) (1.11) 9. \( (x, y) \cdot (a, b) = (x, y) \) (1.12) 10. \( (x, y) \cdot (1, 0) = (x, y) \) (1.13) 11. For all \( (x, y) \in \mathbb{C} \setminus \{(0, 0)\} \): \( (x, y) \cdot \left( \frac{(y, -x)}{x^2 + y^2} \right) = (1, 0) \) (1.14) What we are stating here can be compressed in the language of algebra: equations (1.4)–(1.8) say that \( (\mathbb{C}, +) \) is an Abelian group with identity \( (0, 0) \); equations (1.10)–(1.14) say that \( (\mathbb{C} \setminus \{(0, 0)\}, \cdot) \) is an Abelian group with identity \( (1, 0) \). The proof of Proposition 1.1 is straightforward but nevertheless makes for good practice (Exercise 1.14). We give one sample: **Proof (1.18)**. By our definition for complex addition and properties of additive inverse in \( \mathbb{R} \): \[ (x, y) + (-x, -y) = (x + (-x), y + (-y)) = (0, 0). \] The definition of our multiplication implies the innocent-looking statement: \[ (0, 1) \cdot (1, 0) = (1, 0). \quad (1.15) \] #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 10 Context: ``` # Complex Numbers This identity together with the fact that $$ (x_0, \cdots, x_n) \cdot (x, y) = (ax, az) $$ allows an alternative notation for complex numbers. The latter implies that we can write $$ (x, y) = (x_0, 0) + (0, y) = (x_0, 0) + (0, 1)(0, y). $$ If we think—in the spirit of our remark about embedding R into C—of (x, 0) and (y, 0) as the real numbers x and y, then this means that we can write any complex number (x, y) as a linear combination of (1, 0) and (0, 1), with the real coefficients x and y. Now (1, 0), in turn, can be thought of as the real number 1. So if we give (0, 1) a special name, say i, then the complex number that we used to call (x, y) can be written as $$ x + iy. $$ ### Definition The number x is called the **real part** and y the **imaginary part** of the complex number x + iy, often denoted as Re(x + iy) = x and Im(x + iy) = y. The identity (1.15) then reads $$ i^2 = -1. $$ In fact, much more can now be said with the introduction of the square root of -1. It is not just that (1.1) has a root, but every nonconstant polynomial has roots in C: ### Fundamental Theorem of Algebra (see Theorem 5.11). Every nonconstant polynomial of degree d has d roots (counting multiplicity) in C. The proof of this theorem requires some (important) machinery, so we defer its proof and an extended discussion of it to Chapter 5. We invite you to check that the definitions of our binary operations and Proposition 1.1 are coherent with the usual real arithmetic rules if we think of complex numbers as given in the form x + iy. *The names have historical reasons: people thought of complex numbers as unreal, imagined.* ``` #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 12 Context: # COMPLEX NUMBERS Aside from the exceptional case of 0, for any complex number \( z \), the arguments of \( z \) all differ by a multiple of \( 2\pi \), just as we saw for the example \( z = 1 \). The absolute value of the difference of two vectors has a nice geometric interpretation: ## Proposition 1.2 Let \( z_1, z_2 \in \mathbb{C} \) be two complex numbers, thought of as vectors in \( \mathbb{R}^2 \), and let \( d(z_1, z_2) \) denote the distance between the endpoints of the two vectors in \( \mathbb{R}^2 \) (see Figure 1.2). Then \[ d(z_1, z_2) = |z_1 - z_2| = |z_2 - z_1|. \] **Proof:** Let \( z_1 = x_1 + iy_1 \) and \( z_2 = x_2 + iy_2 \). From geometry, we know that \[ d(z_1, z_2) = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}. \] This is the definition of \( |z_1 - z_2| \). Since \( (x_1 - x_2) = (x_2 - x_1) \) and \( (y_1 - y_2) = (y_2 - y_1) \), this is also equal to \( |z_2 - z_1| \. \qed ![Figure 1.2: Geometry behind the distance between two complex numbers.](path_to_image) That \( |z_1 - z_2| = |z_2 - z_1| \) simply says that the vector from \( z_2 \) to \( z_1 \) has the same length as the vector from \( z_1 \) to \( z_2 \). One reason to introduce the absolute value and argument of a complex number is that they allow us to give a geometric interpretation for the multiplication of two complex numbers. Let’s say we have two complex numbers \( z_1 = x_1 + iy_1 \), with absolute value \( r_1 \) and argument \( \phi_1 \), and \( z_2 = x_2 + iy_2 \), with absolute value \( r_2 \) and argument \( \phi_2 \). #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 13 Context: # FROM ALGEBRA TO GEOMETRY AND BACK ## 7 ϕ₂. This means we can write \(x_1 + iy_1 = (r_1 \cos \phi_1) + i(r_1 \sin \phi_1)\) and \(x_2 + iy_2 = (r_2 \cos \phi_2) + i(r_2 \sin \phi_2)\). To compute the product, we make use of some classic trigonometric identities: \[ (x_1 + iy_1)(x_2 + iy_2) = (r_1 \cos \phi_1 + ir_1 \sin \phi_1)(r_2 \cos \phi_2 + ir_2 \sin \phi_2) \] \[ = r_1 r_2 (\cos \phi_1 \cos \phi_2 - \sin \phi_1 \sin \phi_2) + i(r_1 r_2 \cos \phi_2 \sin \phi_1 + r_1 r_2 \sin \phi_2 \cos \phi_1) \] \[ = r_1 r_2 \left( \cos(\phi_1 + \phi_2) + i \sin(\phi_1 + \phi_2) \right). \] So the absolute value of the product is \(r_1 r_2\) and one of its arguments is \(ϕ_1 + ϕ_2\). Geometrically, we are multiplying the lengths of the two vectors representing our two complex numbers and adding their angles measured with respect to the positive real axis. ![Figure 1.3: Multiplication of complex numbers](#) In view of the above calculation, it should come as no surprise that we will have to deal with quantities of the form \(x + iy\) (where \(y\) is some real number) quite a bit. To save space, bytes, ink, etc., and because "Mathematics is for lazy people" we introduce a shortcut notation and define: \[ e^{i\phi} = \cos \phi + i \sin \phi. \] ^1 You should convince yourself that there is no problem with this fact since there are many possible arguments for complex numbers, as both cosine and sine are periodic functions with period \(2\pi\). ^2 Peter Halton (unpublished articles, Hudson River Undergraduate Mathematics Conference 2000). #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 14 Context: # COMPLEX NUMBERS Figure 1.4 shows three examples. At this point, this exponential notation is indeed ![Figure 1.4: Three sample complex numbers of the form \( e^{x} \).](image-path) purely a notation. We will later see in Chapter 3 that it has an intimate connection to the complex exponential function. For now, we motivate this maybe strange seeming definition by collecting some of its properties: ## Proposition 1.3. For any \( x_1, x_2 \in \mathbb{R} \): 1. \( e^{(x_1 + x_2 i)} = e^{x_1} e^{x_2 i} \) 2. \( e^{0} = 1 \) 3. \( \frac{d}{dx} e^{x} = e^{x} \) 4. \( |e^{x}| = 1 \) 5. \( e^{i\pi} = -1 \) You are encouraged to prove them (Exercise 1.16); again, we give a sample. ## Proof of \( e^{x} \). By definition of \( e^{x} \): \[ \frac{d}{dx} e^{(x + i)} = -\sin{(x)} + \cos{(x)} = i ( \cos{(x)} + i\sin{(x)}) = i e^{(x+i)} \] #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 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: 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: 27 Context: # Elementary Topology of the Plane 21 (c) Determine the boundary points of \( G \). (d) Determine the isolated points of \( G \). ## 1.30 The set \( G \) in the previous exercise can be written in three different ways as the union of two disjoint nonempty separated subsets. Describe them, and in each case say briefly why the subsets are separated. ## 1.31 Show that the union of two regions with nonempty intersection is itself a region. ## 1.32 Show that if \( A \subset B \) and \( B \) is closed, then \( A \subset B \). Similarly, if \( A \subset B \) and \( A \) is open, show that \( A \) is contained in the interior of \( B \). ## 1.33 Find a parameterization for each of the following paths: (a) The circle \( C(1 + i, 1) \), oriented counter-clockwise (b) The line segment from \( -1 - i \) to \( 2i \) (c) The top half of the circle \( C(0, \frac{3}{4}) \), oriented clockwise (d) The rectangle with vertices \( \pm 1 \pm 2i \), oriented counter-clockwise (e) The ellipse \( \{ z \in \mathbb{C} : |z - 1| + |z + 1| = 4 \} \), oriented counter-clockwise ## 1.34 Draw the path parameterized by \[ \gamma(t) = \cos(t) \cdot \cosh(t) + i \cdot \sin(t) \cdot \sinh(t), \quad 0 \leq t \leq 2\pi. \] ## 1.35 Let \( G \) be the annulus determined by the inequalities \( 2 < |z| < 3 \). This is a connected open set. Find the maximum number of horizontal and vertical segments in \( G \) needed to connect two points of \( G \). --- ### Optional Lab Open your favorite web browser and search for the complex function grapher for the open-source software GeoGebra. #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 28 Context: # Complex Numbers 1. Convert the following complex numbers into their polar representation, i.e., give the absolute value and the argument of the number: \[ 3 + 4i = r e^{i\theta} = 2 + 2i = \frac{1}{\sqrt{3 + i}} = \] After you have finished computing these numbers, check your answers with the program. 2. Convert the following complex numbers given in polar representation into their rectangular representation: \[ 2e^{i\theta} = 3e^{i\phi} = \frac{1}{e^{i\varphi}} = -\sqrt{2}e^{2i\phi} = \] After you have finished computing these numbers, check your answers with the program. 3. Pick your favorite five numbers from the ones that you’ve played around with and put them in the tables below, in both rectangular and polar form. Apply the functions listed to your numbers. Think about which representation is more helpful in each instance. | rectangular | polar | |-------------|------------| | z + 1 | | | z + 2 - i | | | 2z | | | -z | | | z^2 | | | Re(z) | | | Im(z) | | | /Im(z) | | | |z| | | | 1/2 | | 4. Play with other examples until you get a feel for these functions. #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 29 Context: # Chapter 2 ## Differentiation Mathematical study and research are very suggestive of generalizing. Whispers make several efforts before he climbed the Mathematics in the 1860s and even then it cut the life short of this party. Now, however, any source can be looked up for a small act, and perhaps one does not appreciate the difficulty of the original account. So in mathematics, it may be found hard to realize the great initial difficulty of making a little step which may seem so natural and obvious, and it may not be surprising if such a step has been found and lost again. **Louis Nodel (1888–1972)** We will now start our study of complex functions. The fundamental concept on which all of calculus is based is that of a limit—it allows us to develop the central properties of continuity and differentiability of functions. Our goal in this chapter is to do the same for complex functions. ### 2.1 Limits and Continuity **Definition.** A (complex) function \( f \) is a map from a subset \( G \subset \mathbb{C} \) to \( \mathbb{C} \); in this situation we will write \( f : G \to \mathbb{C} \) and call \( G \) the domain of \( f \). This means that each element \( z \in G \) gets mapped to exactly one complex number, called the image of \( z \) and usually denoted by \( f(z) \). So far there is nothing that makes complex functions any more special than, say, functions from \( \mathbb{R} \) to \( \mathbb{R} \). In fact, we can construct many familiar looking functions from the standard calculus repertoire, such as \( f(z) = z \) (the identity map), \( f(z) = z^2 + 1 \), or \( f(z) = \frac{1}{z} \). The former three could be defined on all of \( \mathbb{C} \), whereas for the latter we have to exclude the origin \( z = 0 \) from the domain. On the other hand, we could construct some functions that make use of a certain representation of \( z \), for example, \( f(x, y) = x - 2iy \), \( f(x, y) = y - i \), or \( f(z, \varphi) = 2r e^{i\varphi} \). #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 30 Context: Next we define limits of a function. The philosophy of the following definition is not restricted to complex functions, but for sake of simplicity we limit it only for those functions. ## Definition Suppose \( f : G \to \mathbb{C} \) and \( z_0 \) is an accumulation point of \( G \). If \( \epsilon \) is a complex number such that for every \( \delta > 0 \) we can find \( \delta > 0 \) such that \( 0 < |z - z_0| < \delta \), then \[ \lim_{z \to z_0} f(z) = w_0. \] This definition is the same as is found in most calculus texts. The reason we require that \( z_0 \) is an accumulation point of the domain is just that we need to be sure that there are points \( z \) of the domain that are arbitrarily close to \( z_0 \). Just as in the real case, our definition (i.e., the part that says \( 0 < |z - z_0| \)) does not require that \( z_0 \) is in the domain of \( f \); the definition explicitly ignores the value of \( f(z_0) \). ## Example 2.1 Let's prove that \( \lim_{z \to 2} z^2 = 4 \). Given \( \epsilon > 0 \), we need to determine \( \delta > 0 \) such that \( 0 < |z - 2| < \delta \) implies \( |z^2 - 4| < \epsilon \). We rewrite: \[ |z^2 - 4| = |z - 2||z + 2|. \] If we choose \( \delta < 1 \), then the factor \( |z + 2| \) on the right can be bounded by 3 (draw a picture). This means that any \( \delta < \min\{1, \frac{\epsilon}{3}\} \) should do the trick; in this case, \( 0 < |z - 2| < \delta \) implies: \[ |z^2 - 4| < 3 |z - 2| < 3 \delta < \epsilon. \] This proves \( \lim_{z \to 2} z^2 = 4 \). #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 32 Context: ``` # Differentiation Proof of **Theorem**. Assume that \( x_0 \in G \) (otherwise there is nothing to prove), and let \( L = \lim_{x \to x_0} f(x) \) and \( M = \lim_{x \to x_0} g(x) \). Then we know that given \( \epsilon > 0 \), we can find \( \delta_1, \delta_2 > 0 \) such that \[ 0 < |x - x_0| < \delta_1 \implies |f(x) - L| < \frac{\epsilon}{2} \] and \[ 0 < |x - x_0| < \delta_2 \implies |g(x) - M| < \frac{\epsilon}{2}. \] Thus, choosing \( \delta = \min(\delta_1, \delta_2) \), we infer that \( 0 < |x - x_0| < \delta \) implies \[ |f(x) + g(x) - (L + M)| \leq |f(x) - L| + |g(x) - M| < \epsilon. \] Here we used the triangle inequality (Proposition 1.6). This proves that \[ \lim_{x \to x_0} (f(x) + g(x)) = L + M, \] which was our claim. Because the definition of the limit is somewhat elaborate, the following fundamental definition looks almost trivial. ## Definition Suppose \( f : G \to \mathbb{C} \). If \( x_0 \in G \) and either \( x_0 \) is an isolated point of \( G \) or \[ \lim_{x \to x_0} f(x) = f(x_0) \] then \( f \) is continuous at \( x_0 \). More generally, \( f \) is continuous on \( E \subset G \) if it is continuous at every \( x \in E \). However, in almost all proofs using continuity it is necessary to interpret this in terms of \( \epsilon \) and \( \delta \). ## Definition Suppose \( f : G \to \mathbb{C} \) and \( x_0 \in G \). Then \( f \) is continuous at \( x_0 \) if, for every positive real number \( \epsilon \) there is a positive real number \( \delta > 0 \) such that \[ |f(x) - f(x_0)| < \epsilon \quad \text{for all } x \in E \text{ satisfying } |x - x_0| < \delta. \] See Exercise 2.11 for a proof that these definitions are equivalent. ``` #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 33 Context: # LIMITS AND CONTINUITY ## Example 2.5 We already proved (in Example 2.1) that the function \( f : C \to C \) given by \( f(x) = x^2 \) is continuous at \( x = 1 \). You're invited (Exercise 2.8) to extend our proof to show that, in fact, this function is continuous on \( C \). On the other hand, let \( g : C \to C \) be given by \[ g(x) = \begin{cases} \frac{1}{x} & \text{if } x \neq 0 \\ 1 & \text{if } x = 0 \end{cases} \] In Example 2.3 we proved that \( g \) is not continuous at \( x = 0 \). However, this is its only point of discontinuity (Exercise 2.9). Just as in the real case, we can "take the limit inside" a continuous function, by considering composition of functions. ## Definition The image of the function \( g : G \to C \) is the set \( \{ g(x) : x \in G \} \). If the image of \( g \) is contained in the domain of another function \( f : H \to C \), we define the composition \( f \circ g : G \to C \) through \[ (f \circ g)(x) = f(g(x)). \] ## Proposition 2.6 Let \( g : G \to C \) with image contained in \( H \), and let \( f : H \to C \). Suppose \( x_n \) is an accumulation point of \( G \), \( \lim_{n \to \infty} g(x_n) = w_0 \in H \), and \( f \) is continuous at \( w_0 \). Then \[ \lim_{n \to \infty} f(g(x_n)) = f\left(\lim_{n \to \infty} g(x_n)\right), \] in short, \[ \lim_{n \to \infty} f(g(x_n)) = f\left( \lim_{n \to \infty} g(x_n) \right). \] **Proof.** Given \( \epsilon > 0 \), we know there is \( \delta > 0 \) such that \[ |w - w_0| < \delta \implies |f(w) - f(w_0)| < \epsilon. \] For this, we also know there is a \( \eta > 0 \) such that \[ 0 < |x - z_0| < \eta \implies |g(x) - w| < \delta. \] Stringing these two implications together gives that \[ 0 < |x - z_0| < \eta \implies |f(g(x)) - f(w_0)| < \epsilon. \] #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 36 Context: # Differentiation The basic properties for derivatives are similar to those we know from real calculus. In fact, the following rules follow mostly from properties of the limit. ## Proposition 2.10 Suppose \( f \) and \( g \) are differentiable at \( c \in \mathbb{C} \) and \( g \) is differentiable at \( g(c) \). Then: 1. \( (f + g)'(z) = f'(z) + g'(z) \) for any \( z \in \mathbb{C} \) 2. \( (f \cdot g)'(z) = f'(z)g(z) + f(z)g'(z) \) 3. \( \left( \frac{f}{g} \right)'(z) = \frac{f'(z)g(z) - f(z)g'(z)}{(g(z))^2} \) provided that \( g(z) \neq 0 \) 4. \( (z^n)' = n z^{n-1} \) for any nonzero integer \( n \) 5. \( g \) is continuous at \( c \) 6. \( (h \circ g)'(z) = h'(g(z))g'(z) \) **Proof of (2):** \[ (fg)'(z) = \lim_{h \to 0} \frac{f(z + h)g(z + h) - f(z)g(z)}{h} \] \[ = \lim_{h \to 0} \left( f(z + h)g(z + h) - f(z)g(z + h) + f(z)g(z + h) - f(z)g(z) \right) \frac{1}{h} \] \[ = \lim_{h \to 0} \left( \frac{f(z + h) - f(z)}{h}g(z + h) + f(z)\frac{g(z + h) - g(z)}{h} \right) \] \[ = \lim_{h \to 0} \frac{f(z + h) - f(z)}{h} \cdot g(z) + f(z) \cdot \lim_{h \to 0} \frac{g(z + h) - g(z)}{h} \] \[ = f'(z)g(z) + f(z)g'(z) \] Note that we have used the definition of the derivative and Proposition 2.4(a) & (b) (the addition and multiplication rules for limits). A prominent application of the differentiation rules is the composition of a complex function \( f(z) \) with a path \( \gamma(t) \). The proof of the following result gives the reverse. ## Proposition 2.11 Suppose \( f \) is holomorphic at \( c \in \mathbb{C} \) with \( f(\gamma(0)) \) and \( \gamma_1 \) and \( \gamma_2 \) are two smooth paths that pass through \( c \), making an angle of \(\theta\) with each other. Then if \( \gamma_1 \) transforms \( z \) and \( \gamma_2 \) into smooth paths which meet \( f(\gamma) \), the transformed paths make an angle of \( \alpha \) with each other. In words, a holomorphic function with nonzero derivative preserves angles. Functions that preserve angles in this way are called conformal. #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 38 Context: # 2 Differentiation Now define \( z_0 = g(x_0) \) and set \[ \phi(w) = \begin{cases} \frac{f(w) - f(w_0)}{w - w_0} & \text{if } w \neq w_0 \\ f'(w_0) & \text{if } w = w_0 \end{cases} \] This is continuous at \( w_0 \) and \(\lim_{w \to w_0} \phi(w) = f'(w_0)\), so we can apply Proposition 2.6: \[ g'(x_0) = \lim_{h \to 0} \frac{1}{\phi(g(x_0 + h))} \cdot \frac{1}{f'(w_0)} \] ## 2.3 The Cauchy–Riemann Equations When considering a real-valued function \( f: \mathbb{R}^2 \to \mathbb{R} \) of two variables, there is no notion of the derivative of a function. For such a function, we instead only have partial derivatives \( \frac{\partial f}{\partial x}(x_0, y_0) \) and \( \frac{\partial f}{\partial y}(x_0, y_0) \) (and also directional derivatives) which depend on the way in which we approach a point \( (x_0, y_0) \in \mathbb{R}^2 \). For a complex-valued function \( f(z) \), we now have a new concept of the derivative \( f'(z_0) \), which definition cannot depend on the way in which we approach a point \( z_0 = (x_0, y_0) \in \mathbb{C} \). It is logical, then, that there should be a relationship between the complex derivative \( f'(z_0) \) and the partial derivatives \[ \frac{\partial f}{\partial x}(z_0) = \lim_{h \to 0} \frac{f(x_0 + h) - f(x_0, y_0)}{h} \] and \[ \frac{\partial f}{\partial y}(z_0) = \lim_{k \to 0} \frac{f(x_0, y_0 + k) - f(x_0, y_0)}{k} \] (this definition is exactly as in the real-valued case). This relationship between the complex derivative and partial derivatives is very strong, and it is a powerful computational tool. It is described by the Cauchy–Riemann equations, named after Augustin Louis Cauchy (1789–1857) and Georg Friedrich Bernhard Riemann (1826–1866), even though the equations appeared already in the works of Jean le Rond d'Alembert (1717–1783) and Leonhard Euler (1707–1783). #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 41 Context: In the second case, \( \Delta z = i \Delta y \) and \[ f'(z_0) = \lim_{y_0 \to y_0} \frac{f(x_0 + i y) - f(x_0)}{i y} = \lim_{y_0 \to 0} \frac{f(x_0 + i y) - f(x_0)}{y} \] Thus, we have shown that \( f'(z_0) = f_x(z_0) - i f_y(z_0) \). (b) Suppose the Cauchy-Riemann equation (2.2) holds and the partial derivatives \( f_x \) and \( f_y \) are continuous in an open disk centered at \( z_0 \). Our goal is to prove that \( f'(z_0) = f_x(z_0) + i f_y(z_0) \). By (2.2), \[ f(z_0) = x_0 + i y_0 \quad f(z_0 + \Delta z) = f(z_0 + \Delta x + i \Delta y) \] On the other hand, we can rewrite the difference quotient for \( f'(z_0) \) as \[ f(z_0 + \Delta z) - f(z_0) = \frac{f(z_0 + \Delta z) - f(z_0)}{\Delta z} = \frac{f(z_0 + \Delta x + i \Delta y) - f(z_0)}{\Delta z} = \frac{f(z_0 + \Delta x) - f(z_0)}{\Delta z} + \frac{f(z_0 + \Delta y) - f(z_0)}{\Delta z} \] Thus, \[ \lim_{\Delta z \to 0} \frac{f(z_0 + \Delta z) - f(z_0)}{\Delta z} = \lim_{\Delta y \to 0} \frac{f(z_0 + i \Delta y) - f(z_0)}{\Delta y} = \lim_{\Delta x \to 0} \left( \frac{f(z_0 + \Delta x + i \Delta y) - f(z_0)}{\Delta z} \right). \] We claim that both limits on the right-hand side are 0, so we have achieved our set goal. The fractions \( \frac{\Delta y}{\Delta z} \) and \( \frac{\Delta x}{\Delta z} \) are bounded in absolute value by 1, so we just need to see that the differences in parentheses are 0. The second term on the right-hand side of (2.5) has a limit of 0 since, by definition, \[ f(z) = \lim_{\Delta z \to 0} \frac{f(z + \Delta z) - f(z)}{\Delta z} \] and taking the limit here as \( \Delta z \to 0 \) is the same as taking the limit as \( \Delta x \to 0 \). #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 42 Context: ```markdown 36 # Differentiation We cannot do something equivalent for the first term in (2.5), since now both \( \Delta x \) and \( \Delta y \) are involved, and both change as \( \Delta x \to 0 \). Instead, we apply the Mean-Value Theorem A.2 for real functions\(^1\) to the real and imaginary parts \( u(x, y) \) and \( v(x, y) \) of \( f(z) \). Theorem A.2 gives real numbers \( \xi \) and \( \eta \) such that \[ \frac{u(x_0 + \Delta x, y_0 + \Delta y) - u(x_0, y_0)}{\Delta y} = u_y(x_0, y_0 + \eta) \] and \[ \frac{v(x_0 + \Delta x, y_0 + \Delta y) - v(x_0, y_0)}{\Delta y} = v_y(x_0, y_0 + \xi). \] Thus, \[ \frac{f(x_0 + \Delta x + i \Delta y) - f(x_0 + i y)}{\Delta y} = \frac{u(x_0 + \Delta x, y_0 + \Delta y) - u(x_0, y_0)}{\Delta y} - iv_x(x_0, y_0) \] \[ = \left( \frac{u(x_0 + \Delta x, y_0 + \Delta y) - u(x_0, y_0)}{\Delta y} \right) - i v_y(x_0, y_0) + \left( \frac{u(x_0 + \Delta x, y_0 + \Delta y) - u(x_0, y_0)}{\Delta y} - iv_y(x_0, y_0) \right) = u_y(x_0, y_0) - iv_y(x_0, y_0). \tag{2.6} \] Because \( u \) and \( v \) are continuous at \( (x_0, y_0) \), \[ \lim_{\Delta y \to 0} u(x_0 + \Delta x, y_0 + \Delta y) = u(x_0, y_0) \] and \[ \lim_{\Delta y \to 0} v(x_0 + \Delta x, y_0 + \Delta y) = v(x_0, y_0), \] and so (2.6) goes to \( \Delta z \to 0 \), which we set out to prove. ## 2.4 Constant Functions As a simple application of the definition of the derivative of a complex function, we consider functions that have a derivative of 0. In a typical calculus course, one of the first applications of the Mean-Value Theorem for real-valued functions (Theorem A.1) is to constant functions. ``` #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 43 Context: # CONSTANT FUNCTIONS ## Proposition 2.16 If \( I \) is an interval and \( f : I \rightarrow \mathbb{R} \) is a real-valued function with \( f'(x) \) defined and equal to 0 for all \( x \in I \), then there is a constant \( c \in \mathbb{R} \) such that \( f(x) = c \) for all \( x \in I \). **Proof:** The Mean-Value Theorem A.2 says that for any \( x, y \in I \), \[ f(y) - f(x) = f' \left( c \right) (y - x) \] for some \( 0 < a < 1 \). Now \( f' (x + a(y - x)) = 0 \), so the above equation yields \( f(y) = f(x) \). Since this is true for any \( x, y \in I \), the function \( f \) must be constant on \( I \). We do not (yet) have a complex version of the Mean-Value Theorem, and so we will use a different argument to prove that a complex function whose derivative is always 0 must be constant. Our proof of Proposition 2.16 required two key features of the function \( f \), both of which are somewhat obviously necessary. The first is that \( f \) be differentiable everywhere in its domain. In fact, if \( f \) is not differentiable everywhere, we can construct functions that have zero derivative almost everywhere but that have infinitely many values in their image. The second key feature is that the interval \( I \) is connected. It is certainly important for the domain to be connected in both the real and complex cases. For instance, if we define the function \[ f(z) = \begin{cases} 1 & \text{if } \text{Re}(z) \geq 0 \\ 2 & \text{if } \text{Re}(z) < 0 \end{cases} \] then \( f(z) \) is 0 for all \( z \in I \), but \( f \) is not constant. This may seem like a silly example, but it illustrates a pivotal to proving a function is constant that we must be careful of. Recall that a region of \( \mathbb{C} \) is an open connected subset. ## Theorem 2.17 If \( G \subset \mathbb{C} \) is a region and \( f : G \rightarrow \mathbb{C} \) is a complex-valued function with \( f'(z) \) defined and equal to 0 for all \( z \in G \), then \( f \) is constant. **Proof:** We will show that \( f \) is constant along horizontal segments and along vertical segments in \( G \). #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 45 Context: # CONSTANT FUNCTIONS ## 2.6. Proposition 2.2 is useful for showing that limits do not exist, but it is not at all useful for showing that a limit does exist. For example, define $$ f(x,y) = \frac{xy}{x^2 + y^2} \quad \text{where } (x,y) \neq (0,0). $$ Show that the limits of \( f \) along all straight lines through the origin exist and are equal, but \( \lim_{(x,y) \to (0,0)} f(x,y) \) does not exist. (Hint: Consider the limit along the parabola \( y = x^2 \).) ## 2.7. Suppose that \( f(z) = u(x,y) + iv(x,y) \) and \( u_0 = u(x_0,y_0) \) and \( v_0 = v(x_0,y_0) \). Prove that $$ \lim_{(x,y) \to (x_0,y_0)} f(z) = u_0 + iv_0 $$ if and only if $$ \lim_{(x,y) \to (x_0,y_0)} u'(x,y) = u_0 \quad \text{and} \quad \lim_{(x,y) \to (x_0,y_0)} v'(x,y) = v_0. $$ ## 2.8. Show that the function \( f : C \to \mathbb{C} \) given by \( f(z) = z^2 \) is continuous on \( C \). ## 2.9. Show that the function \( g : C \to \mathbb{C} \) given by $$ g(z) = \begin{cases} \frac{1}{z} & \text{if } z \neq 0 \\ 1 & \text{if } z = 0 \end{cases} $$ is continuous on \( C \setminus \{0\} \). ## 2.10. Determine where each of the following functions \( f : C \to C \) is continuous: (a) $$ f(z) = \begin{cases} 0 & \text{if } z = 0 \text{ or } |z| \text{ is irrational} \\ 1 & \text{if } |z| \in \mathbb{Q} \setminus \{0\} \end{cases} $$ (b) $$ f(z) = \begin{cases} 0 & \text{if } z = 0 \\ \sin z & \text{if } z \neq 0 \text{ and } z \neq re^{i\theta} \end{cases} $$ #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 47 Context: ```markdown # Constant Functions ## 2.19 Define \( f(z) = 0 \) if \( \text{Re}(z) - \text{Im}(z) = 0 \), and \( f(z) = 1 \) if \( \text{Re}(z) - \text{Im}(z) \neq 0 \). Show that \( f \) satisfies the Cauchy–Riemann equation (2.2) at \( z = 0 \), yet \( f \) is not differentiable at \( z = 0 \). Why doesn't this contradict Theorem 2.13(b)? ## 2.20 Prove: If \( f \) is holomorphic in the region \( G \subset \mathbb{C} \) and takes real-valued, then \( f' \) is constant in \( G \). (Hint: Use the Cauchy–Riemann equations (2.3) to show that \( f' = 0 \).) ## 2.21 Prove: If \( f(z) \) and \( \overline{f}(z) \) are both holomorphic in the region \( G \subset \mathbb{C} \), then \( f(z) \) is constant in \( G \). ## 2.22 Suppose \( f \) is entire and can be written as \( f(z) = u(x,y) + iv(x,y) \), that is, the real part of \( f \) depends only on \( x \) and the imaginary part of \( f \) depends only on \( y \). Assume that \( f(z) = az + b \) for some \( a \in \mathbb{R} \) and \( b \in \mathbb{C} \). ## 2.23 Suppose \( f \) is entire, with real and imaginary parts \( u \) and \( v \) satisfying \[ u(x,y) v(x,y) = 3 \] for all \( z = x + iy \). Show that \( f \) is constant. ## 2.24 Prove that the Cauchy–Riemann equations take on the following form in polar coordinates: \[ \frac{\partial u}{\partial r} = \frac{1}{r} \frac{\partial v}{\partial \theta} \quad \text{and} \quad \frac{1}{r} \frac{\partial v}{\partial r} = -\frac{\partial u}{\partial \theta}. \] ## 2.25 For each of the following functions, find a function \( f \) such that \( u + iv \) is holomorphic in some region. Maximize that region. (a) \( u(x,y) = x^2 - y^2 \) (b) \( u(x,y) = \cosh(y) \) (c) \( u(x,y) = 2x^2 + x + 1 - 2y^2 \) (d) \( u(x,y) = \frac{xy}{x^2 + y^2} \) ## 2.26 Is \( u(x,y) = \frac{x}{x^2+y^2} \) harmonic on \( \mathbb{C} \)? What about \( v(x,y) = \frac{y}{x^2+y^2} \)? ``` #################### 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: 53 Context: # INFINITY AND THE CROSS RATIO ### Example 3.6 We claim that \(\lim_{z \to \infty} f(z) = \infty\): Given \(M > 0\), let \(\delta = \frac{1}{M}\). Then \(0 < |z| < \delta\) implies \[ |f(z)| = \left| \frac{1}{z} \right| = \frac{1}{|z|} > M. \] ### Example 3.7 Let \(f(z) = \frac{az + b}{cz + d}\) be a Möbius transformation with \(c \neq 0\). Then \(\lim_{z \to \infty} f(z) = \frac{a}{c}\). To simplify the notation, introduce the constant \(L = |az - b| \geq |c| |z| - |d|.\) Given \(\epsilon > 0\), let \(N = \frac{|d|}{|a| - |c|}\). Then \(|z| > N\) implies, with the reverse triangle inequality (Corollary 1.7(b)), \[ |z + d| \geq ||c| z - |d| \geq |c| |z| - |d| \] and so \[ f(z) - \frac{a}{c} = \frac{(az + b) - \frac{a}{c}(cz + d)}{cz + d} = \frac{(ad - bc)}{c(cz + d)} \cdot L < \epsilon. \] We stress that \(\infty\) is not a number in \(\mathbb{C}\), just as \(0\) are not numbers in \(\mathbb{R}\). However, we can extend \(\mathbb{C}\) by adding on \(\infty\) if we are careful. We do so by realizing that we are always talking about a limit when handling \(\infty\). It turns out (Exercise 3.11) that the usual limit rules behave well when we mix complex numbers and zero. For example, if \(\lim_{z \to c} f(z) = L\) and \(\lim_{z \to c} g(z) = \infty\), then the usual limit of \(f(z) + g(z)\) still holds and gives \[ \lim_{z \to c} (f(z) + g(z)) = \infty. \] The following definition captures the philosophy of this paragraph. ### Definition The extended complex plane is the set \(\hat{\mathbb{C}} = \mathbb{C} \cup \{\infty\}\), together with the following algebraic properties: - (a) \(0 + \infty = \infty\) - (b) \(x + \infty = \infty\) for all \(x \in \mathbb{C}\) - (c) \(0 \cdot \infty = 0\) - (d) \(\frac{a}{0} = \infty\) - (e) if \(a \neq 0\) then \(\frac{0}{a} = 0\). The extended complex plane is also called the Riemann sphere or the complex projective line, denoted \(\text{CP}^1\). If a calculation involving \(\infty\) is not covered by the rules above then we must investigate the limit more carefully. For example, it may seem strange that \(0 + \infty\) is defined as \(\infty\). #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 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: 58 Context: # Examples of Functions It is bijective, with inverse map \[ \varphi^{-1}(p,q,0) = \left( \frac{2p}{p^2 + q^2 + 1}, \frac{2q}{p^2 + q^2 + 1}, \frac{p^2 + q^2 - 1}{p^2 + q^2 + 1} \right) \] and \(\varphi^{-1}(\infty) = (0,0,1)\). **Proof:** Given \( P = (x,y,z) \in S^3 \setminus \{N\} \), the straight line through \( N \) and \( P \) is parametrized by \[ r(t) = N + t(P - N) = (0,0,1) + t((x,y,z) - (0,0,1)) = (tx, ty, t(z + 1)). \] where \( t \in \mathbb{R} \). When \( r(t) \) hits \( C \), the third coordinate is \( 0 \), so it must be that \( t = \frac{1}{1-z} \). Plugging this value of \( t \) into the formula for \( r \) yields \( x \) as stated. To see the formula for the inverse map, we begin with a point \( Q = (p,q,0) \in C \) and solve for a point \( P = (x,y,z) \in S^3 \) such that \( \varphi(Q) = P \). The point \( P \) satisfies the equation \( x^2 + y^2 + z^2 = 1 \). The equation \( \varphi(Q) \) tells us that \( z = \frac{p^2 + q^2 - 1}{p^2 + q^2 + 1} \). Thus, we solve three equations for three unknowns. The latter two equations yield: \[ p^2 + q^2 = x^2 + y^2 \cdot \frac{1-x^2}{(1-x^2)} + \frac{1 - z}{1 - z} \] Solving \( p^2 + q^2 = \frac{1}{z^2} \) and then plugging this into the identities \( x = \varphi(1) \) and \( y = \varphi(1) \) proves the desired formula. It is easy to check that \( \varphi \circ \varphi^{-1} \) and \( \varphi^{-1} \circ \varphi \) are both the identity map; see Exercise 3.25. ## Theorem 3.15 The stereographic projection takes the set of circles in \( S^2 \) bijectively to the set of circles in \( C \), where for a circle \( C \subset S^2 \) we have \( \varphi(C) \) that is a line in \( C \) if and only if \( N \in C \). **Proof:** A circle in \( S^2 \) is the intersection of \( S^2 \) with some plane \( H \). If \( (x_0,y_0,z_0) \) is a normal vector for \( H \), then there is a unique point \( k \) so that \( H \) is given by \[ \{(x,y,z) \in \mathbb{R}^3 : (x,y,z) \cdot (x_0,y_0,z_0) = k\}. \] #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 60 Context: # Examples of Functions Project it to the plane by the stereographic projection \( \pi \), apply \( f \) to the point that results, and then pull this point back to \( S^2 \) by \( \pi^{-1} \). We know \( \varphi(x,y,z) = \left( \frac{x}{1-z}, \frac{y}{1-z} \right) \) which we now regard as the complex number \[ \rho + i \gamma = \frac{-x}{1-z} + i \frac{y}{1-z}. \] We know from a previous calculation that \( p^2 + q^2 = \frac{1}{z^2} \), which gives \( x^2 + y^2 = (1 + z)(1 - z) \). Thus, \[ f\left( \frac{x}{1-z} + i \frac{y}{1-z} \right) = \frac{1 - z}{x + iy} = \frac{1 - z}{(1 - z) x + i (1 - z) y} = \frac{1 - 2X - iY}{x^2 + y^2}. \] Rather than plug this result into the formulas for \( \pi^* \), we can just ask what triple of numbers will be mapped to this particular pair using the formulas \( \varphi(x,y,z) = \left( \frac{x}{1-z}, \frac{y}{1-z} \right) \). Thus, we have shown that the effect of \( f \) on \( S^2 \) is to take \( (x,y,z) \) to \( (x, -y, -z) \). This is a rotation around the \( x \)-axis by 180 degrees. We now have a second argument that \( f(z) \) takes circles and lines to circles and lines. A circle in \( C \) is taken to a circle on \( S^2 \) by \( \pi \). Then \( f(z) = \frac{1}{z} \) rotates the sphere which certainly takes circles to circles. Now \( g \) takes circles back to circles and lines. We can also say that the circles that go to lines under \( f(z) = \frac{1}{z} \) are the circles through 0 because 0 is mapped to \( (0, 0) \) under \( \pi \), and so a circle through 0 in \( C \) goes to a circle through the south pole on \( S^2 \). Now 180-degree rotation about the \( x \)-axis takes the south pole to the north pole, and our circle is now passing through \( N \). But we know that \( g \) will take this circle to a line in \( C \). We end by mentioning that there is, in fact, a way of putting the complex metric on \( S^2 \). It is certainly not the (finite) distance function induced by \( \pi \). Indeed, the origin in the complex plane corresponds to the south pole of \( S^2 \). We have to be able to arbitrarily far away from the origin in \( C \), so the complex distance function has to increasingly grow with the \( z \)-coordinate. The closest points are to the north pole \( N \) corresponding to 0 in \( C \), the larger their distance to the origin, and to each other! In this light, a 'line' in the Riemann sphere \( S^2 \) corresponds to a circle in \( S^2 \) through \( N \). In the regular sphere, the circle has finite lengths, but as a line on the Riemann sphere with the complex metric, it has infinite length. #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 62 Context: # EXAMPLES OF FUNCTIONS They are continuous in \( \mathbb{C} \) and satisfy the Cauchy–Riemann equation (2.2): \[ \frac{\partial f}{\partial x} (z) = \frac{\partial f}{\partial y} (z) \] for all \( z \in \mathbb{C} \). Thus Theorem 2.13 says that \( f(z) = \exp(z) \) is entire with derivative \[ f'(z) = \frac{\partial f}{\partial x} (z) = \exp(z). \] We should make sure that the complex exponential function specializes to the real exponential function for real arguments: indeed, if \( z = x \in \mathbb{R} \), then \[ \exp(x) = e^x = \left( \cos(0) + i\sin(0) \right) = e^x. \] The trigonometric functions—sine, cosine, tangent, cotangent, etc.—also have complex analogues; however, they do not play the same prominent role as in the real case. In fact, we can define them as merely being special combinations of the exponential function. ## Definition The (complex) sine and cosine are defined as: \[ \sin z = \frac{1}{2i} \left( \exp(iz) - \exp(-iz) \right) \] and \[ \cos z = \frac{1}{2} \left( \exp(iz) + \exp(-iz) \right), \] respectively. The tangent and cotangent are defined as: \[ \tan z = \frac{\sin z}{\cos z} = \frac{e^{iz} - e^{-iz}}{i(e^{iz} + e^{-iz}) - 1} \] and \[ \cot z = \frac{\cos z}{\sin z} = \frac{e^{iz} + 1}{e^{iz} - e^{-iz}}, \] respectively. Note that to write tangent and cotangent in terms of the exponential function, we used the fact that \( \exp(iy) - \exp(-iy) = 0 \). Because \( \exp \) is entire, so are sine and cosine. #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 63 Context: # Exponential and Trigonometric Functions ![Image properties of the exponential function](path/to/image.png) Figure 3.2: Image properties of the exponential function. As with the exponential function, we should make sure that we're not redefining the real sine and cosine: if \( z \in \mathbb{R} \) then \[ \sin x = \frac{1}{2i} \left( \exp(ix) - \exp(-ix) \right) \] \[ = \frac{1}{2} \left( \cos x + i \sin x - \cos(-x) - i \sin(-x) \right) = \sin x. \] #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 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: 71 Context: # LOGARITHMS AND COMPLEX EXPONENTIALS ### 3.18 Find a Möbius transformation that maps the unit disk to \(\{x + iy \in \mathbb{C} : x + y > 0\}\). ### 3.19 The Jacobian of a transformation \(z = u(x, y)\), \(w = v(x, y)\) is the determinant of the matrix \[ \begin{bmatrix} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} \\ \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y} \end{bmatrix}. \] Show that if \(f = u + iv\) is holomorphic then the Jacobian equals \(|f'(z)|^2\). ### 3.20 Find the fixed points in \(C\) of \(f(z) = \frac{az + b}{cz + d}\). ### 3.21 Find each Möbius transformation \(f\): (a) \(f\) maps \(0 \mapsto 1\), \(1 \mapsto \infty\), \(\infty \mapsto 0\). (b) \(f\) maps \(1 \mapsto -1\), \(-1 \mapsto i\), and \(i \mapsto -1\). (c) \(f\) maps the \(x\)-axis to \(y = x\), the \(y\)-axis to \(y = -x\), and the unit circle to itself. ### 3.22 (a) Find a Möbius transformation that maps the unit circle to \(\{x + iy \in \mathbb{C} : x + y = 0\}\). (b) Find two Möbius transformations that map the unit disk \[ \{z \in \mathbb{C} : |z| < 1\} \] to \(\{x + iy \in \mathbb{C} : x + y > 0\}\) and \(\{x + iy \in \mathbb{C} : x + y < 0\}\), respectively. ### 3.23 Given \(a \in \mathbb{R} \setminus \{0\}\), show that the image of the line \(y = a\) under inversion is the circle with center \(\frac{1}{a}\) and radius \(\frac{1}{|a|}\). ### 3.24 Suppose \(z_1, z_2\), and \(z_3\) are distinct points in \(\mathbb{C}\). Show that \(z\) is on the circle passing through \(z_1, z_2\), and \(z_3\) if and only if \([z, z_1, z_2, z_3]\) is real or \(\infty\). #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 72 Context: # EXAMPLES OF FUNCTIONS ## 3.25 Prove that the stereographic projection of Proposition 3.14 is a bijection by verifying that \( \phi \circ \psi^{-1} \) and \( \psi \circ \phi^{-1} \) are the identity map. ## 3.26 Find the image of the following points under the stereographic projection \( \phi \): - \( (0, 0, -1) \) - \( (0, 0, 1) \) - \( (1, 0, 0) \) - \( (1, 1, 0) \) - \( (1, 1, 1) \) ## 3.27 Consider the plane \( H \) determined by \( x + y - z = 0 \). What is a unit normal vector to \( H \)? Compute the image of \( H \cap S^2 \) under the stereographic projection \( \phi \). ## 3.28 Prove that every circle in the extended complex plane \( \mathbb{C} \) is the image of some circle in \( S^2 \) under the stereographic projection \( \phi \). ## 3.29 Describe the effect of the basic Möbius transformations: rotation, real dilation, and translation on the Riemann sphere. *Hint:* For the first two, consider all circles in \( S^2 \) centered on the \( N \) axis, and all circles through both \( N \) and \( S \). For translation, consider two families of circles through \( N \), orthogonal to and perpendicular to the translation. ## 3.30 Show that \( \sin(i) = \sin(E) \) and \( \cos(i) = \cos(E) \). ## 3.31 Let \( z = x + iy \) and show that: (a) \( \sin z = \sin x \cosh y + i \cos x \sinh y \). (b) \( \cos z = \cos x \cosh y - i \sin x \sinh y \). ## 3.32 Prove that the zeros of \( \sin z \) are all real valued. Conclude that they are precisely the integer multiples of \( \pi \). ## 3.33 Describe the images of the following sets under the exponential function \( \exp(z) \): (a) The line segment defined by \( z = 1 + iy, \, 0 \leq y \leq 2\pi \) (b) The line segment defined by \( z = i + x, \, 0 \leq x \leq 2\pi \) (c) The rectangle defined by \( z = x + iy \in \mathbb{C}: \, 0 \leq x \leq 1, \, 0 \leq y \leq 2\pi \) #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 74 Context: # EXAMPLES OF FUNCTIONS ## 3.41. Convert the following expressions to the form \( z = x + iy \). (Reason carefully.) 1. \( e^z \) 2. \( e^{\cos(\log(3 + 4i))} \) 3. \( i^z \) 4. \( e^{i \pi} \) 5. \( e^{z} \) 6. \( e^{(1 + i) z} \) 7. \( \sqrt{3(1 - i)} \) 8. \( \left( \frac{1}{1} \right)^4 \) ## 3.42. Is \( \arg(E) = -\arg(z) \) true for the multiple-valued argument? What about \( \arg(E) = \Arg(z) \) for the principal branch? ## 3.43. For the multiple-valued logarithm, is there a difference between the set of all values of \( \log(z^2) \) and the set of all values of \( 2 \log(z) \)? (Hint: Try some fixed numbers for \( z \).) ## 3.44. For each of the following functions, determine all complex numbers for which the function is holomorphic. If you run into a logarithm, use the principal value unless otherwise stated. (a) \( \frac{z^2}{2} \) (b) \( \frac{1}{z^2} \) (c) \( \Log(z - 2i + 1) \) where \( \Log(z) = \ln |z| + i \arg(z) \) with \( 0 < \arg(z) < 2\pi \) (d) \( \exp(z) \) (e) \( z - 3i \) (f) \( i^z \) ## 3.45. Find all solutions to the following equations: (a) \( \Log(z) = \frac{1}{2} \) \( \cos(z) = 0 \) (b) \( \Log(z) = \frac{1}{2} \) (c) \( \exp(z) = \pi \) (d) \( \sin(z) = \cos(4) \) (e) \( \exp(z) = \exp(\sqrt{E}) \) (f) \( z = 1 + i \) #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 75 Context: # Logarithms and Complex Exponentials ## 3.46 Find the image of the annulus \( 1 < |z| < u \) under the principal value of the logarithm. ## 3.47 Use Exercise 2.24 to give an alternative proof that \( \log \) is holomorphic in \( \mathbb{C} \setminus \{0\} \). ## 3.48 Let \( \log_G \) be a branch of the logarithm on \( G \), and let \( H = \{ e^{g(z)} : z \in G \} \), the image of \( \log_G \). Show that \( \log_G : G \to H \) is a bijection whose inverse map is \( f(z) : H \to G \) given by \( f(z) = \exp(z) \) (i.e., \( f \) is the exponential function restricted to \( H \)). ## 3.49 Show that \( |z| = e^{\nu} \) if \( \nu \) is a positive real constant. ## 3.50 Let \( z \in \mathbb{C} \setminus \{0\} \). Find the derivative of \( f(z) = z^t \). ## 3.51 Prove that \( \exp(b \log(z)) \) is single valued if and only if \( b \) is an integer. (Note that this means that complex exponentials do not clash with monomials \( z^n \), no matter which branch of the logarithm is used.) What can you say if \( b \) is rational? ## 3.52 Describe the image under \( f \) of the line with equation \( y = x \). To do this you should find an equation (at least parametrically) for the image (you can start with the parameter form \( x = r, y = r \)), plot it reasonably carefully, and explain what happens in the limits as \( r \to -\infty \) and \( r \to \infty \). ## 3.53 For this problem, \( f(z) = z^2 \). ### (a) Show that the image under \( f \) of a circle centered at the origin is a circle centered at the origin. ### (b) Show that the image under \( f \) of a ray starting at the origin is a ray starting at the origin. ### (c) Let \( T \) be the figure formed by the horizontal segment from \( 0 \) to \( 2 \), the arc from \( 2 \) to \( 2i \), and then the vertical segment from \( 2i \) to \( 0 \). Draw \( f(T) \). ### (d) Is the right angle at the origin in part (c) preserved? Is something wrong here? (Hint: Use polar coordinates.) #################### 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: 89 Context: ```markdown # CAUCHY'S THEOREM For \( 0 \leq s \leq 1 \), let \( \gamma \) be the path parametrized by \( h(t, s) \), \( 0 \leq t \leq 1 \). Consider the function \( I : [0, 1] \to \mathbb{C} \) given by \[ I(s) := \int_{\gamma} f. \] Thus, \( I(0) = \int_{\gamma(0)} f \) and \( I(1) = \int_{\gamma(1)} f \). We will show that \( I \) is constant; in particular, \( I(0) = I(1) \), which proves the theorem. By Leibniz rule (Theorem A.9), \[ \frac{d}{ds} I(s) = \int_{\gamma(h(t,s))} f \frac{\partial h}{\partial s} \, dt = \int_{0}^{1} f\left(h(t, s)\right) \frac{\partial h(t, s)}{\partial s} \, dt \] \[ = \int_{0}^{1} \left( f\left(h(t,s)\right) \frac{\partial h(t,s)}{\partial s}\right) \, dt. \] Note that we used Theorem A.7 to switch the order of the second partials in the penultimate step—here is where we need our assumption that \( h \) has continuous second partials. Also, we needed continuity of \( f \) in order to apply Leibniz's rule. If \( b \) is piecewise defined, we split up the integral accordingly. Finally, by the Fundamental Theorem of Calculus (Theorem A.3), applied separately to the real and imaginary parts of the above integral, \[ \frac{d}{ds} I(s) = \int_{\gamma(h(t, s))} f \frac{\partial h}{\partial s} \, dt = f\left(h(t,s)\right) \bigg|_{t=0}^{t=1} = f(h(1,s)) - f(h(0,s)) \frac{\partial h}{\partial s} = 0, \] where the last step follows from \( h(0,s) = h(1,s) \) for all \( s \). **Definition.** Let \( G \subset \mathbb{C} \) be a region. If the closed path \( \gamma \) is \( G \)-homotopic to a point (that is, a constant path) then it is \( G \)-contractible, and we write \( \gamma \simeq_G \). (See Figure 4.2 for an example.) The fact that an integral over a point is zero has the following immediate consequence. ``` #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 92 Context: # INTEGRATION Corollary 4.25. If \( f \) is holomorphic in an open set containing \( D[w, R] \), then \[ f(w) = \frac{1}{2\pi i} \int_{\mathcal{C}[w, R]} (f + R e^{i\theta}) \, d\theta, \] \[ u(w) = \frac{1}{2\pi i} \int_{0}^{2\pi} (w + R e^{i\theta}) \, d\theta \] and \[ v(w) = \frac{1}{2\pi i} \int_{0}^{2\pi} (w + R e^{i\theta}) \, d\theta. \] **Proof of Theorem 4.24 and Corollary 4.25.** By assumption, \( f \) is holomorphic in an open set \( G \) that contains \( D[w, R] \), and so \( f(z) \) is holomorphic in \( H = G \setminus \{w\} \). For any \( 0 < r < R \), \[ C[w, r] \mapsto C[w, R]. \] And so Cauchy's Theorem 4.18 and Exercise 4.4 give \[ \left| \frac{f(z)}{C[w, R]} \right|_{C[w, R]} = \frac{1}{2\pi i} \int_{C[w, R]} \frac{f(z) \, dx}{C[w, R] - z} \] \[ = \frac{1}{2\pi i} \int_{C[w, R]} \frac{f(z) - f(w)}{C[w, R] - z} \, dz \] \[ \leq S_{\max} \, \text{within} \, C[w, R] \quad (4.6) \] \[ \leq \max_{z \in C[w, R]} |f(z) - f(w)| \cdot \text{length}(C[w, r]) \cdot \frac{1}{2\pi R}. \] Here the inequality comes from Proposition 4.6(4). Now let \( \epsilon > 0 \). Because \( f \) is continuous at \( w \), there exists \( \delta > 0 \) such that \( |z - w| < \delta \) implies \[ |f(z) - f(w)| < \frac{\epsilon}{2\pi R}. \] #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 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: 102 Context: ``` 4.38 Let \( f(z) = \frac{1}{z} \) and define the two paths \( \gamma = C_{1.1} \) oriented counterclockwise and \( a = C_{1,-1} \) oriented clockwise. Show that \( f \circ \gamma = \int_{\gamma} f \, dz \) even though \( \gamma \) goes through \( G = \mathbb{C} \setminus \{ 0 \} \), the region of holomorphicity of \( f \). 4.39 This exercise gives an alternative proof of Cauchy's Integral Formula (Theorem 4.27) that does not depend on Cauchy's Theorem (Theorem 4.18). Suppose the region \( G \) is convex; this means that, whenever \( u \) and \( w \) are in \( G \), the line segment between them is also in \( G \). Suppose \( f \) is holomorphic in \( G \), \( f' \) is continuous, and \( \gamma \) is a positively oriented, simple, closed, piecewise smooth path, such that \( w \) is inside \( \gamma \) and \( \gamma \) is simple. (a) Consider the function \( g: [0,1] \to \mathbb{C} \) given by \[ g(t) = \int_{\gamma} \frac{f(u + t(u - w))}{z - w} \, dz. \] Show that \( g'(t) = 0 \). (Hint: Use Theorem A.9 (Leibniz's rule) and then find an antiderivative for \( \frac{f(u + t(u - w))}{z - w} \).) (b) Prove Theorem 4.27 by evaluating \( g(0) \) and \( g(1) \). (c) Why did we assume \( G \) is convex? ``` #################### 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: 107 Context: Proof: Theorem 4.15 yields an antiderivative \( F \) for \( f \) in \( G \). Because \( F \) is holomorphic in \( G \), Corollary 5.5 implies that \( f \) is also holomorphic in \( G \). Just like there are several variations of Theorem 4.15, we have variations of Corollary 5.6. For example, by Corollary 4.16, we can replace the condition for all piecewise smooth closed paths \( \gamma \) in the statement of Corollary 5.6 by the condition for every closed path in \( G \) (which, in fact, gives a stronger version of this result). A special case of Theorem 4.15 applies to regions in which every closed path is contractible. ### Definition A region \( G \subset \mathbb{C} \) is **simply connected** if \( \gamma \) is contractible for every closed path \( \gamma \) in \( G \). ### Example 5.7 Any disk \( D_r \) is simply connected, as is \( \mathbb{C} \setminus \{0\} \). (You should draw a few closed paths in \( \mathbb{C} \setminus \{0\} \) to convince yourself that they are all contractible.) The region \( \{0\} \) is not simply connected, as e.g., the unit circle is not \( \{(0)\} \)-contractible. If \( f \) is holomorphic in a simply-connected region then Corollary 4.20 implies that \( f \) satisfies the conditions of Theorem 4.15, hence we conclude: ### Corollary 5.8 Every holomorphic function on a simply-connected region \( G \subset \mathbb{C} \) has an antiderivative on \( G \). Note that this corollary gives us no indication of how to compute an antiderivative. For example, it says that the (entire) function \( f : \mathbb{C} \to \mathbb{C} \) given by \( f(z) = \exp(z^2) \) has an antiderivative \( F \) in \( C \); it is an entirely different matter to derive a formula for \( F \). Corollary 5.8 also illustrates the role played by two of the regions in Example 5.7, in connection with the function \( f(z) = \frac{1}{z} \). This function has no antiderivative on \( \mathbb{C} \setminus \{0\} \), as was proved in Example 4.14. Consequently, since \( \mathbb{C} \setminus \{0\} \) is not simply connected, however, the function \( f(z) = \frac{1}{z} \) does not illustrate the instance implied by Corollary 5.8. Finally, Corollary 5.8 implies that, if we have two paths in a simply-connected region with the same endpoints, we can concatenate them—changing direction on one—to form a closed path, which proves: #################### 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: 114 Context: 5.12. Show that a polynomial of odd degree with real coefficients must have a real zero. *(Hint: Use Exercise 1.24.)* 5.13. Suppose \( f \) is entire and \( |f(z)| \leq \sqrt{|z|} \) for all \( z \in \mathbb{C} \). Prove that \( f \) is identically 0. *(Hint: Show first that \( f \) is constant.)* 5.14. Suppose \( f \) is entire and there exists \( M > 0 \) such that \( |f(z)| \leq M \) for all \( z \in \mathbb{C} \). Prove that \( f \) is constant. 5.15. Suppose \( f \) is entire with bounded real part, i.e., writing \( f(z) = u(z) + iv(z) \), there exists \( M > 0 \) such that \( |u(z)| \leq M \) for all \( z \in \mathbb{C} \). Prove that \( f \) is constant. *(Hint: Consider the function \( \exp(f(z)) \).)* 5.16. Suppose \( f \) is entire and there exist constants \( a \) and \( b \) such that \( |f(z)| \leq |z|^a + b \) for all \( z \in \mathbb{C} \). Prove that \( f \) is polynomial of degree at most 1. *(Hint: Use Theorem 5.1 and Exercise 2.29.)* 5.17. Suppose \( f: D(0, 1) \to D(0, 1) \) is holomorphic. Prove that for \( |z| < 1 \), \[ |f'(z)| \leq \frac{1}{1 - |z|}. \] 5.18. Compute \[ \int_0^{\infty} \frac{dx}{x^4 + 1}. \] 5.19. In this problem \( f(z) = \frac{e^{iz}}{e^{iz} + 1} \) and \( R > 1 \). Modify our computations in Example 5.14 as follows: (a) Show that \( f^* = f \) where \( f^* \) is again (as in Figure 5.2) the counterclockwise semicircle formed by the segment \([-R, R]\) on the real axis, followed by the circular arc \( \gamma_R \) of radius \( R \) in the upper half plane from \( R \) to \(-R\). (b) Show that \( |f(z)| \leq \frac{1}{\sqrt{2}} \) for \( z \) in the upper half plane, and conclude that \( \lim_{|z| \to \infty} f(z) = 0 \) and hence \( \lim_{R \to \infty} f_{R} = 0 \). #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 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: 121 Context: Theorem 6.10. Suppose \( u \) is harmonic in the region \( G \) and \( \overline{D(u, r)} \subset G \). Then \[ u(x) = \frac{1}{2\pi} \int_{D(u,r)} u(y + r e^{i\theta}) d\theta. \] **Proof.** Exercise 6.14 provides \( x \in D(u,r) \subset D[u,r] \subset G \). The open disk \( D[u,r] \) is simply connected, so by Theorem 6.6 there is a function \( f \) holomorphic in \( D[u,r] \) such that \( u = \operatorname{Re} f \) on \( D[u,r] \). Now we apply Corollary 4.25 to \( f \): \[ f(x) = \frac{1}{2\pi} \int_{0}^{2\pi} f(x + r e^{i\theta}) d\theta. \] Theorem 6.10 follows by taking the real part on both sides. **Corollary** 4.25 and Theorem 6.10 say that holomorphic and harmonic functions have the mean-value property. Our next result is an important consequence of this property to extreme values of a function. **Definition.** Let \( G \) be a region. The function \( u : G \to \mathbb{R} \) has a strong relative maximum at \( x_0 \in G \) if there exists a disk \( D[u, r] \subset G \) such that \( u(x) \leq u(x_0) \) for all \( x \in D[u, r] \) and \( u(x_0) < u(x) \) for some \( y \in D[u, r] \). The definition of a strong relative minimum is analogous. Theorem 6.11. If \( u \) is harmonic in the region \( G \), then it does not have a strong relative maximum or minimum in \( G \). **Proof.** Assume, by way of contradiction, that \( u \) has a strong relative maximum. Then there is a disk in \( G \) centered at \( x_0 \) containing a point \( x \) with \( u(x) < u(x_0) \). Let \( r = |x - x_0| \) and apply Theorem 6.10: \[ u(x) = \frac{1}{2\pi} \int_{0}^{2\pi} u(x_0 + r e^{i\theta}) d\theta. \] Intuitively, this cannot hold, because some of the function values we're integrating are smaller than \( u(x_0) \), contradicting the mean-value property. To make this into a thorough argument, suppose that \( u(x) < u(x_0) \) for \( r \leq 5 \) and \( x \in [0, 2r] \). Now we split up the mean-value... #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 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: 133 Context: # SERIES ## 127 form a nondecreasing sequence if the terms of the series are nonnegative, and this observation immediately yields the following: ### Corollary 7.10. If \( b_k \in \mathbb{R}^+ \) then \( \sum_{k=1}^{\infty} b_k \) converges if and only if the partial sums are bounded. ### Example 7.11. With this new terminology, we can revisit Example 7.4: Let \( b_k = \frac{1}{k} \). In Example 7.4 we showed that the partial sums \[ \sum_{k=1}^{n} b_k = \sum_{k=1}^{n} \frac{1}{k} \] are bounded, and \( \sum_{k=1}^{\infty} \frac{1}{k} = c - 1 \). Although Corollary 7.10 is a mere direct consequence of the completeness property of \( \mathbb{R} \), it is surprisingly useful. Here is one application, sometimes called the Comparison Test: ### Corollary 7.12. If \( b_k \geq c_k \geq 0 \) for all \( k \geq 1 \) and \( \sum_{k=1}^{\infty} b_k \) converges then so does \( \sum_{k=1}^{\infty} c_k \). **Proof.** By Corollary 7.10, the partial sums \( \sum_{k=1}^{n} b_k \) are bounded, and thus so are \[ \sum_{k=1}^{n} c_k \leq \sum_{k=1}^{n} b_k. \] But this means, again by Corollary 7.10, that \( \sum_{k=1}^{\infty} c_k \) converges. ### Proposition 7.13. If \( \sum_{k=1}^{\infty} b_k \) converges then \( \lim_{k \to \infty} b_k = 0 \). The contrapositive of this proposition is often called the Test for Divergence: ### Corollary 7.14. If \( \lim_{k \to \infty} b_k \neq 0 \) or \( \lim_{k \to \infty} b_k = \infty \), then \( \sum_{k=1}^{\infty} b_k \) diverges. ### Example 7.15. Continuing Example 7.8, for \( |z| \geq 1 \) the geometric series \( \sum_{k=1}^{\infty} z^k \) diverges since in this case \( \lim_{k \to \infty} z^k \) either does not exist or is not 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: 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: 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: 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: 160 Context: # TAYLOR AND LAURENT SERIES Thus we can define a function \( g : G \rightarrow \mathbb{C} \) through \( g(z) = \begin{cases} \sum_{n=0}^{\infty} a_n (z - a)^n & \text{if } z \in D(a, R), \\ f(z) & \text{if } z \in G \setminus \{ a \}. \end{cases} \) (According to our calculations above, the two definitions give the same value when \( z \in D(a, R) \). The function \( g \) is holomorphic in \( D(a, R) \) by the first definition, and \( g \) is holomorphic in \( G \setminus \{ a \} \) by the second definition. Note that \( g(a) = c \neq 0 \) and, by construction, \( f(z) = (z - a)^n s(z) \quad \text{for all } z \in G. \) Since \( s(a) \neq 0 \) by continuity, \( r > 0 \) so that \( s(z) \neq 0 \) for all \( z \in D(a, r) \), so \( D(a, r) \) contains no other zero of \( f \). The integer \( r \) is unique, since it is defined in terms of the power series expansion of \( f \) at \( a \), which is unique by Corollary 8.6. Theorem 8.14 gives rise to the following result, which is sometimes called the identity principle or the uniqueness theorem. ## Theorem 8.15 Suppose \( G \) is a region, \( f : G \rightarrow \mathbb{C} \) is holomorphic, and \( f(a) = 0 \) where \( (a_k) \) is a sequence of distinct numbers that converges in \( G \). Then \( f \) is the zero function on \( G \). Applying this theorem to the difference of two functions immediately gives the following variant. ## Corollary 8.16 Suppose \( f \) and \( g \) are holomorphic in a region \( G \) and \( f(a) = g(a) = 0 \) at a sequence that converges to \( a \in G \) with \( a_k \neq a \) for all \( k \). Then \( f(z) = g(z) \) for all \( z \in G \). ### Proof of Theorem 8.15 Consider the following two subsets of \( G \): - \( X := \{ z \in G : \text{ there exists } z \text{ such that } F(z) = 0 \text{ for all } z \in D(a, r) \} \) - \( Y := \{ z \in G : \text{ there exists } z \text{ such that } F(z) \neq 0 \text{ for all } z \in D(a, r) \setminus \{ a \} \} \) If \( f(a) \neq 0 \) then, by continuity of \( f \), there exists a disk centered at \( a \) in which \( f \) is nonzero, and so \( a \in Y \). #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 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: 165 Context: # LAURENT SERIES ## Theorem 8.24 Suppose \( f \) is a function that is holomorphic in \( A = \{ z \in \mathbb{C} : R_1 < |z - z_0| < R_2 \} \). Then \( f \) can be represented in \( A \) as a Laurent series centered at \( z_0 \): \[ f(z) = \sum_{n=-\infty}^{\infty} c_n (z - z_0)^n \] with \[ c_n = \frac{1}{2 \pi i} \int_{C_{r_1}} \frac{f(w)}{(w - z_0)^{n+1}} \, dw, \] where \( R_1 < r < R_2 \). By Cauchy's Theorem 4.18, we can replace the circle \( C_{r_1} \) in the formula for the Laurent coefficients by any path \( \gamma \subset A [C_{r_1}, r_2] \). ![Figure 8.1: The path γ in our proof of Theorem 8.24.](path_to_figure) ## Proof Let \( g(z) = f(z + z_0) \), so \( g \) is a function holomorphic in \( \{ z \in \mathbb{C} : R_1 < |z| < R_2 \} \). Fix \( R_1 < r_1 < r_2 < R_2 \), and let \( \gamma \) be the path in Figure 8.1, where \( \gamma_1 = C[r_1, r_2] \) and \( \gamma_2 = C[0, r_2] \). By Cauchy's Integral Formula (Theorem 4.27), \[ g(z) = \frac{1}{2 \pi i} \int_{\gamma} \frac{g(w)}{w - z} \, dw. \tag{8.1} \] Complete any missing details from the original image as necessary while ensuring the accuracy of the markup. #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 170 Context: 8.11. Suppose \( G \) is a region and \( f : G \to \mathbb{C} \) is holomorphic. Prove that the sets - \( X = \{ a \in G : \text{there exists } r \text{ such that } f(a) = 0 \text{ for all } z \in D(a, r) \} \) - \( Y = \{ a \in G : \text{there exists } r \text{ such that } f(z) \neq 0 \text{ for all } z \in D(a, r) \} \) in our proof of Theorem 8.15 are open. 8.12. Prove the Minimum-Modulus Theorem (Corollary 8.19): Suppose \( f \) is holomorphic and nonconstant in a region \( G \). Then \( |f| \) does not attain a weak relative minimum at a point \( a \in G \) unless \( f(a) = 0 \). 8.13. Prove Corollary 8.20: Assume that \( u \) is harmonic in a region \( G \) and has a weak local maximum at \( a \in G \). (a) If \( G \) is simply connected then apply Theorem 8.17 to \( u = \operatorname{Re}(f) \) where \( f \) is a harmonic conjugate of \( u \). Conclude that \( u \) is constant on \( G \). (b) If \( G \) is not simply connected, then the above argument applies to \( u \) on any disk \( D(a, r) \cap G \). Conclude that the partials \( u_x \) and \( u_y \) are zero on \( G \), and adapt the argument of Theorem 2.17 to show that \( u \) is constant. 8.14. Let \( C : \mathbb{C} \to \mathbb{C} \) be given by \( C(z) = z^2 - 2 \). Find the maximum and minimum of \( |f(z)| \) on the closed unit disk. 8.15. Give another proof of the Fundamental Theorem of Algebra (Theorem 5.11), using the Minimum-Modulus Theorem (Corollary 8.19). (Hint: Use Proposition 5.10 to show that a polynomial does not achieve its minimum modulus on a large circle; then use the Minimum-Modulus Theorem to deduce that the polynomial has a zero.) 8.16. Give another proof of (a variant of) the Maximum-Modulus Theorem 8.17 via Corollary 8.11, as follows: Suppose \( f \) is holomorphic in a region containing \( \overline{D(a, r)} \) and \( |f(z)| \leq M \) for each point \( z \in D(a, r) \). Show (e.g., by Corollary 8.11) that there is a constant \( c \in \mathbb{C} \ such that \[ |f(z)| \leq c < M. \] Conclude that \( |f(a)| \leq M \). #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 171 Context: # LAURENT SERIES ## 8.17 Find a Laurent series for \[ \frac{1}{(z-1)(z+1)} \] centered at \( z = 1 \) and specify the region in which it converges. ## 8.18 Find a Laurent series for \[ \frac{1}{z(z-2)} \] centered at \( z = 2 \) and specify the region in which it converges. ## 8.19 Find a Laurent series for \[ \frac{z-2}{z+1} \] centered at \( z = -1 \) and the region in which it converges. ## 8.20 Find the terms \( c_n z^n \) in the Laurent series for \[ \frac{1}{\sin(z)} \] centered at \( z = 0 \), for \( -4 \leq n \leq 4 \). ## 8.21 Find the first four nonzero terms in the power series expansion of \( \tan(e) \) centered at the origin. What is the radius of convergence? ## 8.22 (a) Find the power series representation for \( \exp(a z) \) centered at 0, where \( a \in \mathbb{C} \) is any constant. (b) Show that \[ \exp(z) \cos(z) = \frac{1}{2} \left( \exp\left({(1+i)z}\right) + \exp\left({(1-i)z}\right) \right). \] (c) Find the power series expansion for \( \exp(z) \cos(z) \) centered at 0. ## 8.23 Show that \[ \frac{z-1}{z-2} = \sum_{n=0}^{\infty} \frac{1}{(z-1)^n} \] for \( |z| > 1 \). #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 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: 187 Context: # ARGUMENT PRINCIPLE AND ROUCHE'S THEOREM where \( g \) is also holomorphic in \( G \) and never zero. Let's compute the logarithmic derivative of \( f \) and play the same remarkable cancellation game as above: \[ f'(z) = \frac{f(z)}{g(z)} \] \[ n(z - z_j)^{-1}(z - z_j)^m g'(z) + (z - z_j)^m g(z)(z - z_j)^{-1} = \frac{m}{z - z_j} + \frac{n}{z - z_j} \frac{g'(z)}{g(z)}\tag{9.1} \] Something similar happens if \( f \) has finitely many poles in \( G \). In Exercise 9.19, we invite you to prove that, if \( p_1, \ldots, p_k \) are all the poles of \( f \) in \( G \) with order \( m_1, \ldots, m_k \), respectively, then the logarithmic derivative of \( f \) can be expressed as \[ \frac{f'(z)}{f(z)} = -\sum_{j=1}^{k} \frac{m_j}{z - p_j} - \sum_{j=1}^{k} \frac{m_j}{z - p_k} + \frac{g'(z)}{g(z)} \tag{9.2} \] where \( g \) is a function without poles in \( G \). Naturally, we can combine the expressions for zeros and poles, as we will do in a moment. ## Definition A function \( f \) is **meromorphic** in the region \( G \) if it is holomorphic in \( G \) except for poles. ### Theorem 9.17 (Argument Principle) Suppose \( f \) is meromorphic in a region \( G \) and \( \gamma \) is a positively oriented, simple, closed, piecewise smooth path that does not pass through any zero or pole of \( f \), and let \( \gamma^c \) denote \( Z(f; \gamma) \) the number of zeros of \( f \) inside \( \gamma \) counted according to multiplicity and \( P(f; \gamma) \) the number of poles of \( f \) inside \( \gamma \) counted according to multiplicity. Then \[ \frac{1}{2 \pi i} \int_{\gamma} \frac{f'(z)}{f(z)} \, dz = Z(\gamma) - P(f; \gamma) \] **Proof**. Suppose the zeros of \( f \) inside \( \gamma \) are \( z_1, \ldots, z_n \) of multiplicities \( m_1, \ldots, m_n \), respectively, and the poles inside \( \gamma \) are \( p_1, \ldots, p_k \) with order \( m_1, \ldots, m_k \), respectively. (You may meditate about the fact that there can be only finitely many zeros and poles inside \( \gamma \).) In fact, we may shrink \( \gamma \) if necessary, so that these are the only #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 193 Context: # Argument Principle and Rouche's Theorem ## 9.21. Find the number of zeros of 1. \( 3z^2 - z \) in \( D[0, 1] \) 2. \( \{ z \} \in D[0, 1] \) 3. \( e^{-z} - 5 + 1 \) in \( \{ z \in \mathbb{C} : |z| \leq 2 \} \) ## 9.22. Give another proof of the Fundamental Theorem of Algebra (Theorem 5.11), using Rouche's Theorem 9.18. *(Hint: If \( p(z) = z^n + a_{n-1}z^{n-1} + \ldots + a_0 \), let \( f(z) = a_n z^n \) and \( g(z) = a_{n-1}z^{n-1} + a_{n-2}z^{n-2} + \ldots + a_0 \), and choose \( r \) a circle that is large enough to make the condition of Rouche's theorem work. You might want to first apply Proposition 5.10 to \( g(z) \).)* ## 9.23. Suppose \( S \subset C \) is closed and bounded and all points of \( S \) are isolated points of \( S \). Show that \( S \) is finite, as follows: (a) For each \( x \in S \), we can choose \( \varepsilon > 0 \) so that \( D[x, \varepsilon] \) contains no points of \( S \) except \( x \). Show that \( f \) is continuous. *(Hint: This is really easy if you use the first definition of continuity in Section 2.1.)* (b) Assume \( S \) is non-empty. By the Extreme Value Theorem A.1, \( S \) has a minimum value, \( r_0 > 0 \). Let \( r = r_0/2 \). Since \( S \) is bounded, it lies in a disk \( D[0, M] \) for some \( M > 0 \). Show that the small disks \( D[x, r] \) for \( x \in S \) are disjoint and lie in \( D[0, M+r] \). (c) Find a bound on the number of such small disks. *(Hint: Compare the areas of \( D[x, r] \) and \( D[0, M + r] \).)* #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 195 Context: # Binomial Coefficients (4) Evaluate \(\sum_{n=1}^{\infty} \frac{1}{n^2}\). (5) Repeat the exercise with the function \(f(z) = \frac{z^{-1}}{e^{z} - 1}\) to arrive at an evaluation of \[ \sum_{n=1}^{\infty} \frac{(-1)^{n}}{n^2}. \] (Hint: To bound this function, you may use the fact that \(\frac{1}{e^z - 1} = 1 + \cos(z_n)\).) (6) Evaluate \(\sum_{n=1}^{\infty} n\) and \(\sum_{n=1}^{\infty} n^{-1}\). We remark that, in the language of Example 7.21, you have computed the evaluations \(\zeta(2)\) and \(\zeta(4)\) of the Riemann zeta function. The function \(\zeta'(s) := \sum_{n=1}^{\infty} \frac{1}{n^s}\) is called the alternating zeta function. ## 10.2 Binomial Coefficients The binomial coefficient \(\binom{n}{k}\) is a natural candidate for being explored analytically, as the binomial theorem states: \[ (x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{k} y^{n-k} \] (for \(x, y \in \mathbb{C}\) and \(n \in \mathbb{Z}_{\geq 0}\)) tells us that \(\binom{n}{k}\) is the coefficient of \(x^{k}\) in \((x + y)^{n}\). You will derive two simple identities in the course of the exercises below. 1. Convince yourself that \[ \binom{n}{k} = \frac{1}{2\pi i} \oint_{C} \frac{(x + 1)^{n}}{z^{k + 1}} dz, \] where \(C\) is any simple closed piecewise smooth path such that \(0\) is inside \(r\). 2. Derive a recurrence relation for binomial coefficients from the fact that \[ \frac{1}{z^{n}} = \frac{1}{2^{n}} \cdot \left( \text{Multiply both sides by } \frac{1}{z^{n}}\right). \] 3. Now suppose \(z\) is such that \(|z| < 1/4\). Find a simple path \(P\) surrounding the origin such that \[ \sum_{n=0}^{\infty} \left( \frac{(z + 1)^{k}}{k} \right) \] converges uniformly on \(y\) as a function of \(\epsilon\). Evaluate this sum. #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 196 Context: ``` Keeping \( x \) and \( y \) from (3), convince yourself that \[ \sum_{k=0}^{\infty} \left( \frac{2k}{k} \right) k^k = \frac{1}{2\pi i} \int \left( \frac{x^{n+1}}{z^{n+1}} \right) dz, \] use (3) to interchange summation and integral, and use the Residue Theorem 9.10 to evaluate the integral, giving an identity for \( \sum_{k=0}^{\infty} \left( \frac{1}{2} \right)^k \). ### 10.3 Fibonacci Numbers The Fibonacci\(^1\) numbers are a sequence of integers defined recursively through \[ f_0 = 0, \] \[ f_1 = 1, \] \[ f_n = f_{n-1} + f_{n-2} \quad \text{for } n \geq 2. \] Let \( F(z) = \sum_{n=0}^{\infty} f_n z^n \). 1. Show that \( F \) has a positive radius of convergence. 2. Show that the recurrence relation among the \( f_n \) implies that \( F(z) = \frac{z}{1 - z - z^2} \). (Hint: Write down the power series for \( F(z) \) and \( z^2 F(z) \) and rearrange both so that you can easily add.) 3. Verify that \[ \text{Res}_{z=0} \left( \frac{1}{z(1 - z - z^2)} \right) = f_0. \] 4. Use the Residue Theorem 9.10 to derive an identity for \( f_n \). (Hint: Integrate \[ \frac{1}{z(1 - z - z^2)} \] around \( C[0, R] \) and show that this integral vanishes as \( R \to \infty \). \(^1\) Named after Leonardo Pisano Fibonacci (1170–1250). ``` #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 197 Context: # The Coin-Exchange Problem (5) Generalize to other sequences defined by recurrence relations, e.g., the Fibonacci numbers - \( t_0 = 0 \) - \( t_1 = 0 \) - \( t_2 = 1 \) - \( t_n = t_{n-1} + t_{n-2} + t_{n-3} \) for \( n \geq 3 \). ## 10.4 The Coin-Exchange Problem In this exercise, we will solve and extend a classical problem of Ferdinand Georg Frobenius (1849–1917). Suppose \( a \) and \( b \) are relatively prime positive integers, and suppose \( r \) is a positive integer. Consider the function \[ f(c) = \frac{1}{(1 - z)(1 - z^2)z^r} \] 1. Compute the residues at all nonzero poles of \( f \). 2. Verify that \( Res_{z = 0}(f) = N(r) \), where \[ N(r) = \{(m, n) \in \mathbb{Z}^2 : m \geq 0, n \geq 0, m + nb = r\} \]. 3. Use the Residue Theorem, Theorem 9.10, to derive an identity for \( N(r) \). (Hint: Integrate \( f \) around \( [0, R] \) and show that this integral vanishes as \( R \to \infty \).) 4. Use the following three steps to simplify this identity to \[ N(t) = \frac{1}{ab} \left( \frac{b-1}{a} - \left( \frac{a-1}{b} \right) \right) + 1. \] *This means that the integers do not have any common factor.* #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 198 Context: # Discrete Applications of the Residue Theorem Here, \(\{x_n\}\) denotes the fractional part of \(x\): \(a_n \equiv 1 \,(\text{mod } b)\), and \(b \cdot 6 \equiv 1 \,(\text{mod } d)\). ## (a) Verify that for \(b = 1\), \[ N(n) = \left| \{(m,n) \in \mathbb{Z}^2 : m \geq 0, n + n \leq s \} \right| = \left| \{ m \in \mathbb{Z} : m \geq 0, m \cdot s \leq 1 \} \right| = \left[ \frac{0}{d} \right] n^2 = \left\lfloor \frac{1}{d} \right\rfloor. \] ## (b) Use this together with the identity found in (3) to obtain: \[ -\frac{1}{2} \sum_{k=1}^{\infty} \frac{1}{(1 - 2e^{2\pi i k})\cdot 2k} = \left( -\frac{1}{2} + \frac{1}{2} \right) + \frac{1}{z} \, \] ## (c) Verify that \[ \sum_{k=1}^{\infty} \frac{1}{(1 - 2e^{2\pi i k})} = \sum_{k=1}^{\infty} \frac{1}{(1 - 2e^{2\pi i k})} \] ## (5) Prove that \(N(a - b) = 0\), and \(N(n) > 0\) for all \(x > a\) and \(x < b\). ### Historical Remark Given relatively prime positive integers \(a_1, a_2, \ldots\), let’s call an integer \(t\) representable if there exist nonnegative integers \(m_1, m_2, \ldots, m_k\) such that \[ t = m_1 a_1 + m_2 a_2 + \cdots + m_k a_k. \] (There are many scenarios in which you may ask whether or not \(t\) is representable, given fixed \(a_1, a_2, \ldots, a_k\); for example, if the \(a_i\) are co-prime, this question asks whether you can give exact change for \(t\). In the late 19th century, Frobenius raised the problem of finding the largest integer that is not representable. We will call this largest integer the Frobenius number \(g(a_1, a_2, \ldots, a_k)\). It is well known (probably at least since the 1880s, when James Joseph Sylvester (1814–1897) studied the Frobenius problem) that \(g(a_1, a_2) = a_1 a_2 - a_1 - a_2\). You verified this result in (5). For \(x > 2\), there is no closed formula for \(g(a_1, a_2, \ldots, a_k)\). The formula in (4) states that The fractional part of a real number \(x\) is loosely speaking the part after the decimal point. More thoroughly, the principal integer function of \(x\), denoted by \(\lfloor x \rfloor\), is the largest integer not exceeding \(x\). The fractional part is defined as \( \{x\} = x - \lfloor x \rfloor \). **"This means that \(c\) is an integer such that \(c = a + 1 + b + k\) for some \(k \in \mathbb{Z}\)."** #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 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: 206 Context: 200 # Solutions to Selected Exercises ## 4.1 (a) \(6\) (b) \(\pi\) (c) \(4\) (d) \(\sqrt{7} + \frac{1}{2} \sinh^{-1}(4)\) ## 4.5 (a) \(8i\) (b) \(0\) (c) \(0\) ## 4.6 (a) \(1-i\) (b) \(i\pi, -\pi, 0, 2\pi\) (c) \(i^2, -2, 0, 2r^2\) ## 4.7 (a) \(\frac{1}{3} (e^{3z} - e^{-3z})\) (b) \(0\) (c) \(\{g(\exp(3 + 3i)) - 1\}\) ## 4.12 \(\sqrt{3} i\) ## 4.18 (a) \(-4 + i\left(4 + \frac{1}{3}\right)\) (b) \((\ln(17) - 1) + i \left(\frac{\pi}{4} + 1\right)\) (c) \(2\sqrt{2} - 1 + 2i\sqrt{2}\) (d) \(\frac{1}{4} \sin(8) - 2 + i \left(2 - \frac{1}{4} \sinh(8)\right)\) ## 4.27 For \(|r| < |z| \Rightarrow 2\pi for \ |r| < |z|\) ## 4.30 \(\frac{24}{30}\) ## 4.34 \(0\) ## 4.35 For \(r = 1\): \( \frac{-1}{3} \); for \(r = 3\): \(0\); for \(r = 5\) ## 4.37 (a) \(2\pi\) (b) \(0\) (c) \(-\frac{3}{2}\) (d) \(\frac{2}{2r^2 - 1}\) ## 5.1 (a) \(2\pi\) (b) \(-6i\) (c) \(\pi\) (d) \(0\) ## 5.3 (a) \(0\) (b) \(2\pi i\) (c) \(0\) (d) \(\pi\) (e) \(f(0)\) ## 5.4 \(2\pi \exp{(iw)}\) ## 5.18 \(\frac{\sqrt{3}}{2}\) ## 5.20 \(\frac{\sqrt{z}}{\sin{\left(\frac{1}{\sqrt{2}} z\right)}}\) ## 7.1 (a) Divergent (b) Convergent (limit \(0\)) (c) Divergent (d) Convergent (limit \(2 -\frac{1}{2}\)) (e) Convergent (limit \(0\)) ## 7.3 \(\sum_{n=2}^{\infty} (-4)^n z^n = \sum_{n=3}^{\infty} \frac{1}{6} z^n = \frac{k + 1}{2} z^{n+2}\) #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 209 Context: # Table of Contents 1. **Convergence** - Sequence - Series - Cosine - Cotangent - Cross Ratio - Curve - Cycloid 2. **Numerical Concepts** - Dedekind Sum - Dense - Derivative - Partial - Difference Quotient - Differentiable - Dilation - Discriminant 3. **Geometric Concepts** - Disk - Closed - Open - Punctured - Unit - Distance of Complex Numbers - Divergent - Domain - Double Series 4. **Mathematical Constants** - e - Embedding of R into C - Empty Set - Entire Functions - Essential Singularity - Euclidean Plane - Euler’s Formula 5. **Functions and Theorems** - Even - Exponential Function - Exponential Rules - Extended Complex Plane - Fibonacci Numbers - Field - Fixed Point - Frobenius Problem - Functions - Conformal - Even - Exponential - Logarithmic - Odd - Trigonometric - Fundamental Theorem - Of Algebra - Of Calculus 6. **Applications** - Geogebra - Geometric Interpretation of Multiplication - Geometric Series - Group 7. **Harmonic Analysis** - Harmonic - Harmonic Conjugate - Holomorphic - Homotopy - Hyperbolic Trig Functions #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 210 Context: ``` # Index - i, 4 - identity, 3 - identity map, 23 - identity principle, 154 - image - of a function, 27 - of a point, 23 - imaginary axis, 5 - imaginary part, 4 - improper integral, 104, 186 - infinity, 46 - inside, 88 - integral, 71 - path independent, 102 - integral test, 129 - integration by parts, 92 - interior point, 13 - inverse function, 31 - of a Möbius transformation, 43 - inverse parametrization, 74 - inversion, 45 - isolated point, 13 - isolated singularity, 170 - Jacobian, 65 - Jordan curve theorem, 88 - L'Hôpital's rule, 197 - Laplace, 110 - Laurent series, 157 - least upper bound, 124, 137 - Leibniz's rule, 83, 196 - length, 73 - limit - infinity, 46 - of a function, 24 - of a sequence, 122 - of a series, 126 - linear fractional transformation, 43 - Liouville's theorem, 104 - Log, 59 - log, 61 - logarithm, 59 - logarithmic derivative, 180 - max/min property for harmonic functions, 115, 155 - maximum - strong relative, 115 - weak relative, 116, 155 - mean-value theorem - for harmonic functions, 114 - for holomorphic functions, 86 - for real functions, 195 - meromorphic, 181 - minimum - strong relative, 115 - weak relative, 155 - Möbius transformation, 43 - modulus, 5 - monotone, 124 - monotone sequence property, 124 - Morera's theorem, 100 - multiplication, 2 - north pole, 51 ``` #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 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 5 sentences can you explain the life of albert einstein""" Consider the chat history for relevant information. 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Page 209, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 210, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 212 ================================================== **Elapsed Time: 6.61 seconds** ================================================== FINAL ANSWER Answer: Albert Einstein was born on March 14, 1879, in Ulm, Germany, and he later moved to Munich, where he began his education in physics and mathematics. He studied at the Polytechnic Institute in Zurich, graduating in 1900, and initially struggled to find a teaching position, eventually working at the Swiss Patent Office. In 1905, often referred to as his "miracle year," he published four groundbreaking papers, including his theory of special relativity and the famous equation \(E = mc^2\), which established him as a leading physicist. Einstein's work continued to evolve, and he received the Nobel Prize in Physics in 1921 for his explanation of the photoelectric effect, further solidifying his reputation. He spent the latter part of his life in the United States, where he became a professor at Princeton University and remained an influential figure in both science and social issues until his death on April 18, 1955. ================================================== **Elapsed Time: 5.32 seconds** ==================================================