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Is there a particular topic or question on your mind?" }, { "role": "user", "content": "can you tell me the contents of your context in 250 words?" }, { "role": "assistant", "content": "" } ], "stream": true, "language_level": "", "chat_channel": ":4185:1727847802559", "language": "German", "tone": "neutral", "writing_style": "standard", "model": "gpt-4o-mini-2024-07-18", "knowledgebase": "ki-dev-large", "seed": 1055, "client_id": 1727847802559, "all_context": false, "follow_up_for": null, "knowledgebase_files_count": 3, "override_command": null, "disable_clarity_check": false, "custom_primer": "", "logging": true, "query_route": "" } INITIALIZATION Knowledgebase: ki-dev-large Base Query: can you tell me the contents of your context in 250 words? Model: gpt-4o-mini-2024-07-18 **Elapsed Time: 0.00 seconds** ROUTING Query type: summary **Elapsed Time: 1.82 seconds** RAG PARAMETERS Max Context To Include: 100 Lowest Score to Consider: 0.1 ================================================== **Elapsed Time: 0.14 seconds** ================================================== VECTOR SEARCH ALGORITHM TO USE Use MMR search?: True Use Similarity search?: False ================================================== **Elapsed Time: 0.10 seconds** ================================================== VECTOR SEARCH DONE ================================================== **Elapsed Time: 6.27 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: 4 Context: # Robert Chaffee Robert Chaffee (cover art) is a professor emeritus at Central Michigan University. His academic interests are in abstract algebra, combinatorics, geometry, and computer applications. Since retirement from teaching, he has devoted much of his time to applying those interests to creation of art images. ## A Note to Instructors The material in this book should be more than enough for a typical semester-long undergraduate course in complex analysis; our experience teaches us that there is more content in this book than fits into one semester. Depending on the nature of your course and its place in your department's overall curriculum, some sections can be either partially omitted or their definitions and theorems can be assumed true without delving into proofs. Chapter 10 contains optional longer homework problems that could also be used as group projects at the end of a course. We would be happy to hear from anyone who has adopted our book for their course, as well as suggestions, corrections, or other comments. ## Acknowledgments We thank our students who made many suggestions for and found errors in the text. Special thanks go to Sheldon Axler, Collin Bleak, Pierre-Alexandre Bliman, Matthew Brin, Andrew Huang, John McCleary, Sharma Pallakonda, Joshua Palamater, and Dmytro Savchuk for comments, suggestions, and additions after teaching from this book. We thank Lon Mitchell for his initiative and support for the print version of our book with Orthogonal Publishing, and Bob Chaffee for allowing us to feature his art on the book’s cover. We are grateful to the American Institute of Mathematics for including our book in their Open Textbook Initiative [https://ainth.org/textbooks](https://ainth.org/textbooks). #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 5 Context: # Contents 1. **Complex Numbers** 1.1. Definitions and Algebraic Properties . . . . . . . . 2 1.2. From Algebra to Geometry and Back . . . . . . . . 5 1.3. Geometric Properties . . . . . . . . . . . . . . . . 9 1.4. Elementary Topology of the Plane . . . . . . . . . 12 Optional Lab . . . . . . . . . . . . . . . . . . . . . . 21 2. **Differentiation** 2.1. Limits and Continuity . . . . . . . . . . . . . . . . 23 2.2. Differentiability and Holomorphicity . . . . . . . 28 2.3. The Cauchy–Riemann Equations . . . . . . . . . . . 32 2.4. Constant Functions . . . . . . . . . . . . . . . . . 36 3. **Examples of Functions** 3.1. Möbius Transformations . . . . . . . . . . . . . . . 43 3.2. Infinity and the Cross Ratio . . . . . . . . . . . 46 3.3. Stereographic Projection . . . . . . . . . . . . . . 48 3.4. Exponential and Trigonometric Functions . . . . . 55 3.5. Logarithms and Complex Exponentials . . . . . . . 59 4. **Integration** 4.1. Definition and Basic Properties . . . . . . . . . . 71 4.2. Antiderivatives . . . . . . . . . . . . . . . . . . 76 4.3. Cauchy’s Theorem . . . . . . . . . . . . . . . . . 80 4.4. Cauchy’s Integral Formula . . . . . . . . . . . . 85 5. **Consequences of Cauchy’s Theorem** 5.1. Variations of a Theme . . . . . . . . . . . . . . . 97 5.2. Antiderivatives Again . . . . . . . . . . . . . . 100 5.3. Taking Cauchy's Formulas to the Limit . . . . . . 102 #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 6 Context: ``` # Contents 6. Harmonic Functions ................................. 110 6.1 Definitions and Basic Properties ................. 114 6.2 Mean-Value Principle ............................ 114 7. Power Series ....................................... 121 7.1 Sequences and Completeness ...................... 122 7.2 Series .......................................... 125 7.3 Sequences and Series of Functions ............... 131 7.4 Regions of Convergence .......................... 135 8. Taylor and Laurent Series .......................... 146 8.1 Power Series and Holomorphic Functions .......... 146 8.2 Classification of Zeros and the Identity Principle 152 8.3 Laurent Series .................................. 156 9. Isolated Singularities and the Residue Theorem..... 169 9.1 Classification of Singularities ................... 169 9.2 Residues ........................................ 176 9.3 Argument Principle and Rouché’s Theorem ........ 180 10. Discrete Applications of the Residue Theorem ...... 188 10.1 Infinite Sums ................................... 188 10.2 Binomial Coefficients ........................... 189 10.3 Fibonacci Numbers ............................... 190 10.4 The Coin-Exchange Problem ...................... 193 10.5 Dedekind Sums .................................. 193 ## Theorems From Calculus ## Solutions to Selected Exercises ## Index ``` #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 7 Context: # Chapter 1 ## Complex Numbers Die ganzen Zahlen hat der liebe Gott geschaffen, alles andere ist Menschenwerk. (God created the integers, everything else is made by humans.) Leopold Kronecker (1823–1891) The real numbers have many useful properties. There are operations such as addition, subtraction, and multiplication, as well as division by any nonzero number. There are useful laws that govern these operations, such as the commutative and distributive laws. We can take limits and do calculus, differentiating and integrating functions. But you cannot take a square root of \(-1\); that is, you cannot find a real root of the equation $$ x^2 + 1 = 0. \tag{1.1} $$ Most of you have heard that there is a "new" number that is a root of (1.1); that is, \(x^2 + 1 = 0\) or \(x^2 = -1\). We will show that when the real numbers are enlarged to a new system called the **complex numbers**, which includes \(i\), not only do we gain numbers with interesting properties, but we do not lose many of the nice properties that we had before. The complex numbers, like the real numbers, will have the operations of addition, subtraction, multiplication, as well as division by any complex number except zero. These operations will follow all the laws that we are used to, such as the commutative and distributive laws. We will also be able to take limits and do calculus. And there will be a root of (1.1). As a brief historical aside, complex numbers did not originate with the search for a square root of \(-1\); rather, they were introduced in the context of cubic equations. Scipione del Ferro (1465–1526) and Niccolò Tartaglia (1499–1557) discovered a way to find a root of any cubic polynomial, which was publicized by Gerolamo Cardano (1501–1576) and is often referred to as **Cardano's formula**. For the cubic polynomial \(x^3 + px + q\), Cardano's formula involves the quantity $$ \sqrt[3]{-\frac{q}{2} + \sqrt{\left(\frac{q}{2}\right)^2 + \left(\frac{p}{3}\right)^3}} + \sqrt[3]{-\frac{q}{2} - \sqrt{\left(\frac{q}{2}\right)^2 + \left(\frac{p}{3}\right)^3}}. $$ #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 9 Context: So we can think of the real numbers being embedded in \( \mathbb{C} \) as those complex numbers whose second coordinate is zero. The following result states the algebraic structure that we established with our definitions. ## Proposition 1.1 Let \( (x, y) \) be a field, that is, for all \( (a, b) \in \mathbb{C} \): 1. \( (x, y) + (a, b) \in \mathbb{C} \) (1.4) 2. \( (x, y) + (a, d) + (b, c) = (x, y) + ((a + b), c) \) (1.5) 3. \( (x, y) + (0, 0) = (x, y) \) (1.6) 4. \( (x, y) + (-x, -y) = (0, 0) \) (1.7) 5. \( (a, b) + (c, d) = (x, y) \cdot (a, b) + (c, d) \) (1.8) 6. \( (x, y) \in \mathbb{C} \) (1.9) 7. \( (x, y) \cdot (a, b) = (x \cdot a, y \cdot b) \) (1.10) 8. \( (x, y) \cdot (b, d) = (x \cdot b, (a \cdot b)(c, d)) \) (1.11) 9. \( (x, y) \cdot (a, b) = (x, y) \) (1.12) 10. \( (x, y) \cdot (1, 0) = (x, y) \) (1.13) 11. For all \( (x, y) \in \mathbb{C} \setminus \{(0, 0)\} \): \( (x, y) \cdot \left( \frac{(y, -x)}{x^2 + y^2} \right) = (1, 0) \) (1.14) What we are stating here can be compressed in the language of algebra: equations (1.4)–(1.8) say that \( (\mathbb{C}, +) \) is an Abelian group with identity \( (0, 0) \); equations (1.10)–(1.14) say that \( (\mathbb{C} \setminus \{(0, 0)\}, \cdot) \) is an Abelian group with identity \( (1, 0) \). The proof of Proposition 1.1 is straightforward but nevertheless makes for good practice (Exercise 1.14). We give one sample: **Proof (1.18)**. By our definition for complex addition and properties of additive inverse in \( \mathbb{R} \): \[ (x, y) + (-x, -y) = (x + (-x), y + (-y)) = (0, 0). \] The definition of our multiplication implies the innocent-looking statement: \[ (0, 1) \cdot (1, 0) = (1, 0). \quad (1.15) \] #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 10 Context: ``` # Complex Numbers This identity together with the fact that $$ (x_0, \cdots, x_n) \cdot (x, y) = (ax, az) $$ allows an alternative notation for complex numbers. The latter implies that we can write $$ (x, y) = (x_0, 0) + (0, y) = (x_0, 0) + (0, 1)(0, y). $$ If we think—in the spirit of our remark about embedding R into C—of (x, 0) and (y, 0) as the real numbers x and y, then this means that we can write any complex number (x, y) as a linear combination of (1, 0) and (0, 1), with the real coefficients x and y. Now (1, 0), in turn, can be thought of as the real number 1. So if we give (0, 1) a special name, say i, then the complex number that we used to call (x, y) can be written as $$ x + iy. $$ ### Definition The number x is called the **real part** and y the **imaginary part** of the complex number x + iy, often denoted as Re(x + iy) = x and Im(x + iy) = y. The identity (1.15) then reads $$ i^2 = -1. $$ In fact, much more can now be said with the introduction of the square root of -1. It is not just that (1.1) has a root, but every nonconstant polynomial has roots in C: ### Fundamental Theorem of Algebra (see Theorem 5.11). Every nonconstant polynomial of degree d has d roots (counting multiplicity) in C. The proof of this theorem requires some (important) machinery, so we defer its proof and an extended discussion of it to Chapter 5. We invite you to check that the definitions of our binary operations and Proposition 1.1 are coherent with the usual real arithmetic rules if we think of complex numbers as given in the form x + iy. *The names have historical reasons: people thought of complex numbers as unreal, imagined.* ``` #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 12 Context: # COMPLEX NUMBERS Aside from the exceptional case of 0, for any complex number \( z \), the arguments of \( z \) all differ by a multiple of \( 2\pi \), just as we saw for the example \( z = 1 \). The absolute value of the difference of two vectors has a nice geometric interpretation: ## Proposition 1.2 Let \( z_1, z_2 \in \mathbb{C} \) be two complex numbers, thought of as vectors in \( \mathbb{R}^2 \), and let \( d(z_1, z_2) \) denote the distance between the endpoints of the two vectors in \( \mathbb{R}^2 \) (see Figure 1.2). Then \[ d(z_1, z_2) = |z_1 - z_2| = |z_2 - z_1|. \] **Proof:** Let \( z_1 = x_1 + iy_1 \) and \( z_2 = x_2 + iy_2 \). From geometry, we know that \[ d(z_1, z_2) = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}. \] This is the definition of \( |z_1 - z_2| \). Since \( (x_1 - x_2) = (x_2 - x_1) \) and \( (y_1 - y_2) = (y_2 - y_1) \), this is also equal to \( |z_2 - z_1| \. \qed ![Figure 1.2: Geometry behind the distance between two complex numbers.](path_to_image) That \( |z_1 - z_2| = |z_2 - z_1| \) simply says that the vector from \( z_2 \) to \( z_1 \) has the same length as the vector from \( z_1 \) to \( z_2 \). One reason to introduce the absolute value and argument of a complex number is that they allow us to give a geometric interpretation for the multiplication of two complex numbers. Let’s say we have two complex numbers \( z_1 = x_1 + iy_1 \), with absolute value \( r_1 \) and argument \( \phi_1 \), and \( z_2 = x_2 + iy_2 \), with absolute value \( r_2 \) and argument \( \phi_2 \). #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 13 Context: # FROM ALGEBRA TO GEOMETRY AND BACK ## 7 ϕ₂. This means we can write \(x_1 + iy_1 = (r_1 \cos \phi_1) + i(r_1 \sin \phi_1)\) and \(x_2 + iy_2 = (r_2 \cos \phi_2) + i(r_2 \sin \phi_2)\). To compute the product, we make use of some classic trigonometric identities: \[ (x_1 + iy_1)(x_2 + iy_2) = (r_1 \cos \phi_1 + ir_1 \sin \phi_1)(r_2 \cos \phi_2 + ir_2 \sin \phi_2) \] \[ = r_1 r_2 (\cos \phi_1 \cos \phi_2 - \sin \phi_1 \sin \phi_2) + i(r_1 r_2 \cos \phi_2 \sin \phi_1 + r_1 r_2 \sin \phi_2 \cos \phi_1) \] \[ = r_1 r_2 \left( \cos(\phi_1 + \phi_2) + i \sin(\phi_1 + \phi_2) \right). \] So the absolute value of the product is \(r_1 r_2\) and one of its arguments is \(ϕ_1 + ϕ_2\). Geometrically, we are multiplying the lengths of the two vectors representing our two complex numbers and adding their angles measured with respect to the positive real axis. ![Figure 1.3: Multiplication of complex numbers](#) In view of the above calculation, it should come as no surprise that we will have to deal with quantities of the form \(x + iy\) (where \(y\) is some real number) quite a bit. To save space, bytes, ink, etc., and because "Mathematics is for lazy people" we introduce a shortcut notation and define: \[ e^{i\phi} = \cos \phi + i \sin \phi. \] ^1 You should convince yourself that there is no problem with this fact since there are many possible arguments for complex numbers, as both cosine and sine are periodic functions with period \(2\pi\). ^2 Peter Halton (unpublished articles, Hudson River Undergraduate Mathematics Conference 2000). #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 14 Context: # COMPLEX NUMBERS Figure 1.4 shows three examples. At this point, this exponential notation is indeed ![Figure 1.4: Three sample complex numbers of the form \( e^{x} \).](image-path) purely a notation. We will later see in Chapter 3 that it has an intimate connection to the complex exponential function. For now, we motivate this maybe strange seeming definition by collecting some of its properties: ## Proposition 1.3. For any \( x_1, x_2 \in \mathbb{R} \): 1. \( e^{(x_1 + x_2 i)} = e^{x_1} e^{x_2 i} \) 2. \( e^{0} = 1 \) 3. \( \frac{d}{dx} e^{x} = e^{x} \) 4. \( |e^{x}| = 1 \) 5. \( e^{i\pi} = -1 \) You are encouraged to prove them (Exercise 1.16); again, we give a sample. ## Proof of \( e^{x} \). By definition of \( e^{x} \): \[ \frac{d}{dx} e^{(x + i)} = -\sin{(x)} + \cos{(x)} = i ( \cos{(x)} + i\sin{(x)}) = i e^{(x+i)} \] #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 15 Context: # Geometric Properties Proposition 1.3 implies that \( z^{n} = 1 \) for any integers \( n \) and \( r > 0 \). Thus, numbers of the form \( z^{n} \) with \( z \in \mathbb{Q} \) play a pivotal role in solving equations of the form \( z^{n} = 1 \)—plenty of reason to give them a special name. **Definition.** A root of unity is of the form \( z^{n} \) for some integers \( n \) and \( r > 0 \). A root of unity \( z \) is a complex number \( z \) such that \( z^{n} = 1 \) for some positive integer \( n \). In this case, we call \( z \) an \( n \)th root of unity. If \( n \) is the smallest positive integer with the property \( z^{n} = 1 \), then \( z \) is a primitive \( n \)th root of unity. **Example 1.4.** The 4th roots of unity are \( 1 \) and \( i \) and \( e^{i\frac{\pi}{2}} \). The latter two are primitive 4th roots of unity. With our new notation, the sentence "the complex number \( x + iy \) has absolute value \( r \) and argument \( \theta \) now becomes the identity \[ x + iy = r e^{i\theta}. \] The left-hand side is often called the **rectangular form**, the right-hand side the **polar form** of this complex number. We now have five different ways of thinking about a complex number: the formal definition, in rectangular form, in polar form, and geometrically, using Cartesian coordinates or polar coordinates. Each of these five ways is useful in different situations, and translating between them is an essential ingredient in complex analysis. The five ways and their corresponding notation are listed in Figure 1.5. This list is not exhaustive; see, e.g., Exercise 1.21. ## 1.3 Geometric Properties From the chain of basic inequalities \[ -\sqrt{x^{2} + y^{2}} \leq -|z| \leq \sqrt{x^{2} + y^{2}} \leq |z| \leq \sqrt{x^{2} + y^{2}}, \] (or, alternatively, by arguing with basic geometric properties of triangles), we obtain the inequalities \[ -|z| \leq \text{Re}(z) \leq |z| \] and \[ -|z| \leq \text{Im}(z) \leq |z|. \tag{1.16} \] The square of the absolute value has the nice property \[ |x + iy|^{2} = x^{2} + y^{2} = (x + iy)(x - iy). \] #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 16 Context: # COMPLEX NUMBERS | | Algebraic | Geometric | Exponential | |------------|------------------|------------------|---------------| | Formal | \(x + iy\) | Cartesian | \(re^{i\theta}\) | | | Rectangular | \(x\) | Polar | | | | \(y\) | | **Figure 1.5:** Five ways of thinking about a complex number. This is one of many reasons to give the process of passing from \(x + iy\) to \(x - iy\) a special name. ## Definition The number \(x - iy\) is the (complex) conjugate of \(x + iy\). We denote the conjugate by \[ x + iy = x - iy. \] Geometrically, conjugating \(z\) means reflecting the vector corresponding to \(z\) with respect to the real axis. The following collects some basic properties of the conjugate. ## Proposition 1.5 For any \(z_1, z_2, z_3 \in \mathbb{C}\): 1. \(z_1 \overline{z_2} = \overline{z_1 z_2}\) 2. \(\overline{z_1 z_2} = \overline{z_2} \overline{z_1}\) 3. \(\overline{\left(\frac{z_1}{z_2}\right)} = \frac{\overline{z_2}}{\overline{z_1}}\) 4. \(\overline{z} = z\) 5. \(|z| = |z|\) The proofs of these properties are easy (Exercise 1.22); once more we give a sample. ### Proof of (b) Let \(z_1 = x_1 + iy_1\) and \(z_2 = x_2 + iy_2\). Then \[ z_1 z_2 = (x_1 + iy_1)(x_2 + iy_2) = (x_1x_2 - y_1y_2) + i(x_1y_2 + y_1x_2) = \overline{z_1} \overline{z_2}. \] Image Analysis: ### 1. Localization and Attribution - **Page**: This is a page from a document or book, numbered at the top left as "10." - **Image 1**: An illustrated table at the top of the page labeled as "Figure 1.5." - **Text Block**: Multiple paragraphs of text underneath the table, including definitions, propositions, and proofs. ### 2. Object Detection and Classification - **Image 1:** - **Objects**: A table with three rows and three columns. - **Categories**: Types of complex number representations. - **Key Features**: - **Row 1**: "Formal" corresponds to (x, y). - **Row 2**: "Algebraic" includes rectangular (x + iy) and exponential (re^iθ). - **Row 3**: "Geometric" includes cartesian and polar representations. ### 3. Scene and Activity Analysis - **Scene**: - The scene depicts a page from a textbook or academic document. - The key activity is educational content delivery, specifically discussing complex numbers. ### 4. Text Analysis - **Detected Text**: - Title: "COMPLEX NUMBERS." - Sections: - "Figure 1.5: Five ways of thinking..." - Definitions and propositions related to complex numbers. - Proof of a property. - **Analysis**: - The page discusses different representations of complex numbers, highlights their properties, and presents relevant mathematical proofs. ### 5. Diagram and Chart Analysis - **Image 1:** - **Content**: The table categorizes the representations of complex numbers. - **Axes/Scales**: Not applicable. - **Legends**: Explanatory text within the cells. - **Key Insights**: - Complex numbers can be represented in multiple forms: formal, algebraic, geometric. - The table is a clear and concise way to present these different perspectives. ### 6. Product Analysis - **Not applicable**: No products were depicted in the image. ### 7. Anomaly Detection - **Not applicable**: No anomalies in the visual content. ### 8. Color Analysis - **Dominant Colors**: Black text on a white background. - **Impact**: Standard colors for printed or digital educational content, aiding readability and focus. ### 9. Perspective and Composition - **Perspective**: Straight-on view, typical for document pages. - **Composition**: - The table (Image 1) is at the top, drawing initial attention. - Subsequent text follows a structured format, typical of academic documents. ### 10. Contextual Significance - **Contribution**: - The image and text provide fundamental knowledge about complex numbers. - It enriches the educational material by categorizing and defining complex number representations. ### 11. Metadata Analysis - **Not available**: No metadata provided from the image. ### 12. Graph and Trend Analysis - **Not applicable**: No graphs present. ### 13. Graph Numbers - **Not applicable**: No graphs present. ### Additional Aspects - **Process Flows and Descriptions**: - A defined process for understanding the conjugate of a complex number. - Steps to prove a proposition about complex numbers. - **Type Designations**: - Designates formal, algebraic, and geometric types of complex number representations. - **Trend and Interpretation**: - Emphasizes the versatility and various intuitions behind complex number representations. - **Tables**: - The table encapsulates the five types of complex number representations succinctly. This comprehensive examination focuses on the detailed aspects and provides a thorough understanding of the educational content on the described page. #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 18 Context: # 1.4 Elementary Topology of the Plane In Section 1.2 we saw that the complex numbers \( C \), which were initially defined algebraically, can be identified with the points in the Euclidean plane \( \mathbb{R}^2 \). In this section we collect some definitions and results concerning the topology of the plane. \[ \begin{array}{c} y \\ \text{C}(2+i,2) \\ \hspace{-30pt} \text{D}(-2, 1) \\ \hspace{80pt} 1 \\ \hspace{-30pt} x \end{array} \] **Figure 1.6:** Sample circle and disk. In Proposition 1.2, we interpreted \( |z - w| \) as the distance between the complex numbers \( z \) and \( w \), viewed as points in the plane. So if we fix a complex number \( a \) and a positive real number \( r \), then all \( z \in C \) satisfying \( |z - a| = r \) form the set of points at distance \( r \) from \( a \); this set is the circle with center \( a \) and radius \( r \), which we denote by \[ C(a, r) = \{ z \in C : |z - a| = r \} \] The inside of this circle is called the open disk with center \( a \) and radius \( r \); we use the notation \[ D(a, r) = \{ z \in C : |z - a| < r \} \] Note that \( D(a, r) \) does not include the points on \( C(a, r) \). Figure 1.6 illustrates these definitions. Next we need some terminology for talking about subsets of \( C \). **Definition.** Suppose \( G \) is a subset of \( C \). #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 20 Context: # Complex Numbers ![Figure 1.7: The intervals [0,1) and (1,2) are separated.](image_link) ## Definition Two sets \( X, Y \subset \mathbb{C} \) are separated if there are disjoint open sets \( A, B \subset C \) so that \( X \subset A \) and \( Y \subset B \). A set \( G \subset \mathbb{C} \) is connected if it is impossible to find two separated nonempty sets whose union is \( G \). A region is a connected open set. The idea of separation is that the two open sets \( A \) and \( B \) ensure that \( X \) and \( Y \) cannot just "stick together." It is usually easy to check that a set is not connected. On the other hand, it is hard to use the above definition to show that a set is connected, since we have to rule out any possible separation. ## Example 1.10 The intervals \( X = [0,1) \) and \( Y = (1,2) \) on the real axis are separated: There are infinitely many choices for \( A \) and \( B \) that work; one choice is \( A = D[0,1) \) and \( B = D(1,2) \), depicted in Figure 1.7. Hence \( X \cup Y = [0,1) \cup (1,2) \) is not connected. One type of connected set that we will use frequently is a path. ## Definition A path (or curve) in \( \mathbb{C} \) is a continuous function \( \gamma : [a,b] \to \mathbb{C} \), where \([a,b]\) is a closed interval in \( \mathbb{R} \). We may think of \( \gamma \) as a parametrization of the image that is painted by the path and will often write this parametrization as \( \gamma(t) \), for \( a \leq t \leq b \). The path is smooth if it is differentiable and the derivative \( \gamma' \) is continuous and nonzero. #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 21 Context: # ELEMENTARY TOPOLOGY OF THE PLANE This definition uses the calculus notions of continuity and differentiability; that is, \( \gamma: [a, b] \to C \) being continuous means that for all \( t_0 \in [a, b] \) \[ \lim_{t \to t_0} \gamma(t) = \gamma(t_0) \] and the derivative of \( \gamma \) at \( t_0 \) is defined by \[ \gamma'(t_0) = \lim_{t \to t_0} \frac{\gamma(t) - \gamma(t_0)}{t - t_0} \] \[ \gamma_1(t) = -2 + 2e^{it}, \quad 0 \leq t \leq 2\pi \] \[ \gamma_2(t) = \begin{cases} 3 + i(t - 2) & \text{if } 0 \leq t \leq 3 \\ 6 - i + i(t - 1) & \text{if } 3 < t \leq 5 \end{cases} \] Figure 1.8: Two paths and their parameterizations: \( \gamma_1 \) is smooth and \( \gamma_2 \) is continuous and piecewise smooth. Figure 1.8 shows two examples. We remark that each path comes with an orientation, i.e., a sense of direction. For example, the path \( \gamma_1 \) in Figure 1.8 is different from \[ \gamma_1(t) = -2 + 2e^{it}, \quad 0 \leq t \leq \frac{3\pi}{2} \] even though both \( \gamma_1 \) and \( \gamma_3 \) yield the same picture; \( \gamma_1 \) features a counter-clockwise orientation, where as that of \( \gamma_2 \) is clockwise. It is a customary and practical abuse of notation to use the same letter for the path and its parameterization. We emphasize that a path must have a parameterization, and that the parameterization must be defined and continuous on a closed and bounded interval \([a, b]\). Since topologically we may identify \( C \) with \( \mathbb{R}^2 \), a path can be specified. #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 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: 26 Context: # COMPLEX NUMBERS ## 1.23 Sketch the following sets in the complex plane: - (a) \( \{ z \in \mathbb{C} : |z - 1 + i| = 1 \} \) - (b) \( \{ z \in \mathbb{C} : |z - 1 + i| \leq 2 \} \) - (c) \( \{ z \in \mathbb{C} : \text{Re}(z^2 + 2 - 2i) = 3 \} \) - (d) \( \{ z \in \mathbb{C} : |z - i| + |z + i| = 3 \} \) - (e) \( \{ z \in \mathbb{C} : |z| = |z - 1| \} \) - (f) \( \{ z \in \mathbb{C} : |z - 1| = 2 |z + 1| \} \) - (g) \( \{ z \in \mathbb{C} : \text{Re}(z) = 1 \} \) - (h) \( \{ z \in \mathbb{C} : \text{Im}(z) = 1 \} \) ## 1.24 Suppose \( \rho \) is a polynomial with real coefficients. Prove that: - (a) \( \rho(\overline{z}) = \overline{\rho(z)} \). - (b) \( \rho(z) = 0 \) if and only if \( \rho(\overline{z}) = 0 \). ## 1.25 Prove the reverse triangle inequality (Proposition 1.7(b)): \[ |z_1 - z_2| \leq |z_1| + |z_2|. \] ## 1.26 Use the previous exercise to show that \[ \left| \frac{1}{z - 2} \right| < \frac{1}{3} \] for every \( z \) on the circle \( C[0,2] \). ## 1.27 Sketch the sets defined by the following constraints and determine whether they are open, closed, or neither; bounded; connected. - (a) \( |z + 3| < 2 \) - (b) \( \text{Im}(z) < 1 \) - (c) \( 0 < |z - 1| < 2 \) - (d) \( |z| = |1 + i| + 2 \) - (e) \( |z| = |1 + i| + 3 \) ## 1.28 What are the boundaries of the sets in the previous exercise? ## 1.29 Let \( G \) be the set of points \( z \in \mathbb{C} \) satisfying either \( z = r \) and \( -2 < r < -1 \), or \( |z| < 1 \), or \( z = \sigma \) where \( \sigma = -2 \). - (a) Sketch the set \( G \), being careful to indicate exactly the points that are in \( G \). - (b) Determine the interior points of \( G \). #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 27 Context: # Elementary Topology of the Plane 21 (c) Determine the boundary points of \( G \). (d) Determine the isolated points of \( G \). ## 1.30 The set \( G \) in the previous exercise can be written in three different ways as the union of two disjoint nonempty separated subsets. Describe them, and in each case say briefly why the subsets are separated. ## 1.31 Show that the union of two regions with nonempty intersection is itself a region. ## 1.32 Show that if \( A \subset B \) and \( B \) is closed, then \( A \subset B \). Similarly, if \( A \subset B \) and \( A \) is open, show that \( A \) is contained in the interior of \( B \). ## 1.33 Find a parameterization for each of the following paths: (a) The circle \( C(1 + i, 1) \), oriented counter-clockwise (b) The line segment from \( -1 - i \) to \( 2i \) (c) The top half of the circle \( C(0, \frac{3}{4}) \), oriented clockwise (d) The rectangle with vertices \( \pm 1 \pm 2i \), oriented counter-clockwise (e) The ellipse \( \{ z \in \mathbb{C} : |z - 1| + |z + 1| = 4 \} \), oriented counter-clockwise ## 1.34 Draw the path parameterized by \[ \gamma(t) = \cos(t) \cdot \cosh(t) + i \cdot \sin(t) \cdot \sinh(t), \quad 0 \leq t \leq 2\pi. \] ## 1.35 Let \( G \) be the annulus determined by the inequalities \( 2 < |z| < 3 \). This is a connected open set. Find the maximum number of horizontal and vertical segments in \( G \) needed to connect two points of \( G \). --- ### Optional Lab Open your favorite web browser and search for the complex function grapher for the open-source software GeoGebra. #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 28 Context: # Complex Numbers 1. Convert the following complex numbers into their polar representation, i.e., give the absolute value and the argument of the number: \[ 3 + 4i = r e^{i\theta} = 2 + 2i = \frac{1}{\sqrt{3 + i}} = \] After you have finished computing these numbers, check your answers with the program. 2. Convert the following complex numbers given in polar representation into their rectangular representation: \[ 2e^{i\theta} = 3e^{i\phi} = \frac{1}{e^{i\varphi}} = -\sqrt{2}e^{2i\phi} = \] After you have finished computing these numbers, check your answers with the program. 3. Pick your favorite five numbers from the ones that you’ve played around with and put them in the tables below, in both rectangular and polar form. Apply the functions listed to your numbers. Think about which representation is more helpful in each instance. | rectangular | polar | |-------------|------------| | z + 1 | | | z + 2 - i | | | 2z | | | -z | | | z^2 | | | Re(z) | | | Im(z) | | | /Im(z) | | | |z| | | | 1/2 | | 4. Play with other examples until you get a feel for these functions. #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 29 Context: # Chapter 2 ## Differentiation Mathematical study and research are very suggestive of generalizing. Whispers make several efforts before he climbed the Mathematics in the 1860s and even then it cut the life short of this party. Now, however, any source can be looked up for a small act, and perhaps one does not appreciate the difficulty of the original account. So in mathematics, it may be found hard to realize the great initial difficulty of making a little step which may seem so natural and obvious, and it may not be surprising if such a step has been found and lost again. **Louis Nodel (1888–1972)** We will now start our study of complex functions. The fundamental concept on which all of calculus is based is that of a limit—it allows us to develop the central properties of continuity and differentiability of functions. Our goal in this chapter is to do the same for complex functions. ### 2.1 Limits and Continuity **Definition.** A (complex) function \( f \) is a map from a subset \( G \subset \mathbb{C} \) to \( \mathbb{C} \); in this situation we will write \( f : G \to \mathbb{C} \) and call \( G \) the domain of \( f \). This means that each element \( z \in G \) gets mapped to exactly one complex number, called the image of \( z \) and usually denoted by \( f(z) \). So far there is nothing that makes complex functions any more special than, say, functions from \( \mathbb{R} \) to \( \mathbb{R} \). In fact, we can construct many familiar looking functions from the standard calculus repertoire, such as \( f(z) = z \) (the identity map), \( f(z) = z^2 + 1 \), or \( f(z) = \frac{1}{z} \). The former three could be defined on all of \( \mathbb{C} \), whereas for the latter we have to exclude the origin \( z = 0 \) from the domain. On the other hand, we could construct some functions that make use of a certain representation of \( z \), for example, \( f(x, y) = x - 2iy \), \( f(x, y) = y - i \), or \( f(z, \varphi) = 2r e^{i\varphi} \). #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 30 Context: Next we define limits of a function. The philosophy of the following definition is not restricted to complex functions, but for sake of simplicity we limit it only for those functions. ## Definition Suppose \( f : G \to \mathbb{C} \) and \( z_0 \) is an accumulation point of \( G \). If \( \epsilon \) is a complex number such that for every \( \delta > 0 \) we can find \( \delta > 0 \) such that \( 0 < |z - z_0| < \delta \), then \[ \lim_{z \to z_0} f(z) = w_0. \] This definition is the same as is found in most calculus texts. The reason we require that \( z_0 \) is an accumulation point of the domain is just that we need to be sure that there are points \( z \) of the domain that are arbitrarily close to \( z_0 \). Just as in the real case, our definition (i.e., the part that says \( 0 < |z - z_0| \)) does not require that \( z_0 \) is in the domain of \( f \); the definition explicitly ignores the value of \( f(z_0) \). ## Example 2.1 Let's prove that \( \lim_{z \to 2} z^2 = 4 \). Given \( \epsilon > 0 \), we need to determine \( \delta > 0 \) such that \( 0 < |z - 2| < \delta \) implies \( |z^2 - 4| < \epsilon \). We rewrite: \[ |z^2 - 4| = |z - 2||z + 2|. \] If we choose \( \delta < 1 \), then the factor \( |z + 2| \) on the right can be bounded by 3 (draw a picture). This means that any \( \delta < \min\{1, \frac{\epsilon}{3}\} \) should do the trick; in this case, \( 0 < |z - 2| < \delta \) implies: \[ |z^2 - 4| < 3 |z - 2| < 3 \delta < \epsilon. \] This proves \( \lim_{z \to 2} z^2 = 4 \). #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 32 Context: ``` # Differentiation Proof of **Theorem**. Assume that \( x_0 \in G \) (otherwise there is nothing to prove), and let \( L = \lim_{x \to x_0} f(x) \) and \( M = \lim_{x \to x_0} g(x) \). Then we know that given \( \epsilon > 0 \), we can find \( \delta_1, \delta_2 > 0 \) such that \[ 0 < |x - x_0| < \delta_1 \implies |f(x) - L| < \frac{\epsilon}{2} \] and \[ 0 < |x - x_0| < \delta_2 \implies |g(x) - M| < \frac{\epsilon}{2}. \] Thus, choosing \( \delta = \min(\delta_1, \delta_2) \), we infer that \( 0 < |x - x_0| < \delta \) implies \[ |f(x) + g(x) - (L + M)| \leq |f(x) - L| + |g(x) - M| < \epsilon. \] Here we used the triangle inequality (Proposition 1.6). This proves that \[ \lim_{x \to x_0} (f(x) + g(x)) = L + M, \] which was our claim. Because the definition of the limit is somewhat elaborate, the following fundamental definition looks almost trivial. ## Definition Suppose \( f : G \to \mathbb{C} \). If \( x_0 \in G \) and either \( x_0 \) is an isolated point of \( G \) or \[ \lim_{x \to x_0} f(x) = f(x_0) \] then \( f \) is continuous at \( x_0 \). More generally, \( f \) is continuous on \( E \subset G \) if it is continuous at every \( x \in E \). However, in almost all proofs using continuity it is necessary to interpret this in terms of \( \epsilon \) and \( \delta \). ## Definition Suppose \( f : G \to \mathbb{C} \) and \( x_0 \in G \). Then \( f \) is continuous at \( x_0 \) if, for every positive real number \( \epsilon \) there is a positive real number \( \delta > 0 \) such that \[ |f(x) - f(x_0)| < \epsilon \quad \text{for all } x \in E \text{ satisfying } |x - x_0| < \delta. \] See Exercise 2.11 for a proof that these definitions are equivalent. ``` #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 33 Context: # LIMITS AND CONTINUITY ## Example 2.5 We already proved (in Example 2.1) that the function \( f : C \to C \) given by \( f(x) = x^2 \) is continuous at \( x = 1 \). You're invited (Exercise 2.8) to extend our proof to show that, in fact, this function is continuous on \( C \). On the other hand, let \( g : C \to C \) be given by \[ g(x) = \begin{cases} \frac{1}{x} & \text{if } x \neq 0 \\ 1 & \text{if } x = 0 \end{cases} \] In Example 2.3 we proved that \( g \) is not continuous at \( x = 0 \). However, this is its only point of discontinuity (Exercise 2.9). Just as in the real case, we can "take the limit inside" a continuous function, by considering composition of functions. ## Definition The image of the function \( g : G \to C \) is the set \( \{ g(x) : x \in G \} \). If the image of \( g \) is contained in the domain of another function \( f : H \to C \), we define the composition \( f \circ g : G \to C \) through \[ (f \circ g)(x) = f(g(x)). \] ## Proposition 2.6 Let \( g : G \to C \) with image contained in \( H \), and let \( f : H \to C \). Suppose \( x_n \) is an accumulation point of \( G \), \( \lim_{n \to \infty} g(x_n) = w_0 \in H \), and \( f \) is continuous at \( w_0 \). Then \[ \lim_{n \to \infty} f(g(x_n)) = f\left(\lim_{n \to \infty} g(x_n)\right), \] in short, \[ \lim_{n \to \infty} f(g(x_n)) = f\left( \lim_{n \to \infty} g(x_n) \right). \] **Proof.** Given \( \epsilon > 0 \), we know there is \( \delta > 0 \) such that \[ |w - w_0| < \delta \implies |f(w) - f(w_0)| < \epsilon. \] For this, we also know there is a \( \eta > 0 \) such that \[ 0 < |x - z_0| < \eta \implies |g(x) - w| < \delta. \] Stringing these two implications together gives that \[ 0 < |x - z_0| < \eta \implies |f(g(x)) - f(w_0)| < \epsilon. \] #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 37 Context: # Differentiability and Holomorphicity ## Proof Let \( \gamma_1(t) \) and \( \gamma_2(t) \) be parametrizations of the two paths such that \( \gamma_1(0) = \gamma_2(0) = \gamma(0) \). Then \( \gamma_2(t) \) (considered as a vector) is the tangent vector of \( \gamma_1 \) at the point \( a \), and \( \gamma_2'(0) \) is the tangent vector of \( \gamma_2 \) at the point \( f(a) \) given by: \[ \frac{d}{dt} \bigg|_{t=0} \gamma_1(t) = f'(\gamma_1(0)) \cdot \gamma_1'(0) \] and similarly, the tangent vector of \( f(\gamma_2) \) at the point \( f(a) \) is \( f'(\gamma_2'(0)) = f'[\gamma_2'(0)] \). This means that the action of \( f \) multiplies the two tangent vectors \( \gamma_1'(0) \) and \( \gamma_2'(0) \) by the same nonzero complex number \( f'(a) \), and so the two tangent vectors get dilated by \( |f'(a)| \) (which does not affect their direction) and rotated by the same angle (an argument of \( f(a) \)). We end this section with yet another differentiation rule, that for inverse functions. As in the real case, this rule is only defined for functions that are bijections. ## Definition A function \( f: G \to H \) is one-to-one if for every image \( w \in H \) there is a unique \( z \in G \) such that \( f(z) = w \). The function is onto if every \( w \in H \) has a preimage \( z \in G \) that gives \( f(z) = w \). A bijection is a function that is both one-to-one and onto. If \( f: G \to H \) is a bijection then \( g: H \to G \) is the inverse of \( f \) if \( f(g(z)) = z \) for all \( z \in H \); in other words, the composition \( g \circ f \) is the identity function on \( H \). ## Proposition 12 Suppose \( G, H \subset \mathbb{C} \) are open sets, \( f: G \to H \) is a bijection, \( s: H \to G \) is the inverse function of \( f \), and \( z_0 \in H \). If \( f \) is differentiable at \( s(z_0) \) with \( f'(s(z_0)) \neq 0 \) and \( g \) is continuous at \( z_0 \), then \( g \) is differentiable at \( z_0 \), with \[ g'(z_0) = \frac{1}{f'(s(z_0))} \] ## Proof Since \( f(z_0) = z \) for all \( z \in H \), \[ g'(z_0) = \lim_{z \to f(z_0)} \frac{g(z) - g(f(z_0))}{z - f(z_0)} = \lim_{z \to f(z_0)} \frac{1}{f'(s(z_0))} \cdot \frac{g(f(z_0)) - g(z)}{g'(z_0)} \] \[ = \lim_{z \to f(z_0)} \frac{g(f(z_0)) - g(z)}{g'(z_0)(z - f(z_0))} \] #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 41 Context: In the second case, \( \Delta z = i \Delta y \) and \[ f'(z_0) = \lim_{y_0 \to y_0} \frac{f(x_0 + i y) - f(x_0)}{i y} = \lim_{y_0 \to 0} \frac{f(x_0 + i y) - f(x_0)}{y} \] Thus, we have shown that \( f'(z_0) = f_x(z_0) - i f_y(z_0) \). (b) Suppose the Cauchy-Riemann equation (2.2) holds and the partial derivatives \( f_x \) and \( f_y \) are continuous in an open disk centered at \( z_0 \). Our goal is to prove that \( f'(z_0) = f_x(z_0) + i f_y(z_0) \). By (2.2), \[ f(z_0) = x_0 + i y_0 \quad f(z_0 + \Delta z) = f(z_0 + \Delta x + i \Delta y) \] On the other hand, we can rewrite the difference quotient for \( f'(z_0) \) as \[ f(z_0 + \Delta z) - f(z_0) = \frac{f(z_0 + \Delta z) - f(z_0)}{\Delta z} = \frac{f(z_0 + \Delta x + i \Delta y) - f(z_0)}{\Delta z} = \frac{f(z_0 + \Delta x) - f(z_0)}{\Delta z} + \frac{f(z_0 + \Delta y) - f(z_0)}{\Delta z} \] Thus, \[ \lim_{\Delta z \to 0} \frac{f(z_0 + \Delta z) - f(z_0)}{\Delta z} = \lim_{\Delta y \to 0} \frac{f(z_0 + i \Delta y) - f(z_0)}{\Delta y} = \lim_{\Delta x \to 0} \left( \frac{f(z_0 + \Delta x + i \Delta y) - f(z_0)}{\Delta z} \right). \] We claim that both limits on the right-hand side are 0, so we have achieved our set goal. The fractions \( \frac{\Delta y}{\Delta z} \) and \( \frac{\Delta x}{\Delta z} \) are bounded in absolute value by 1, so we just need to see that the differences in parentheses are 0. The second term on the right-hand side of (2.5) has a limit of 0 since, by definition, \[ f(z) = \lim_{\Delta z \to 0} \frac{f(z + \Delta z) - f(z)}{\Delta z} \] and taking the limit here as \( \Delta z \to 0 \) is the same as taking the limit as \( \Delta x \to 0 \). #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 42 Context: ```markdown 36 # Differentiation We cannot do something equivalent for the first term in (2.5), since now both \( \Delta x \) and \( \Delta y \) are involved, and both change as \( \Delta x \to 0 \). Instead, we apply the Mean-Value Theorem A.2 for real functions\(^1\) to the real and imaginary parts \( u(x, y) \) and \( v(x, y) \) of \( f(z) \). Theorem A.2 gives real numbers \( \xi \) and \( \eta \) such that \[ \frac{u(x_0 + \Delta x, y_0 + \Delta y) - u(x_0, y_0)}{\Delta y} = u_y(x_0, y_0 + \eta) \] and \[ \frac{v(x_0 + \Delta x, y_0 + \Delta y) - v(x_0, y_0)}{\Delta y} = v_y(x_0, y_0 + \xi). \] Thus, \[ \frac{f(x_0 + \Delta x + i \Delta y) - f(x_0 + i y)}{\Delta y} = \frac{u(x_0 + \Delta x, y_0 + \Delta y) - u(x_0, y_0)}{\Delta y} - iv_x(x_0, y_0) \] \[ = \left( \frac{u(x_0 + \Delta x, y_0 + \Delta y) - u(x_0, y_0)}{\Delta y} \right) - i v_y(x_0, y_0) + \left( \frac{u(x_0 + \Delta x, y_0 + \Delta y) - u(x_0, y_0)}{\Delta y} - iv_y(x_0, y_0) \right) = u_y(x_0, y_0) - iv_y(x_0, y_0). \tag{2.6} \] Because \( u \) and \( v \) are continuous at \( (x_0, y_0) \), \[ \lim_{\Delta y \to 0} u(x_0 + \Delta x, y_0 + \Delta y) = u(x_0, y_0) \] and \[ \lim_{\Delta y \to 0} v(x_0 + \Delta x, y_0 + \Delta y) = v(x_0, y_0), \] and so (2.6) goes to \( \Delta z \to 0 \), which we set out to prove. ## 2.4 Constant Functions As a simple application of the definition of the derivative of a complex function, we consider functions that have a derivative of 0. In a typical calculus course, one of the first applications of the Mean-Value Theorem for real-valued functions (Theorem A.1) is to constant functions. ``` #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 43 Context: # CONSTANT FUNCTIONS ## Proposition 2.16 If \( I \) is an interval and \( f : I \rightarrow \mathbb{R} \) is a real-valued function with \( f'(x) \) defined and equal to 0 for all \( x \in I \), then there is a constant \( c \in \mathbb{R} \) such that \( f(x) = c \) for all \( x \in I \). **Proof:** The Mean-Value Theorem A.2 says that for any \( x, y \in I \), \[ f(y) - f(x) = f' \left( c \right) (y - x) \] for some \( 0 < a < 1 \). Now \( f' (x + a(y - x)) = 0 \), so the above equation yields \( f(y) = f(x) \). Since this is true for any \( x, y \in I \), the function \( f \) must be constant on \( I \). We do not (yet) have a complex version of the Mean-Value Theorem, and so we will use a different argument to prove that a complex function whose derivative is always 0 must be constant. Our proof of Proposition 2.16 required two key features of the function \( f \), both of which are somewhat obviously necessary. The first is that \( f \) be differentiable everywhere in its domain. In fact, if \( f \) is not differentiable everywhere, we can construct functions that have zero derivative almost everywhere but that have infinitely many values in their image. The second key feature is that the interval \( I \) is connected. It is certainly important for the domain to be connected in both the real and complex cases. For instance, if we define the function \[ f(z) = \begin{cases} 1 & \text{if } \text{Re}(z) \geq 0 \\ 2 & \text{if } \text{Re}(z) < 0 \end{cases} \] then \( f(z) \) is 0 for all \( z \in I \), but \( f \) is not constant. This may seem like a silly example, but it illustrates a pivotal to proving a function is constant that we must be careful of. Recall that a region of \( \mathbb{C} \) is an open connected subset. ## Theorem 2.17 If \( G \subset \mathbb{C} \) is a region and \( f : G \rightarrow \mathbb{C} \) is a complex-valued function with \( f'(z) \) defined and equal to 0 for all \( z \in G \), then \( f \) is constant. **Proof:** We will show that \( f \) is constant along horizontal segments and along vertical segments in \( G \). #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 44 Context: # Differentiation Suppose that \( H \) is a horizontal line segment in \( G \). Thus there is some number \( y_0 \in \mathbb{R} \) such that the imaginary part of any \( z \in H \) is \( y_0 \). Now consider the real part \( u(x) \) of the function \( f \) for \( x \in H \). Since \( \text{Im}(z) = y_0 \) is constant on \( H \), we can consider \( f(z) = u(x) + iv(y) \) to be just a function of \( x \) for \( z = x + iy \). By assumption, \( f(z) = 0 \) for \( z \in H \) we have \( u(x) = \text{Re}(f(z)) = 0 \). Thus, by Proposition 2.16, \( u(x) \) is constant on \( H \). We can argue the same way to see that the imaginary part \( v(y) \) of \( f(z) \) is constant on \( H \), since \( v(y) = \text{Im}(f(z)) = 0 \) on \( H \). Since both the real and imaginary parts of \( f(z) \) are constant on \( H \), the function \( f(z) \) itself is constant on \( H \). This same argument works for vertical segments, interchanging the roles of the real and imaginary parts. We have thus proved that if \( f \) is constant along horizontal segments and along vertical segments in \( G \), and if \( x \) and \( y \) are two points in \( G \) that can be connected by a path composed of horizontal and vertical segments, we conclude that \( f(x) = f(y) \). But any two points of a region may be connected by finitely many such segments by Theorem 1.12, so \( f \) has the same value at any two points of \( G \), thus proving the theorem. There are a number of surprising applications of Theorem 2.17; see, e.g., Exercises 2.20 and 2.21. ## Exercises 1. Use the definition of limit to show for any \( z \in G \) that \( \lim_{z \to z_0} (az + b) = az_0 + b \). 2. Evaluate the following limits or explain why they don't exist. (a) \( \lim_{x \to 2} \frac{x^2 - 4}{x - 2} \) (b) \( \lim_{x \to 1} \ln(x + (2x + y)) \) 3. Prove that, if a limit exists, then it is unique. 4. Prove Proposition 2.4. 5. Let \( G \subset \mathbb{C} \) and suppose \( z_0 \) is an accumulation point of \( G \). Show that \( \lim_{z \to z_0} f(z) = 0 \) if and only if \( \lim_{z \to z_0} |f(z)| = 0 \). #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 45 Context: # CONSTANT FUNCTIONS ## 2.6. Proposition 2.2 is useful for showing that limits do not exist, but it is not at all useful for showing that a limit does exist. For example, define $$ f(x,y) = \frac{xy}{x^2 + y^2} \quad \text{where } (x,y) \neq (0,0). $$ Show that the limits of \( f \) along all straight lines through the origin exist and are equal, but \( \lim_{(x,y) \to (0,0)} f(x,y) \) does not exist. (Hint: Consider the limit along the parabola \( y = x^2 \).) ## 2.7. Suppose that \( f(z) = u(x,y) + iv(x,y) \) and \( u_0 = u(x_0,y_0) \) and \( v_0 = v(x_0,y_0) \). Prove that $$ \lim_{(x,y) \to (x_0,y_0)} f(z) = u_0 + iv_0 $$ if and only if $$ \lim_{(x,y) \to (x_0,y_0)} u'(x,y) = u_0 \quad \text{and} \quad \lim_{(x,y) \to (x_0,y_0)} v'(x,y) = v_0. $$ ## 2.8. Show that the function \( f : C \to \mathbb{C} \) given by \( f(z) = z^2 \) is continuous on \( C \). ## 2.9. Show that the function \( g : C \to \mathbb{C} \) given by $$ g(z) = \begin{cases} \frac{1}{z} & \text{if } z \neq 0 \\ 1 & \text{if } z = 0 \end{cases} $$ is continuous on \( C \setminus \{0\} \). ## 2.10. Determine where each of the following functions \( f : C \to C \) is continuous: (a) $$ f(z) = \begin{cases} 0 & \text{if } z = 0 \text{ or } |z| \text{ is irrational} \\ 1 & \text{if } |z| \in \mathbb{Q} \setminus \{0\} \end{cases} $$ (b) $$ f(z) = \begin{cases} 0 & \text{if } z = 0 \\ \sin z & \text{if } z \neq 0 \text{ and } z \neq re^{i\theta} \end{cases} $$ #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 46 Context: ``` ## 2.11. Show that the two definitions of continuity in Section 2.1 are equivalent. Consider separately the cases where \(x_0\) is an accumulation point of \(G\) and where \(x_0\) is an isolated point of \(G\). ## 2.12. Consider the function \(f : C \setminus \{0\} \to G\) given by \(f(z) = \frac{1}{z}\). Apply the definition of the derivative to give a direct proof that \(f'(z) = -\frac{1}{z^2}\). ## 2.13. Prove Proposition 2.6. ## 2.14. Prove Proposition 2.10. ## 2.15. Find the derivative of the function \(T(z) = \frac{a z^2 + b z + c}{e^{z}}\), where \(a, b, c \in C\) with \(ad - bc = 0\). When is \(T'(z) = 0\)? ## 2.16. Prove that if \(f(z)\) is given by a polynomial in \(z\) then \(f\) is entire. What can you say if \(f(z)\) is given by a polynomial in \(z = x + iy\) and \(y = \Im z\)? ## 2.17. Prove or find a counterexample. If \(x\) and \(y\) are real valued and continuous, then \(f(z) = u(x,y) + iv(x,y)\) is continuous; if \(u\) and \(v\) are (real) differentiable then \(f\) is (complex) differentiable. ## 2.18. Where are the following functions differentiable? Where are they holomorphic? Determine their derivatives at points where they are differentiable. 1. \(f(z) = e^{z} \cdot e^{c+iy}\) 2. \(f(z) = 2x + iy\) 3. \(f(z) = x^3 + iy\) 4. \(f(z) = e^{c-iy}\) 5. \(f(z) = \cosh x - i \sin y\) 6. \(f(z) = \Im z\) 7. \(f(z) = |z|^2 = x^2 + y^2\) 8. \(f(z) = z^2\) 9. \(f(z) = \Im z\) 10. \(f(z) = \frac{4}{Re z} - i(Re z)^2\) ``` #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 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: 52 Context: # Examples of Functions For real numbers \( a, b, \gamma, y, r \) and \( s \) that satisfy \( \beta^2 + \gamma^2 > 4\alpha \) (Exercise 3.3). The form (3.1) is more convenient for us, because it includes the possibility that the equation describes a line (precisely when \( a = 0 \)). Suppose \( x + iy \) satisfies (3.1); we need to prove that \( x + iy = \frac{x - iy}{x^2 + y^2} \) satisfies a similar equation. \[ u + iv = \frac{x - iy}{x^2 + y^2}. \] We can rewrite (3.1) as: \[ 0 = \alpha + \frac{y}{x^2 + y^2} + \frac{8}{x^2 + y^2}. \] This leads to: \[ 0 = \alpha + \beta u + \gamma (a + b + r^2). \tag{3.2} \] But this equation, in conjunction with Exercise 3.3, says that \( x + iy \) lies on a circle or line. ## 3.2 Infinity and the Cross Ratio In the context of Möbius transformations, it is useful to introduce a formal way of saying that a function \( f \) "blows up" in absolute value, and this gives rise to a notion of infinity. **Definition.** Suppose \( f : G \to \mathbb{C} \). 1. \( \lim_{x \to a} f(x) = \infty \) means that for every \( M > 0 \) we can find \( \delta > 0 \) so that, for all \( x \in G \) satisfying \( 0 < |x - a| < \delta \), we have \( |f(x)| > M \). 2. \( \lim_{x \to b} f(x) = L \) means that for every \( \epsilon > 0 \) we can find \( N > 0 \) so that, for all \( x \in G \) satisfying \( |x| > N \), we have \( |f(x) - L| < \epsilon \). 3. \( \lim_{x \to \infty} f(x) = \infty \) means that for every \( M > 0 \) we can find \( N > 0 \) so that, for all \( x \in G \) satisfying \( |x| > N \), we have \( |f(x)| > M \). In the first definition we require that \( a \) be an accumulation point of \( G \), while in the second and third we require that \( b \) be an "extended accumulation point" of \( G \), in the sense that for every \( B > 0 \) there is some \( z \in G \) with \( |z| > B \). As in Section 2.1, the limit, in any of these senses, is unique if it exists. #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 57 Context: # Stereographic Projection To begin, we think of \(C\) as the \((x,y)\)-plane in \(\mathbb{R}^3\), that is, \(C = \{(x,y,0) \in \mathbb{R}^3\}\). To describe stereographic projection, we will be less concerned with actual complex numbers \(x + iy\) and more concerned with their coordinates. Consider the unit sphere \[ S^2 = \{(x,y,z) \in \mathbb{R}^3 : x^2 + y^2 + z^2 = 1\}. \] The sphere and the complex plane intersect in the set \(\{(x,y,0) : x^2 + y^2 = 1\}\), which corresponds to the equator on the sphere and the unit circle on the complex plane, as depicted in Figure 3.1. Let \(N = (0,0,1)\), the north pole of \(S^2\), and let \(S = (0,0,-1)\), the south pole. ![Figure 3.1: Setting up stereographic projection.](#) ## Definition The stereographic projection of \(S^2\) to \(\hat{C}\) from \(N\) is the map \(\varphi: S^2 \setminus \{N\} \to \hat{C}\) defined as follows. For any point \(P \in S^2 \setminus \{N\}\), let \(x\) be the \(x\)-coordinate of \(P\) and let \(l\) be the line through \(N\) and \(P\). The \(z\)-coordinate of \(P\) is strictly less than \(1\), the line through \(N\) and \(P\) intersects \(C\) in exactly one point \(Q\). Define \(\varphi(P) = Q\). We also declare that \(\varphi(N) = \infty\). ## Proposition 3.14 The map \(\varphi\) is given by \[ \varphi(x,y,z) = \left( \frac{x}{1-z}, \frac{y}{1-z}, 0 \right) \quad \text{if } z \neq 1, \] \[ \varphi(x,y,z) = \infty \quad \text{if } z = 1. \] #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 60 Context: # Examples of Functions Project it to the plane by the stereographic projection \( \pi \), apply \( f \) to the point that results, and then pull this point back to \( S^2 \) by \( \pi^{-1} \). We know \( \varphi(x,y,z) = \left( \frac{x}{1-z}, \frac{y}{1-z} \right) \) which we now regard as the complex number \[ \rho + i \gamma = \frac{-x}{1-z} + i \frac{y}{1-z}. \] We know from a previous calculation that \( p^2 + q^2 = \frac{1}{z^2} \), which gives \( x^2 + y^2 = (1 + z)(1 - z) \). Thus, \[ f\left( \frac{x}{1-z} + i \frac{y}{1-z} \right) = \frac{1 - z}{x + iy} = \frac{1 - z}{(1 - z) x + i (1 - z) y} = \frac{1 - 2X - iY}{x^2 + y^2}. \] Rather than plug this result into the formulas for \( \pi^* \), we can just ask what triple of numbers will be mapped to this particular pair using the formulas \( \varphi(x,y,z) = \left( \frac{x}{1-z}, \frac{y}{1-z} \right) \). Thus, we have shown that the effect of \( f \) on \( S^2 \) is to take \( (x,y,z) \) to \( (x, -y, -z) \). This is a rotation around the \( x \)-axis by 180 degrees. We now have a second argument that \( f(z) \) takes circles and lines to circles and lines. A circle in \( C \) is taken to a circle on \( S^2 \) by \( \pi \). Then \( f(z) = \frac{1}{z} \) rotates the sphere which certainly takes circles to circles. Now \( g \) takes circles back to circles and lines. We can also say that the circles that go to lines under \( f(z) = \frac{1}{z} \) are the circles through 0 because 0 is mapped to \( (0, 0) \) under \( \pi \), and so a circle through 0 in \( C \) goes to a circle through the south pole on \( S^2 \). Now 180-degree rotation about the \( x \)-axis takes the south pole to the north pole, and our circle is now passing through \( N \). But we know that \( g \) will take this circle to a line in \( C \). We end by mentioning that there is, in fact, a way of putting the complex metric on \( S^2 \). It is certainly not the (finite) distance function induced by \( \pi \). Indeed, the origin in the complex plane corresponds to the south pole of \( S^2 \). We have to be able to arbitrarily far away from the origin in \( C \), so the complex distance function has to increasingly grow with the \( z \)-coordinate. The closest points are to the north pole \( N \) corresponding to 0 in \( C \), the larger their distance to the origin, and to each other! In this light, a 'line' in the Riemann sphere \( S^2 \) corresponds to a circle in \( S^2 \) through \( N \). In the regular sphere, the circle has finite lengths, but as a line on the Riemann sphere with the complex metric, it has infinite length. #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 62 Context: # EXAMPLES OF FUNCTIONS They are continuous in \( \mathbb{C} \) and satisfy the Cauchy–Riemann equation (2.2): \[ \frac{\partial f}{\partial x} (z) = \frac{\partial f}{\partial y} (z) \] for all \( z \in \mathbb{C} \). Thus Theorem 2.13 says that \( f(z) = \exp(z) \) is entire with derivative \[ f'(z) = \frac{\partial f}{\partial x} (z) = \exp(z). \] We should make sure that the complex exponential function specializes to the real exponential function for real arguments: indeed, if \( z = x \in \mathbb{R} \), then \[ \exp(x) = e^x = \left( \cos(0) + i\sin(0) \right) = e^x. \] The trigonometric functions—sine, cosine, tangent, cotangent, etc.—also have complex analogues; however, they do not play the same prominent role as in the real case. In fact, we can define them as merely being special combinations of the exponential function. ## Definition The (complex) sine and cosine are defined as: \[ \sin z = \frac{1}{2i} \left( \exp(iz) - \exp(-iz) \right) \] and \[ \cos z = \frac{1}{2} \left( \exp(iz) + \exp(-iz) \right), \] respectively. The tangent and cotangent are defined as: \[ \tan z = \frac{\sin z}{\cos z} = \frac{e^{iz} - e^{-iz}}{i(e^{iz} + e^{-iz}) - 1} \] and \[ \cot z = \frac{\cos z}{\sin z} = \frac{e^{iz} + 1}{e^{iz} - e^{-iz}}, \] respectively. Note that to write tangent and cotangent in terms of the exponential function, we used the fact that \( \exp(iy) - \exp(-iy) = 0 \). Because \( \exp \) is entire, so are sine and cosine. #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 63 Context: # Exponential and Trigonometric Functions ![Image properties of the exponential function](path/to/image.png) Figure 3.2: Image properties of the exponential function. As with the exponential function, we should make sure that we're not redefining the real sine and cosine: if \( z \in \mathbb{R} \) then \[ \sin x = \frac{1}{2i} \left( \exp(ix) - \exp(-ix) \right) \] \[ = \frac{1}{2} \left( \cos x + i \sin x - \cos(-x) - i \sin(-x) \right) = \sin x. \] #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 66 Context: Example 3.18. Here are a few evaluations of Log to illustrate this function: - Log(2) = ln(2) + Arg(2) = ln(2) - Log(-i) = ln(1) + Arg(-1) = \( \frac{3\pi}{2} \) - Log(-3) = ln(3) + Arg(-3) = ln(3) + \( \pi i \) - Log(1 - i) = ln(√2) + Arg(1 - i) = \( \frac{1}{2} \ln(2) - \frac{\pi i}{4} \) The principal logarithm is not continuous on the negative part of the real line, and so Log is a branch of the logarithm on \( \mathbb{C} \setminus \mathbb{R}^+ \). Any branch of the logarithm on a region G can be similarly extended to a function defined on \( G \setminus \{0\} \). Furthermore, the evaluation of any branch of the logarithm at a specific \( z_n \) can differ from Log(z) only by a multiple of \( 2\pi i \); the reason for this is once more the periodicity of the exponential function. So what about the second equation in (3.3), namely, Log(exp(z)) = z? Let’s try the principal logarithm: if \( z = x + iy \) then Log(exp(z)) = ln|\( z \)| + Arg(exp(z)) = ln|\( z \)| + Arg(exp(z)). The right-hand side is equal to \( z = x + iy \) if and only if \( y \in (-\pi, \pi) \). Something similar will happen with any other branch of Log of the logarithm—there will always be many z's for which Log(exp(z)) ≠ \( z \). A warning sign pointing in a similar direction concerns the much-cherished real logarithm: the ln(xy) = ln(x) + ln(y) has no analogue in \( \mathbb{C} \). For example, Log(i) + Log(-1) = \( \frac{1}{2} \ln(2 + \sqrt{2}) = \ln 2 + \frac{pi}{4} \) but Log(i - 1) = Log(-1) = \( \ln 2 - \frac{1}{2} \cdot 2 i \). The problem is that we need to come up with an argument convention to define a logarithm and then stick to this convention. There is quite a bit of subtlety here; e.g., the multi-valued map arg z := all possible arguments of z. #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 71 Context: # LOGARITHMS AND COMPLEX EXPONENTIALS ### 3.18 Find a Möbius transformation that maps the unit disk to \(\{x + iy \in \mathbb{C} : x + y > 0\}\). ### 3.19 The Jacobian of a transformation \(z = u(x, y)\), \(w = v(x, y)\) is the determinant of the matrix \[ \begin{bmatrix} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} \\ \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y} \end{bmatrix}. \] Show that if \(f = u + iv\) is holomorphic then the Jacobian equals \(|f'(z)|^2\). ### 3.20 Find the fixed points in \(C\) of \(f(z) = \frac{az + b}{cz + d}\). ### 3.21 Find each Möbius transformation \(f\): (a) \(f\) maps \(0 \mapsto 1\), \(1 \mapsto \infty\), \(\infty \mapsto 0\). (b) \(f\) maps \(1 \mapsto -1\), \(-1 \mapsto i\), and \(i \mapsto -1\). (c) \(f\) maps the \(x\)-axis to \(y = x\), the \(y\)-axis to \(y = -x\), and the unit circle to itself. ### 3.22 (a) Find a Möbius transformation that maps the unit circle to \(\{x + iy \in \mathbb{C} : x + y = 0\}\). (b) Find two Möbius transformations that map the unit disk \[ \{z \in \mathbb{C} : |z| < 1\} \] to \(\{x + iy \in \mathbb{C} : x + y > 0\}\) and \(\{x + iy \in \mathbb{C} : x + y < 0\}\), respectively. ### 3.23 Given \(a \in \mathbb{R} \setminus \{0\}\), show that the image of the line \(y = a\) under inversion is the circle with center \(\frac{1}{a}\) and radius \(\frac{1}{|a|}\). ### 3.24 Suppose \(z_1, z_2\), and \(z_3\) are distinct points in \(\mathbb{C}\). Show that \(z\) is on the circle passing through \(z_1, z_2\), and \(z_3\) if and only if \([z, z_1, z_2, z_3]\) is real or \(\infty\). #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 74 Context: # EXAMPLES OF FUNCTIONS ## 3.41. Convert the following expressions to the form \( z = x + iy \). (Reason carefully.) 1. \( e^z \) 2. \( e^{\cos(\log(3 + 4i))} \) 3. \( i^z \) 4. \( e^{i \pi} \) 5. \( e^{z} \) 6. \( e^{(1 + i) z} \) 7. \( \sqrt{3(1 - i)} \) 8. \( \left( \frac{1}{1} \right)^4 \) ## 3.42. Is \( \arg(E) = -\arg(z) \) true for the multiple-valued argument? What about \( \arg(E) = \Arg(z) \) for the principal branch? ## 3.43. For the multiple-valued logarithm, is there a difference between the set of all values of \( \log(z^2) \) and the set of all values of \( 2 \log(z) \)? (Hint: Try some fixed numbers for \( z \).) ## 3.44. For each of the following functions, determine all complex numbers for which the function is holomorphic. If you run into a logarithm, use the principal value unless otherwise stated. (a) \( \frac{z^2}{2} \) (b) \( \frac{1}{z^2} \) (c) \( \Log(z - 2i + 1) \) where \( \Log(z) = \ln |z| + i \arg(z) \) with \( 0 < \arg(z) < 2\pi \) (d) \( \exp(z) \) (e) \( z - 3i \) (f) \( i^z \) ## 3.45. Find all solutions to the following equations: (a) \( \Log(z) = \frac{1}{2} \) \( \cos(z) = 0 \) (b) \( \Log(z) = \frac{1}{2} \) (c) \( \exp(z) = \pi \) (d) \( \sin(z) = \cos(4) \) (e) \( \exp(z) = \exp(\sqrt{E}) \) (f) \( z = 1 + i \) #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 76 Context: ``` ## 3.54 As in the previous problem, let \( f(z) = z^2 \). Let \( Q \) be the square with vertices at \( 0, 2, 2+2i, \) and \( 2i \). Draw \( f(Q) \) and identify the types of image curves corresponding to the segments from \( 2 \) to \( 2+2i \) and from \( 2+2i \) to \( 2i \). They are not parts of either straight lines or circles. **Hint:** You can write the vertical segment parametrically as \( z(t) = 2 + it \). Eliminate the parameter in \( u + iv = f(t) \) to get a \( (u, v) \) equation for the image curve. Exercises 3.53 and 3.54 are related to the cover picture of this book. ``` #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 80 Context: ``` 74 # INTEGRATION For any parametrization \( \gamma(t) \), \( a \leq t \leq b \). Naturally, the length of a piecewise smooth path \( \gamma(t) \) is the sum of the lengths of its smooth components. ## Example 4.4 Let \( \gamma \) be the line segment from \( 0 \) to \( 1 + i \), which can be parametrized by \( \gamma(t) = t + it \) for \( 0 \leq t \leq 1 \). Then: \[ \text{length}(\gamma) = \int_{0}^{1} \left| 1 + i \frac{d}{dt} \right| dt = \int_{0}^{1} \sqrt{2} dt = \sqrt{2}. \] ## Example 4.5 Let \( \gamma(t) \) be the unit circle, which can be parametrized by \( \gamma(t) = e^{it} \) for \( 0 \leq t \leq 2\pi \). Then: \[ \text{length}(\gamma) = \int_{0}^{2\pi} \left| \frac{d\gamma}{dt} \right| dt = \int_{0}^{2\pi} dt = 2\pi. \] Now we observe some basic facts about how the line integral behaves with respect to function addition, scalar multiplication, inverse parametrization, and path concatenation; we also give an upper bound for the absolute value of an integral, which we will make use of time and again. ## Proposition 4.6 Suppose \( f \) is a piecewise smooth path, \( f \) and \( g \) are complex functions which are continuous on \( \gamma \) and \( \gamma \subset \mathbb{C} \). (a) \[ \int_{\gamma} (f + g) \, dz = \int_{\gamma} f \, dz + \int_{\gamma} g \, dz. \] (b) If \( \gamma \) is parametrized by \( \gamma(t) \), \( a \leq t \leq b \), we define the path \( \gamma \) from \( \gamma(a+b) \), \( a \leq t \leq b \). Then: \[ \int_{\gamma} f \, dz = \int_{\gamma} f \, dz + \int_{\gamma} g \, dz. \] (c) If \( \gamma_1 \) and \( \gamma_2 \) are piecewise smooth paths such that \( \gamma_2 \) starts where \( \gamma_1 \) ends, we define the path \( \gamma_1 \# \gamma_2 \) by following \( \gamma_1 \) to its end and then continuing on \( \gamma_2 \) to its end. Then: \[ \int_{\gamma_1 \# \gamma_2} f \, dz = \int_{\gamma_1} f \, dz + \int_{\gamma_2} f \, dz. \] (d) \[ \left| \int_{\gamma} f \, dz \right| \leq \max |f(z)| \cdot \text{length}(\gamma). \] The path \( -\gamma \) defined in (b) is the path that we obtain by traveling through \( \gamma \) in the opposite direction. ``` #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 81 Context: # Definition and Basic Properties **Proof.** (a) follows directly from the definition of the integral and Theorem A.4, the analogous theorem from calculus. (b) follows with the real change of variables \( I = a + b - c \): \[ f = \int_{y_1}^{y_2} f(y(a + b - c)) \, dt \] \[ = -\int_{y_1}^{y_2} f(y(a + b - c)) \, dt \] \[ = \int_{y_1}^{y_2} f(y)(y(a + b - c)) \, da - \int_{y_1}^{y_2} f(y)(y) \, da = -\int f. \] (c) We need a suitable parametrization \( \gamma(t) \) for \( \gamma_1\), \( \gamma_2\). If \( \gamma_1\) has domain \([a_1, b_1]\) and \( \gamma_2\) has domain \([b_2, d_2]\), then we can use: \[ \gamma(t) = \begin{cases} \gamma_1(t) & \text{if } a_1 \leq t \leq b_1 \\ \gamma_2(t - b_1 + a_2) & \text{if } b_1 < t \leq b_2 - a_2 \end{cases} \] with domain \([a_1, b_1, b_2 - a_2]\). Now we break the integral over \( \gamma_2\) into two pieces and apply the change of variables \( I = e - b_1 + a_2\): \[ \int_{\gamma_2} f = \int_{a_1}^{b_1} f(y(t)) \, dt \] \[ = \int_{a_1}^{b_1} f(y(t)) \, dt + \int_{b_1}^{b_2} f(y(t)) \, dt. \] \[ = \int f + \int f. \] The last step follows since \( y\) restricted to \([a_1, b_1]\) is \( \gamma\) and \( y\) restricted to \([b_1 + b_2 - a_2]\) is a reparametrization of \( \gamma_2\). #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 85 Context: # Antiderivatives Let \( f \) be defined. By the same argument, \[ F(z+h) - F(z) = \int_{z}^{z+h} f = \int_{z}^{z+h} f \] for any path \( \gamma \) from \( z \) to \( z+h \). The constant function 1 has the antiderivative \( z \in C \), and so \( F = 1 \), by Theorem 4.11. Thus, \[ F(z+h) - F(z) = \frac{1}{h} \int_{z}^{z+h} (f(w) - f(z)) \, dw = \frac{1}{h} \int_{z}^{z+h} f(w) - f(z) \, dw \] If \(|h|\) is sufficiently small, then the line segment from \( z \) to \( z+h \) will be contained in \( G \), and so by applying the assumptions of our theorem for the third time, \[ F(z+h) - F(z) = \frac{1}{h} \int_{z}^{z+h} (f(w) - f(z)) \, dw = \frac{1}{h} \int_{z}^{z+h} (f(w) - f(z)) \, dw. \quad (4.2) \] We will show that the right-hand side goes to zero as \( h \to 0 \), which will conclude the theorem. Given \( \epsilon > 0 \), we can choose \( \delta > 0 \) such that \[ |w - z| < \delta \implies |f(w) - f(z)| < \epsilon \] because \( f \) is continuous at \( z \). (We can also choose \( \delta \) small enough so that (4.2) holds.) Thus, if \( |h| < \delta \), we can estimate with Proposition 6.6(1): \[ \left| \frac{1}{h} \int_{z}^{z+h} (f(w) - f(z)) \, dw \right| \leq \frac{1}{|h|} \max_{|w|<|h|} |f(w) - f(z)| \cdot \text{length}(h) = \max_{|w|<|h|} |f(w) - f(z)| < \epsilon. \] There are several variations of Theorem 4.15, as we can play with the assumptions about paths in the statement of the theorem. We give one such variation, namely, for polygonal paths, i.e., paths that are composed of unions of line segments. You should convince yourself that the proof of the following result is identical to that of Theorem 4.15. #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 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: 95 Context: So all that we need to finish the proof of Theorem 4.27 is one more fact from topology. But we can prove this one: ## Proposition 4.28 Suppose \( G \) is a simple, closed, piecewise smooth path in the region \( D \). Then \( G \) contains the interior of \( y \) if and only if \( r_g \neq 0 \). **Proof.** One direction is easy: If \( G \) contains the interior of \( y \) and \( [D, \overline{R}] \) is any closed disk in the interior of \( y \), then there is a \( G \)-homotopy from \( y \) to \( C[\overline{R}] \), and \( C[\overline{R}] \subseteq G \). In the other direction, we argue by contradiction: Assume \( r_g \cup D \) does not contain the interior of \( y \). So we can find a point \( w \) in the interior of \( y \) which is not in \( G \). Define \( g(z) = \frac{1}{z - w} \) for \( z \neq w \). Now \( g \) is holomorphic on \( G \) and \( r_g \cup D \) so Corollary 4.20 applies, and we have \( f(y, g(z)) dz = 0 \). On the other hand, choose \( R > 0 \) such that \( D[\overline{R}, R] \) is inside \( y \). There is a homotopy in \( C \) from \( y \) to \( [C, R] \), so Cauchy's Theorem 4.18, plus Exercise 4.4, shows that \( \int_{C} g(z) dz = 2\pi i \). This contradiction finishes the proof. --- Notice that, instead of using topology to prove a theorem about holomorphic functions, we just used holomorphic functions to prove a theorem about topology. ## Example 4.29 Continuing Example 4.26, Theorem 4.27 says that \[ \int_{C(2)} \frac{dz}{z^2 + 1} = \pi \] for any positively oriented, simple, closed, piecewise smooth path \( T \) that contains \( i \) on its inside and that \( C \setminus \{(C \setminus i)\} \)-contractible. ## Example 4.30 To compute \[ \int_{C(3)} \frac{\exp(z)}{z^2 - 2z} \, dz \] we use the partial fractions expansion from Example 4.23: \[ \int_{C(3)} \frac{\exp(z)}{z^2 - 2z} \, dz = \frac{1}{2} \int_{C(3)} \frac{\exp(z)}{C(z)} \, dz - \frac{1}{2} \int_{C(3)} \frac{\exp(z)}{C(z)} \, dz. \] #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 98 Context: ``` 4.10. Prove the following integration by parts statement: Let \( f \) and \( g \) be holomorphic in \( G \), and suppose \( \gamma \subset G \) is a piecewise smooth path from \( z_0 \) to \( z_1 \). Then \[ \int_{\gamma} f(z) g'(z) \, dz = f(z_1) g(z_1) - f(z_0) g(z_0) - \int_{\gamma} f'(z) g(z) \, dz. \] 4.11. Let \( I(k) := \int_0^{2\pi} e^{ik\theta} \, d\theta \). (a) Show that \( I(0) = 1 \). (b) Show that \( I(k) = 0 \) if \( k \) is a nonzero integer. (c) What is \( I(k)^2 \)? 4.12. Compute \( \int_{C[0,1]} z^2 \, dz \). 4.13. Show that \( \int_{\gamma} z^{-1} \, dz = 0 \) for any closed piecewise smooth \( \gamma \) and any integer \( n \neq -1 \). (If \( n \) is negative, assume that \( \gamma \) does not pass through the origin, since otherwise the integral is not defined.) 4.14. Exercise 4.13 excluded \( n = -1 \) for a good reason: Exercise 4.4 gives a counterexample. Generalizing these, if \( n \) is any integer, find a closed path \( \gamma \) so that \( \int_{\gamma} z^{-1} \, dz = 2 \pi i \). 4.15. Taking the previous two exercises one step further, fix \( z_0 \in \mathbb{C} \) and let \( \gamma \) be a simple, closed, positively oriented, piecewise smooth path such that \( z_0 \) is inside \( \gamma \). Show that, for any integer \( n \), \[ \int_{\gamma} (z - z_0)^{-n} \, dz = \begin{cases} 2 \pi i & \text{if } n = -1 \\ 0 & \text{otherwise}. \end{cases} \] 4.16. Prove that \( \int_\gamma e^{z^2} \, dz = 0 \) for any closed path \( \gamma \). 4.17. Show that \( F(z) := \log(t + i) - \frac{1}{2} \log(-z - i) \) is an antiderivative of \( \frac{1}{\sqrt{z}} \) for \( \mathrm{Re}(z) > 0 \). Is \( F(z) \) equal to \( \arctan z \)? ``` #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 105 Context: # VARIATIONS OF A THEME Theorem 5.1 has several important consequences. For starters, it can be used to compute certain integrals. ## Example 5.2 \[ \int_{C(0,1)} \frac{\sin(z)}{z^2} \, dz = \frac{2\pi i}{2\pi} \left. \frac{d}{dx} \sin(z) \right|_{z=0} = 2\pi i \cos(0) = 2\pi i. \] ![Figure 5.1: The integration paths in Example 5.3.](path/to/figure5.1.png) ## Example 5.3 To compute the integral \[ \int_{C(0,2)} \frac{dz}{z^2(z-1)}, \] we could employ a partial fractions expansion similar to the one in Example 4.23, or moving the integration path similar to the one in Exercise 4.29. To exhibit an alternative, we split up the integration path as illustrated in Figure 5.1: we introduce an additional path that separates 0 and 1. If we integrate on these two new closed paths (\(Y_1\) and \(Y_2\)) counterclockwise, the two contributions along the new path will cancel each other. The effect is that we transformed an integral for which two singularities were inside the integration path into a sum of two integrals, each of which has only one singularity inside the integration path; these new integrals will... #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 106 Context: # Consequences of Cauchy’s Theorem We know how to deal with, using Theorems 4.24 and 5.1: \[ \int_{\mathcal{C}(0)} \frac{dz}{z^2(z-1)} = \int_{\mathcal{C}(1)} \frac{dz}{z(z-1)^2} = 2\pi i \left( \frac{1}{z} \bigg|_{z=1} + \frac{1}{(-1)^2} \right) = 2\pi i \left( 1 + \frac{1}{2} \right) = 0. \] ## Example 5.4 \[ \int_{\mathcal{C}(1)} \frac{\cos(z)}{z^3} dz = \int_{\mathcal{C}(1)} \frac{d^3}{dz^3} \cos(z) \bigg|_{z=0} = \pi i(-\cos(0)) = -\pi i. \] Theorem 5.1 has another powerful consequence; just from knowing that \( f \) is holomorphic in \( G \), we know of the existence of \( f'' \), that \( f' \) is also holomorphic in \( G \). Repeating this argument for \( f' \), \( f'' \), etc., shows that all derivatives \( f^{(n)} \) exist and are holomorphic. We can translate this into the language of partial derivatives; since the Cauchy–Riemann equations (Theorem 2.13) show that any sequence of partial differentiations of \( f \) results in a constant times \( f^{(n)} \). ## Corollary 5.5 If \( f \) is differentiable in a region \( G \) then \( f \) is infinitely differentiable in \( G \), and all partials of \( f \) with respect to \( x \) and \( y \) exist and are continuous. ## 5.2 Antiderivatives Again Theorem 4.15 gives us an antiderivative for a function that has zero integrals over closed paths in a given region. Now that we have Corollary 5.5, mediating just a bit more over Theorem 4.15 gives a converse of sorts to Corollary 4.20. ### Corollary 5.6 (Morera's Theorem) Suppose \( f \) is continuous in the region \( G \) and \[ \int_{C} f = 0 \] for all piecewise smooth closed paths \( C \subset G \). Then \( f \) is holomorphic in \( G \). [^1]: Named after Giacinto Morera (1856–1907). #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 109 Context: ```markdown # Taking Cauchy’s Formulas to the Limit ## Proof Suppose (by way of contradiction) that \( p(z) \) does not have any roots, that is, \( p(z) \neq 0 \) for all \( z \in \mathbb{C} \). Then \( \frac{1}{p(z)} \) is entire, and so Cauchy’s Integral Formula (Theorem 4.24) gives \[ \frac{1}{p(0)} = \frac{1}{2\pi i} \int_{\mathcal{C}(0, R)} \frac{p'(z)}{p(z)} \, dz, \quad \text{for any } R > 0. \] Let \( d \) be the degree of \( p(z) \) and \( a_k \) its leading coefficient. Propositions 4.6(d) and 5.10 allow us to estimate, for sufficiently large \( R \): \[ \left| \frac{1}{p(0)} \right| = \frac{1}{2\pi} \int_{\mathcal{C}(0, R)} \frac{1}{|p(z)|} \, |dz| \leq \frac{1}{2\pi} \cdot \frac{1}{\text{ext } |a_k| R^{d}} \cdot |2\pi R| = \frac{2 |a_k|}{R^{d}}. \] The left-hand side is independent of \( R \), while the right-hand side can be made arbitrarily small (by choosing \( R \) sufficiently large), and so we conclude that \( \frac{1}{p(0)} = 0 \), which is impossible. ## Theorem 5.11 Theorem 5.11 implies that any polynomial \( p \) can be factored into linear terms of the form \( z - a \) where \( a \) is a root of \( p \) as we can apply the corollary, after getting a root to \( z^n = 0 \) (which is again a polynomial by the division algorithm), etc. (see also Exercise 5.11). A compact reformulation of the Fundamental Theorem of Algebra (Theorem 5.11) is to say that \( \mathbb{C} \) is algebraically closed in contrast, \( \mathbb{R} \) is not algebraically closed. ### Example 5.12 The polynomial \( p(z) = z^4 + 5z^2 + 3 \) has no roots in \( \mathbb{R} \). The Fundamental Theorem of Algebra (Theorem 5.11) states that \( p \) must have a root (in fact, four roots) in \( \mathbb{C} \): \[ p(z) = (z^2 + 1)(z^2 + 3) = (z + i)(z - i)(\sqrt{3} + z)(\sqrt{3} - z) \quad \Box \] Another powerful consequence of Theorem 5.1 is the following result, which again has no counterpart in real analysis (consider, for example, the ratio test). ``` #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 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: 115 Context: # Taking Cauchy's Formulas to the Limit ## 5.20. Compute \[ \int_{-\infty}^{\infty} \frac{\cos(x)}{x^2+1} \, dx. \] ## 5.21. This exercise outlines how to extend some of the results of this chapter to the Riemann sphere as defined in Section 3.2. Suppose \( G \subset \mathbb{C} \) is a region that contains 0, let \( f \) be a continuous function on \( G \), and let \( \gamma \subset G \) be a piecewise smooth path in \( G \) avoiding the origin, parametrized as \( \gamma(t) \), for \( a \leq t \leq b \). ### (a) Show that \[ \int_{\gamma} f(z) \, dz = \int_{a}^{b} \frac{f(\gamma(t))}{\gamma'(t)} \, dt \] where \( o(t) = \gamma(t) \), \( a \leq t \leq b \). Now suppose \(\mathrm{Im}(f) = \{ f(z) \} \) is finite. Let \( H = \{ z \in G \setminus \{ 0 \} \} \) and define the function \( g : H \cup \{ 0 \} \to \mathbb{C} \) by \[ g(z) = \begin{cases} \frac{f(z)}{L} & \text{if } z \in H, \\ 0 & \text{if } z = 0. \end{cases} \] Thus \( g \) is continuous on \( H \cup \{ 0 \} \) and gives the identity \[ \int_{H} f \, dz = \int g. \] In particular, we can transfer certain properties between these two integrals. For example, if \( f \) is path independent, so is \( g \). Here is but one application: ### (a) Show that \[ \int_{\gamma} f(z) \, dz \text{ is path independent for any integer } n \neq -1. \] ### (b) Conclude (once more) that \[ \int z^n \, dz = 0 \text{ for any integer } n \neq -1. \] #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 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: 122 Context: ``` # Harmonic Functions ## Integral \[ u(w) = \frac{1}{2\pi} \int_{0}^{2\pi} \frac{|u(w+r e^{i\theta})|}{r} d\theta \] \[ = \frac{1}{2\pi} \left( \int_{0}^{\pi} |u(w+r e^{i\theta})| d\theta + \int_{\pi}^{2\pi} |u(w+r e^{i\theta})| d\theta \right) \] All the integrands can be bounded by \(u(w)\); for the middle integral we get a strict inequality. Hence \[ a(w) < \frac{1}{2\pi} \left( \int_{0}^{\pi} u(x) dx + \int_{\pi}^{2\pi} u(x) dx \right) = u(w). \] a contradiction. The same argument works if we assume that \(u\) has a relative minimum. But in this case there’s a shortcut argument: if \(u\) has a strong relative minimum then the harmonic function \(-u\) has a strong relative maximum, which we just showed cannot exist. So far, harmonic functions have benefited from our knowledge of holomorphic functions. Here is a result where the benefit goes in the opposite direction. ## Corollary 6.12 If \(f\) is holomorphic and nonzero in the region \(G\), then \(|f|\) does not have a strong relative maximum or minimum in \(G\). ### Proof By Exercise 6.6, the function \(\ln |f(z)|\) is harmonic on \(G\) and so, by Theorem 6.11, does not have a strong relative maximum or minimum in \(G\). But then neither does \(|f(z)|\), because it is monotonic. We finish our excursion about harmonic functions with a preview and its consequences. We say a real valued function on a region \(G\) has a weak relative maximum at \(x_0\) if there exists a disk \(D(x_0, r) \subset G\) such that \[ u(x_0) \geq u(x) \quad \text{for all } x \in D(x_0, r). \] We define weak relative minimum similarly. In Chapter 8 we will strengthen Theorem 6.11 and Corollary 6.12 to Theorem 8.17 and Corollary 8.20 by replacing strong relative extremum in the hypotheses with weak relative extremum. A special but important case is when \(u\) does not assume that \(f\) is nonzero in a region \(G\) to have a strong relative maximum in \(G\). ``` #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 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: 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: 147 Context: # REGIONS OF CONVERGENCE ## 7.9 (a) Show that \( \frac{1}{3} \sum_{k=1}^{\infty} \frac{1}{k} \) for any positive integer \( k \). (b) Conclude with Example 7.9 that for any positive integer \( n \), \[ 1 + \frac{1}{2} + \frac{1}{6} + \ldots + \frac{1}{n!} \geq 3. \] ## 7.10 Derive the Archimedean Property from the Monotone Sequence Property. ## 7.11 Prove Proposition 7.7. ## 7.12 Prove that \( (c_n) \) converges if and only if \( (\text{Re} c_n) \) and \( (\text{Im} c_n) \) converge. ## 7.13 Prove that \( Z \) is complete and that \( Q \) is not complete. ## 7.14 Prove that, if \( a_n \leq b_n \leq c_n \) for all \( n \) and \( \lim_{n \to \infty} a_n = L, \, b_n = \lim_{n \to \infty} b_n = L, \, c_n = L \), then \( \lim_{n \to \infty} b_n = L \). This is called the Squeeze Theorem, and is useful in testing a sequence for convergence. ## 7.15 Find the least upper bound of the set \( \{ \text{Re}(e^{(2\pi i t)}) : t \in Q \} \). ## 7.16 (a) Suppose that the sequence \( (c_n) \) converges to zero. Show that \( \sum_{n=0}^{\infty} c_n \) converges if and only if \( \sum_{n=0}^{\infty}(c_n + c_{n+1}) \) converges. Moreover, if the two series converge then they have the same limit. (b) Give an example where \( (c_n) \) does not converge to 0 and one of the series in (a) diverges while the other converges. ## 7.17 Prove that the series \( \sum b_n \) converges if and only if \( \lim_{n \to \infty} \sum_{k=0}^{n} b_k = 0 \). #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 148 Context: # Power Series ![Functions](https://example.com/path/to/image) **Figure 7.3:** The functions \( f_n(x) = \sin\left(\frac{x}{n}\right) \) in Exercise 7.21. ## 7.18 1. Show that \[ \sum_{k=1}^{\infty} \frac{k}{k+1} \text{ diverges.} \] 2. Show that \[ \sum_{k=1}^{\infty} \frac{k}{k^2 + 1} \text{ converges.} \] ## 7.19 1. Suppose \( G \subset \mathbb{C} \) and \( f_n : G \to \mathbb{C} \) for \( n \geq 1 \). Suppose \( (a_n) \) is a sequence in \( \mathbb{R} \) with \( \lim_{n \to \infty} a_n = 0 \) and, for each \( n \geq 1 \), \[ |f_n(a_n)| \leq a_n \quad \text{for all } a \in G. \] Show that \( f_n \) converges uniformly to the zero function in \( G \). 2. Re-prove the statement of Example 7.24 using part (a). ## 7.20 1. Suppose \( G \subset \mathbb{C}, f_n : G \to \mathbb{C} \) for \( n \geq 1 \), and \( f_n \) converges uniformly to the zero function in \( G \). Show that if \( (a_n) \) is any sequence in \( G \), then \[ \lim_{n \to \infty} f_n(a_n) = 0. \] 2. Apply (a) to the function sequence given in Example 7.23, together with the sequence \( (b_n = \frac{1}{n}) \), to prove that the convergence given in Example 7.23 is not uniform. Image Analysis: ### Comprehensive Examination #### 1. Localization and Attribution - **Image Positioning and Numbering**: - **Image 1 (Top)**: Positioned at the upper part of the page. - **Image 2 (Middle)**: Text content starting from "7.18" to "7.20" located beneath the Image 1. #### 2. Object Detection and Classification - **Image 1**: - **Objects**: Graph/Chart depicting a series of curves. - **Classification**: Mathematical function graphs. #### 3. Scene and Activity Analysis - **Image 1**: - **Scene Description**: Graphical representation of functions \( f_n(x) = \sin^n(x) \) for different values of \( n \). - **Activity**: Visualization of function behavior as \( n \) increases, showing how the functions change shape. #### 4. Text Analysis - **Image 2**: - **Text Content**: - **Figure Caption**: "Figure 7.3: The functions \( f_n(x) = \sin^n(x) \) in Exercise 7.21." - **7.18**: Two parts (a) and (b) discussing convergence and divergence of series. - **7.19**: Two parts (a) and (b), with (a) detailing properties of a function sequence and convergence criteria; (b) asks to re-prove part (a). - **7.20**: Two parts (a) and (b), with (a) expanding on function convergence criteria and (b) applying these criteria to a given example. - **Significance**: The text provides mathematical exercises and explanations related to the visual content (graph of sin functions). #### 5. Diagram and Chart Analysis - **Image 1**: - **Diagram**: Graph of \( f_n(x) = \sin^n(x) \) - **Axes**: - **X-Axis**: Represents the variable \( x \). - **Y-Axis**: Represents the value of \( f_n(x) \). - **Scales**: Standard numerical scales for both x and y axes. - **Key Insights**: Demonstrates how the function \( f_n(x) \) narrows and approaches a delta-like function centered at \( \pi \) and other periodic points as \( n \) increases. #### 8. Color Analysis - **Image 1**: - **Dominant Colors**: Blue curves on a white background. - **Impact on Perception**: The color choice helps emphasize the variation among different curves, making it easy to distinguish between different values of \( n \). #### 9. Perspective and Composition - **Image 1**: - **Perspective**: Straight-on view of the graph. - **Composition**: Centralized graph with labeled axes. The curves are symmetric around the origin, following the properties of sine functions. #### 10. Contextual Significance - **Image 1**: - **Context**: Related to exercises and explanations in the study of power series and function convergence in mathematics. - **Contribution**: Provides visual understanding and concrete examples to complement the theoretical exercises below. #### 12. Graph Numbers - **Image 1**: - **Data Points**: Not explicitly provided, but the curves can be understood as sampled points from the functions \( \sin^n(x) \). - **Trends**: As \( n \) increases, the curve sharply peaks at specific points (multiples of \( \pi \)), demonstrating convergence behavior. #### 13. Graph and Trend Analysis - **Image 1**: - **Trends**: The sequence of functions \( f_n(x) = \sin^n(x) \) becomes more peaked at multiples of \( \pi \) as \( n \) increases, indicating how the function evolves with higher powers. - **Interpretation**: This trend is significant in understanding pointwise convergence and properties of Fourier series in mathematical analysis. #### Additional Aspects - **Prozessbeschreibungen (Process Descriptions)**: - **Series Convergence** (7.18): Analysis of divergence and convergence criteria. - **Function Sequence Convergence** (7.19, 7.20): Conditions and proof of uniform convergence of function sequences on given domains. #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 150 Context: ``` # Power Series ## 7.27 (a) Suppose that the sequence \( (c_n) \) is bounded. Show that the radius of convergence of \( \sum_{n=0}^\infty c_n (z - a)^n \) is at least 1. (b) Suppose that the sequence \( (c_n) \) does not converge to 0. Show that the radius of convergence of \( \sum_{n=0}^\infty c_n (z - a)^n \) is at most 1. ## 7.28 Find the power series centered at 1 and compute its radius of convergence for each of the following functions: (a) \( f(z) = \frac{1}{z} \) (b) \( f(z) = \log(z) \) ## 7.29 Use the Weierstrass M-test to show that each of the following series converges uniformly on the given domain: (a) \( \sum_{k=0}^\infty \frac{z^k}{k!} \) on \( [0, 1] \) (b) \( \sum_{k=0}^\infty z^k \) on \( \{ z \in \mathbb{C} : |z| < 2 \} \) (c) \( \sum_{k=0}^\infty \frac{z^k}{z^k + 1} \) on \( D(0, r) \) where \( 0 < r < 1 \) ## 7.30 Fix \( c \in \mathbb{C} \) and \( r > |z| \). Prove that \( \sum_{k=0}^\infty \left( \frac{z}{w} \right)^k \) converges uniformly in the variable \( z \) for \( |z| > r \). ## 7.31 Complete our proof of Corollary 7.33 by considering the case \( R = \infty \). ## 7.32 Prove that, if \( \lim_{n \to \infty} \left| c_n \right|^{\frac{1}{n}} \) exists then the radius of convergence of the series \( \sum_{n=0}^\infty c_n (z - a)^n \) equals \[ R = \begin{cases} \infty & \text{if } \lim_{n \to \infty} \left| c_n \right|^{\frac{1}{n}} = 0, \\ 0 & \text{otherwise}. \end{cases} \] ## 7.33 Find the radius of convergence for each of the following series: (a) \( \sum_{n=0}^\infty a^n z^n \) where \( a \in \mathbb{C} \) (b) \( \sum_{k=0}^\infty k! z^k \) where \( n \in \mathbb{Z} \) ``` #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 155 Context: # POWER SERIES AND HOLOMORPHIC FUNCTIONS ## Proof For starters, \( f(0) = c_0 \). Theorem 8.2 gives \( f'(0) = c_1 \). Applying the same theorem to \( f' \) gives \[ f''(z) = \sum_{k=2}^{\infty} k(k-1)c_kz^{k-2} \] and so \( f''(c_0) = 2c_2 \). A quick induction game establishes \( f^{(n)}(0) = n!c_n \) for all \( k \geq 0 \). Taylor's formula shows that the coefficients of any power series converging to \( f \) on some open disk \( D \) can be determined from the function \( f \) restricted to \( D \). It follows immediately that the coefficients of a power series are unique: ### Corollary 8.6 If \( \sum_{k=0}^{\infty} a_k(z - z_0)^k \) and \( \sum_{k=0}^{\infty} b_k(z - z_0)^k \) are two power series that both converge to the same function on an open disk centered at \( z_0 \), then \( a_k = b_k \) for all \( k \geq 0 \). ### Example 8.7 We'd like to compute a power series expansion for \( f(z) = \exp(z) \) centered at \( z_0 = 0 \): \[ f^{(k)}(0) = \frac{1}{k!} \exp(0) = 1. \] Corollary 8.5 suggests that this power series is \[ \sum_{k=0}^{\infty} \frac{z^k}{k!} \] which converges for all \( z \in \mathbb{C} \) (essentially by Example 7.35). We now turn to the second cornerstone result of this section, that a holomorphic function can be locally represented by a power series. ### Theorem 8.8 Suppose \( f \) is a function holomorphic in \( D\left(z_0, r\right) \). Then \( f \) can be represented as a power series centered at \( z_0 \) with a radius of convergence \( r \): \[ f(z) = \sum_{k=0}^{\infty} c_k(z - z_0)^k \] with \[ c_k = \frac{1}{2\pi i} \oint_{\gamma} \frac{f(z)}{(z - z_0)^{k+1}} \, dz \] where \( \gamma \) is any positively oriented, simple, closed, piecewise smooth path in \( D\left(z_0, r\right) \) for which \( z_0 \) is inside \( \gamma \). #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 159 Context: # Classification of Zeros and the Identity Principle Continuing in this way, we see that we can factor \( p(z) \) as \( p(z) = (z-a)^m g(z) \) where \( m \) is a positive integer and \( g(z) \) is a polynomial that does not have a zero at \( a \). The integer \( m \) is called the multiplicity of the zero of \( p(z) \). Almost exactly the same thing happens for holomorphic functions. ## Theorem 8.14 (Classification of Zeros) Suppose \( f : G \to \mathbb{C} \) is holomorphic and \( f \) has a zero at \( a \in G \). Then either: 1. \( f \) is identically zero on some disk \( D \) centered at \( a \) (that is, \( f(z) = 0 \) for all \( z \in D \)); or 2. There exists a positive integer \( m \) and a holomorphic function \( g : G \to \mathbb{C} \), such that \( f(z) = (z-a)^m g(z) \) for all \( z \in G \). In this case, the zero is isolated: there is a disk \( D[a,r] \) which contains no other zero of \( f \). The integer \( m \) in the second case is uniquely determined by \( f \) and is called the multiplicity of the zero at \( a \). ### Proof By Theorem 8.8, there exists \( R > 0 \) such that we can expand \[ f(z) = \sum_{k=0}^{\infty} c_k (z-a)^k \text{ for } z \in D[a,R]. \] and \( c_0 = f(a) = 0 \). There are now exactly two possibilities: either 1. \( c_0 = 0 \) for all \( k \geq 0 \); or 2. There is some positive integer \( m \) so that \( c_k = 0 \) for all \( k < m \) but \( c_m \neq 0 \). The first case gives \( f(z) = 0 \) for all \( z \in D[a,R] \). So now consider the second case. We note that for \( z \in D[a,R] \): \[ f(z) = (z-a)^m \left( c_m + c_{m+1} (z-a) + \ldots \right) = (z-a)^m \sum_{n=0}^{\infty} c_{m+n} (z-a)^n. \] #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 160 Context: # TAYLOR AND LAURENT SERIES Thus we can define a function \( g : G \rightarrow \mathbb{C} \) through \( g(z) = \begin{cases} \sum_{n=0}^{\infty} a_n (z - a)^n & \text{if } z \in D(a, R), \\ f(z) & \text{if } z \in G \setminus \{ a \}. \end{cases} \) (According to our calculations above, the two definitions give the same value when \( z \in D(a, R) \). The function \( g \) is holomorphic in \( D(a, R) \) by the first definition, and \( g \) is holomorphic in \( G \setminus \{ a \} \) by the second definition. Note that \( g(a) = c \neq 0 \) and, by construction, \( f(z) = (z - a)^n s(z) \quad \text{for all } z \in G. \) Since \( s(a) \neq 0 \) by continuity, \( r > 0 \) so that \( s(z) \neq 0 \) for all \( z \in D(a, r) \), so \( D(a, r) \) contains no other zero of \( f \). The integer \( r \) is unique, since it is defined in terms of the power series expansion of \( f \) at \( a \), which is unique by Corollary 8.6. Theorem 8.14 gives rise to the following result, which is sometimes called the identity principle or the uniqueness theorem. ## Theorem 8.15 Suppose \( G \) is a region, \( f : G \rightarrow \mathbb{C} \) is holomorphic, and \( f(a) = 0 \) where \( (a_k) \) is a sequence of distinct numbers that converges in \( G \). Then \( f \) is the zero function on \( G \). Applying this theorem to the difference of two functions immediately gives the following variant. ## Corollary 8.16 Suppose \( f \) and \( g \) are holomorphic in a region \( G \) and \( f(a) = g(a) = 0 \) at a sequence that converges to \( a \in G \) with \( a_k \neq a \) for all \( k \). Then \( f(z) = g(z) \) for all \( z \in G \). ### Proof of Theorem 8.15 Consider the following two subsets of \( G \): - \( X := \{ z \in G : \text{ there exists } z \text{ such that } F(z) = 0 \text{ for all } z \in D(a, r) \} \) - \( Y := \{ z \in G : \text{ there exists } z \text{ such that } F(z) \neq 0 \text{ for all } z \in D(a, r) \setminus \{ a \} \} \) If \( f(a) \neq 0 \) then, by continuity of \( f \), there exists a disk centered at \( a \) in which \( f \) is nonzero, and so \( a \in Y \). #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 163 Context: Here \( a_k \in \mathbb{C} \) are terms indexed by the integers. The double series above converges if and only if the two series on the right-hand side do. Absolute and uniform convergence are defined analogously. Equipped with this, we can now introduce the following central concept. ## Definition A Laurent series centered at \( z_0 \) is a double series of the form \[ \sum_{n \in \mathbb{Z}} c_n (z - z_0)^{n}. \] ### Example 8.21 The series that started this section is the Laurent series of \(\exp\left(\frac{1}{z}\right)\) centered at 0. ### Example 8.22 Any power series is a Laurent series (with \( a_k = 0 \) for \( k < 0 \)). We should pause for a minute and ask for which \( z \) a general Laurent series can possibly converge. By definition, \[ \sum_{n \in \mathbb{Z}} c_n (z - z_0)^n = \sum_{n \geq 0} c_n (z - z_0)^n + \sum_{n < 0} c_n (z - z_0)^n. \] The first series on the right-hand side is a power series with some radius of convergence \( R_z \), that is, with Theorem 7.31, it converges in \( \{ z \in \mathbb{C} : |z - z_0| < R_z \} \), and the convergence is uniform in \( \{ z \in \mathbb{C} : |z - z_0| \leq r \} \) for any fixed \( r < R_z \). For the second series, we invite you (in Exercise 8.30) to revise our proof of Theorem 7.31 to show that this series converges for \[ \frac{1}{|z - z_0|} < \frac{1}{R_1} \] for some \( R_1 \), and that the convergence is uniform in \( \{ z \in \mathbb{C} : |z - z_0| \leq r \} \) for any \( r > R_1 \). Thus the Laurent series converges in the annulus \[ A = \{ z \in \mathbb{C} : R_1 < |z - z_0| < R_2 \} \] (assuming this set is not empty, i.e., \( R_1 < R_2 \)), and the convergence is uniform on any set of the form \[ \{ z \in \mathbb{C} : R_1 < |z - z_0| < r \} \] for \( R_1 < r < R_2 \). *Jean-Pierre Antoine Laurent (1813–1894).* #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 168 Context: ```markdown # TAYLOR AND LAURENT SERIES By Examples 8.3 and 8.23, \[ \frac{\exp(z)}{\sin(z)} = \left( 1 + \frac{z}{1!} + \frac{z^2}{2!} + \frac{z^3}{3!} + \cdots \right) \left( -\frac{1}{z} + \frac{7}{360}z + \frac{31}{15120}z^3 + \cdots \right) = z^{-1} + 1 + \frac{2}{3} z + \cdots \] and Corollary 8.27 gives \[ \int_{C[0,1]} \frac{\exp(z)}{\sin(z)} \, dz = 2\pi i. \] For the integral around \( C \), we use the fact that \( \sin(-z) = -\sin(z) \), and so we can compute the Laurent expansion of \( \frac{\exp(z)}{\sin(z)} \) at \( z = 0 \) as Example 8.23: \[ \frac{1}{\sin(z)} = -\frac{1}{z} - \frac{1}{6} (z - z^3) - \frac{7}{360} (z - z^3) - \cdots. \] Adding Example 8.7 to the mix yields \[ \frac{\exp(z)}{\sin(z)} = \left( z^{-1} + \frac{z^{-1}}{2} + \frac{z^{-1}}{2} + \cdots \right) - \left( -\frac{1}{6}(z - z^3) - \cdots \right) \] \[ = -z^{-1} - \frac{z^{-2}}{2} - \frac{z^{-1}}{2} - \cdots. \] And now Corollary 8.27 gives \[ \int_{C[2,3]} \frac{\exp(z)}{\sin(z)} \, dz = -2\pi i e^z. \] Putting it all together yields the integral we've been after for two chapters: \[ \int_{C[1,2]} \frac{\exp(z)}{\sin(z)} \, dz = 2\pi i (1 - e^z). \] ## Exercises ### 8.1 For each of the following series, determine where the series converges absolutely and where it converges uniformly: (a) \[ \sum_{n=0}^{\infty} \frac{2^n}{(n+1)!} z^{2n} \] (b) \[ \sum_{n=0}^{\infty} \left(\frac{1}{(-3)^n}\right)^k \] ### 8.2 What functions are represented by the series in the previous exercise? ``` #################### 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: 172 Context: 8.24. Prove: If \( f \) is entire and \( \text{Im}(f) \) is constant on the closed unit disk then \( f \) is constant. 8.25. (a) Find the Laurent series for \( \frac{1}{z^2} \) centered at \( z = 0 \). (b) Prove that \( f: \mathbb{C} \to \mathbb{C} \) is entire, where \[ f(z) = \begin{cases} \frac{1}{z} & \text{if } z \neq 0 \\ -1 & \text{if } z = 0 \end{cases} \] 8.26. Find the Laurent series for \( \sec(z) \) centered at the origin. 8.27. Suppose that \( f \) is holomorphic at \( z_0 \), \( f(z_0) = 0 \), and \( f'(z_0) \neq 0 \). Show that \( f \) has a zero of multiplicity 1 at \( z_0 \). 8.28. Find the multiplicities of the zeros of each of the following functions: (a) \( f(z) = e^{z} - 1 \), \( z_0 = 2k\pi i \), where \( k \) is any integer. (b) \( f(z) = \sin(z) - \tan(z) \), \( z_0 = 0 \). (c) \( f(z) = \cos(z) - 1 + z^2 \sin'(z_0) \), \( z_0 = 0 \). 8.29. Find the zeros of the following functions and determine their multiplicities: (a) \( 1 + z^2 \) (b) \( \sin(z) \) (c) \( z^2 \cos(z) \) 8.30. Prove that the series of the negative-index terms of a Laurent series \[ \sum_{k < 0} c_k (z - z_0)^k \] converges for \[ \frac{1}{R_1} < |z - z_0| < \frac{1}{R_0} \] for some \( R_1 \), and that the convergence is uniform in \( \{ z \in \mathbb{C} : |z - z_0| \geq r_1 \} \) for any fixed \( r_1 < R_1 \). #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 175 Context: # Chapter 9 ## Isolated Singularities and the Residue Theorem \(\frac{1}{z}\) has a nasty singularity at \(z = 0\), but it did not bother Newton—the reason is far enough. Edward Witten We return one last time to the starting point of Chapters 7 and 8: the quest for \[ \int_{C} \frac{\exp(z)}{\sin(z)} \, dz. \] We computed this integral in Example 8.28 crawling on hands and knees (but we finally computed it), by considering various Taylor and Laurent expansions of \(\exp(z)\) and \(\frac{1}{\sin(z)}\). In this chapter, we develop a calculus for similar integral computations. ## 9.1 Classification of Singularities What are the differences among the functions \(\frac{\exp(z)}{z}\), \(\frac{1}{z}\), and \(\exp(z)\) at \(z = 0\)? None of them are defined at \(0\), but each singularity is of a different nature. We will frequently consider functions in this chapter that are holomorphic in a disk except at its center (usually because that’s where a singularity lies), and it will be handy to define the punctured disk with center \(z_0\) and radius \(R\), \[ D(z_0, R) = \{ z \in \mathbb{C} : 0 < |z - z_0| < R \}. \] We extend this definition naturally with \(D[z_0, R] = \{ z \in \mathbb{C} : |z - z_0| \leq R \}\). For complex functions there are three types of singularities, which are classified as follows: #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 177 Context: # Classification of Singularities show that \( f \) has neither a removable singularity nor a pole. To get a feel for the different types of singularities, we start with the following criteria. **Proposition 9.5.** Suppose \( z_0 \) is an isolated singularity of \( f \). Then 1. \( z_0 \) is removable if and only if \( \lim_{z \to z_0} (z - z_0)f(z) = 0; \) 2. \( z_0 \) is a pole if and only if it is not removable and \( \lim_{z \to z_0} (z - z_0)^n f(z) = 0 \) for some positive integer \( n \). **Proof.** (a) Suppose that \( z_0 \) is a removable singularity of \( f \); so there exists a holomorphic function \( h \) on \( D(z_0, R) \) such that \( f(z) = h(z) \) for all \( z \in D(z_0, R) \). But then \( h \) is continuous at \( z_0 \) and \[ \lim_{z \to z_0} f(z) = \lim_{z \to z_0} h(z) = h(z_0). \] Conversely, suppose that \( \lim_{z \to z_0} (z - z_0)f(z) = 0 \) and \( f \) is holomorphic in \( D(z_0, R) \). We can define the function \( g: D(z_0, R) \to \mathbb{C} \) by \[ g(z) = \begin{cases} (z - z_0)^n f(z) & \text{if } z \neq z_0 \\ 0 & \text{if } z = z_0 \end{cases} \] Then \( g \) is holomorphic in \( D(z_0, R) \) and \[ g'(z_0) = \lim_{z \to z_0} \frac{g(z) - g(z_0)}{z - z_0} = \lim_{z \to z_0} \frac{(z - z_0)^{n-1}f(z)}{z - z_0} = \lim_{z \to z_0} (z - z_0)^{n-1}f(z) = 0, \] so \( g \) is holomorphic in \( D(z_0, R) \). We can then expand \( g \) into a power series: \[ g(z) = \sum_{k \geq 0} a_k(z - z_0)^k \] whose first two terms are zero: \( a_0 = g(z_0) = 0 \) and \( a_1 = g'(z_0) = 0 \). But then we can write \[ f(z) = \frac{g(z)}{(z - z_0)^n} = \sum_{k \geq 0} a_k(z - z_0)^{k - n} \] and so \[ f(z) = \sum_{k \geq 0} a_{k + n}(z - z_0)^k \] for all \( z \in D(z_0, R) \). #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 180 Context: ```markdown # Theorem 9.7 (Casorati–Weierstrass) If \( f \) is an essential singularity of \( r \) and \( r \) is any positive real number, then every \( u \in C \) is arbitrarily close to a point in \( f(B(z_0, r)) \). That is, for any \( x \in C \) and \( \epsilon > 0 \) there exists \( z \in B(z_0, r) \) such that \( |u - f(z)| < \epsilon \). In the language of topology, Theorem 9.7 says that the image of any punctured disk centered at an essential singularity is dense in \( C \). There is a stronger theorem, beyond the scope of this book, which implies the Casorati–Weierstrass Theorem 9.7. It is due to Charles Émile Picard (1856–1941) and says that the image of any punctured disk centered at an essential singularity misses at most one point of \( C \). (It is worth coming up with examples of functions that do not miss any point in \( C \) and functions that miss exactly one point. Try it!) ## Proof Suppose (by way of contradiction) that there exists \( w \in C \) and \( \epsilon > 0 \) such that for all \( z \in B(z_0, r) \) \[ |u - f(z)| \geq \epsilon. \] Then the function \( g(z) = \frac{f(z) - w}{z - z_0} \) stays bounded as \( z \to z_0 \) and so \[ \lim_{z \to z_0} (z - z_0) g(z) = \lim_{z \to z_0} \frac{f(z) - w}{z - z_0} = 0. \] (Proposition 9.5(a) tells us that \( f \) has a removable singularity at \( z_0 \).) Hence \[ \lim_{z \to z_0} \left| \frac{f(z) - w}{z - z_0} \right| = \infty \] and so the function \( f(z) - w \) has a pole at \( z_0 \). By Proposition 9.5(b), there is a positive integer \( n \) so that \[ \lim_{z \to z_0} (z - z_0)^{n+1} f(z) = \lim_{z \to z_0} (z - z_0)^n g(f(z)) = 0. \] Invoking Proposition 9.5 again, we conclude that the function \( f(z) \) has a pole or removable singularity at \( z_0 \), which implies the same holds for \( f(z) \), a contradiction. ``` #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 185 Context: ```markdown # RESIDUES **Proof:** The functions \( f \) and \( g \) have power series centered at \( z_0 \); the one for \( g \) has by assumption no constant term: \[ f(z) = \sum_{n=0}^{\infty} a_n (z - z_0)^n \] \[ g(z) = \sum_{n=1}^{\infty} b_n (z - z_0)^n \] Let \( h(z) = \sum_{n=1}^{\infty} b_n (z - z_0)^{-n} \) and note that \( h(z_0) = b_1 \neq 0 \). Hence, \[ \frac{f(z)}{g(z)} = \frac{f(z)}{(z - z_0) h(z)} \] and the function \( \frac{f(z)}{g(z)} \) is holomorphic at \( z_0 \). By Prop 9.11 and Taylor's formula (Corollary 8.5), \[ \text{Res}_{z \to z_0} \left( \frac{f(z)}{g(z)} \right) = \lim_{z \to z_0} \left( (z - z_0) \frac{f(z)}{g(z)} \right) = \frac{f(z_0)}{h(z_0)} \frac{f(z)}{h(z)} \cdot \frac{d}{dz} \frac{f(z)}{g(z)} \Big|_{z=z_0} \] Example 9.15. Revisiting once more Example 8.28, we note that \( f(z) = e^{z} \) and \( g(z) = \sin(z) \) fill the bill. Thus, \[ \text{Res}_{z \to 0} \left( \frac{e^{z}}{\sin(z)} \right) = \frac{e^0}{\cos(0)} = 1 \] and confirming once more our computations in Examples 8.28 and 9.12. Example 9.16. We compute the residue of \( \frac{e^{z} + z^2}{e^{2z} - z^2} \) at \( z_0 = 2\pi i \), by applying Proposition 9.14 with \( f(z) = e^{z} + z^2 \) and \( g(z) = e^{2z} - z^2 \). Thus, \[ \text{Res}_{z \to 2\pi i} \left( \frac{e^{z} + z^2}{e^{2z} - z^2} \right) = \frac{-4\pi^2 + 2}{\cosh(2\pi)} \] An extension of Proposition 9.14 of sorts is given in Exercise 9.12. ``` #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 189 Context: # ARGUMENT PRINCIPLE AND ROUCHÉ'S THEOREM This theorem is of surprising practicality. It allows us to locate the zeros of a function fairly precisely. Here is an illustration. ## Example 2.19 All the roots of the polynomial \( p(z) = z^5 + z^4 + z^3 + z^2 + z + 1 \) have modulus less than two. To see this, let \( f(z) = z^5 + z^4 + z^3 + z^2 + z + 1 \). For \( z \in [0, 2] \): \[ |f(z)| \leq |z^5| + |z^4| + |z^3| + |z^2| + |z| + 1 = 16 + 8 + 4 + 2 + 1 = 31 < 32 = |z^5| \text{ for } z \in [\sqrt{2}, 2]. \] So \( g \) and \( f \) satisfy the condition of Theorem 9.18. But \( f \) has just one root, of multiplicity 5 at the origin, whence \[ Z(f) \cap (C[0, 2]) = Z(f + g, C[0, 2]) = Z(f, C[0, 2]) = 5. \] ### Proof of Theorem 9.18 By (9.1) and the Argument Principle (Theorem 9.17): \[ Z(f + g, r) = \frac{1}{2\pi i} \int_{\gamma} \frac{f + g}{f + g} \, \frac{(f + g)'}{f + g} \, dz = \frac{1}{2\pi i} \int_{\gamma} \frac{(f + g)'}{f + g} \, dz, \] where \( \gamma \) is a contour around \( r \). We are assuming that \( |g| < |f| \) on \( \gamma \), which means that the function \( \frac{1}{f + g} \) evaluated on \( \gamma \) stays away from \( R_+ \). But then \( \log\left(\frac{1}{f + g}\right) \) is a well-defined holomorphic function on \( \gamma \). Its derivative is \[ \frac{(f + g)'}{f + g}. \] --- > *The Fundamental Theorem of Algebra (Theorem 5.1) asserts that \( p \) has \( n \) roots in \( C \). What's special about the theorem of Example 2.19 is that they all have modulus < 2. Note also that there is no general formula for composing roots of a polynomial of degree \( n \). Although for this \( n \) it’s not hard to find one root—and therefore all of them.* #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 193 Context: # Argument Principle and Rouche's Theorem ## 9.21. Find the number of zeros of 1. \( 3z^2 - z \) in \( D[0, 1] \) 2. \( \{ z \} \in D[0, 1] \) 3. \( e^{-z} - 5 + 1 \) in \( \{ z \in \mathbb{C} : |z| \leq 2 \} \) ## 9.22. Give another proof of the Fundamental Theorem of Algebra (Theorem 5.11), using Rouche's Theorem 9.18. *(Hint: If \( p(z) = z^n + a_{n-1}z^{n-1} + \ldots + a_0 \), let \( f(z) = a_n z^n \) and \( g(z) = a_{n-1}z^{n-1} + a_{n-2}z^{n-2} + \ldots + a_0 \), and choose \( r \) a circle that is large enough to make the condition of Rouche's theorem work. You might want to first apply Proposition 5.10 to \( g(z) \).)* ## 9.23. Suppose \( S \subset C \) is closed and bounded and all points of \( S \) are isolated points of \( S \). Show that \( S \) is finite, as follows: (a) For each \( x \in S \), we can choose \( \varepsilon > 0 \) so that \( D[x, \varepsilon] \) contains no points of \( S \) except \( x \). Show that \( f \) is continuous. *(Hint: This is really easy if you use the first definition of continuity in Section 2.1.)* (b) Assume \( S \) is non-empty. By the Extreme Value Theorem A.1, \( S \) has a minimum value, \( r_0 > 0 \). Let \( r = r_0/2 \). Since \( S \) is bounded, it lies in a disk \( D[0, M] \) for some \( M > 0 \). Show that the small disks \( D[x, r] \) for \( x \in S \) are disjoint and lie in \( D[0, M+r] \). (c) Find a bound on the number of such small disks. *(Hint: Compare the areas of \( D[x, r] \) and \( D[0, M + r] \).)* #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 194 Context: # Chapter 10 ## Discrete Applications of the Residue Theorem *All means (even continuous) sanctify the discrete end.* Doron Zeilberger On the surface, this chapter is just a collection of exercises. They are more involved than any of the ones we've given so far at the end of each chapter, which is one reason why we will lead you through each of the following ones step by step. On the other hand, these sections should really be thought of as a continuation of the book, just in a different format. All of the following problems are of a discrete mathematical nature, and we invite you to solve them using continuous methods—namely, complex integration. There are very few results in mathematics that so intimately combine discrete and continuous mathematics as does the Residue Theorem 9.10. ## 10.1 Infinite Sums In this exercise, we evaluate the sums \(\sum_{n=1}^{\infty} \frac{1}{n}\) and \(\sum_{n=1}^{\infty} \frac{(-1)^{n}}{n}\). We hope the idea of how to compute such sums in general will become clear. 1. Consider the function \(f(z) = \frac{e^{z}}{z}\). Compute the residues at all the singularities of \(f\). (a) Let \(N\) be a positive integer and \(z_{N}\) be the rectangular path from \(N + iN\) to \(N + i(N - 1)\) to \(N - 1 + i(N - 1)\) to \(N - 1 + iN\) back to \(N + iN\). (b) Show that \(\lim_{N \to \infty} \int_{z_{N}} f(z) \, dz = 0\). 2. Use the Residue Theorem 9.10 to arrive at an identity for \(\sum_{n=1}^{\infty} \frac{1}{n^{2}}\). #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 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: 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: 199 Context: # Dedekind Sums This exercise outlines one more nontraditional application of the Residue Theorem 9.10. Given two positive, relatively prime integers \( a \) and \( b \), let \[ f(z) = \cot(\pi z) \cot(b \pi z). \] 1. Choose an \( R > 0 \) such that the rectangular path \( \gamma_R \) from \( -1 - iR \) to \( -1 + iR \) to \( 1 + iR \) to \( 1 - iR \) back to \( -1 - iR \) does not pass through any of the poles of \( f \). (a) Compute the residues for the poles of \( f \) inside \( \gamma \). Hint: Use the periodicity of the cotangent and the fact that \[ \cot z = \frac{1}{z} - \frac{1}{3} z + \text{higher-order terms}. \] (b) Prove that \( \lim_{R \to \infty} \int_{\gamma_R} f \, dz = -2i \) and deduce that for any \( R > 0 \) \[ \int_{-1}^{1} f \, dz = -2i. \] 2. Define \[ t(a, b) = \frac{1}{48} \sum_{k=1}^{b-1} \cot \left( \frac{\pi k a}{b} \right) \left( \frac{k}{b} \right). \] Use the Residue Theorem 9.10 to show that \[ t(a, b) + t(b, a) = -\frac{1}{4} + \frac{1}{12} \left( \frac{a}{b} + \frac{b}{a} \right). \] 3. Generalize (10.1) and (10.2). #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 202 Context: # THEOREMS FROM CALCULUS ## Theorem A.3 (Fundamental Theorem of Calculus). Suppose \( f : [a, b] \to \mathbb{R} \) is continuous. 1. The function \( F : [a, b] \to \mathbb{R} \) defined by \( F(x) = \int_a^x f(t) \, dt \) is differentiable and \( F'(x) = f(x) \). 2. If \( F \) is any antiderivative of \( f \), then \( F' = f \) and \( \int_a^b f(x) \, dx = F(b) - F(a) \). ## Theorem A.4. If \( f, g : [a, b] \to \mathbb{R} \) are continuous functions and \( c \in \mathbb{R} \), then \[ \int_a^b (f(x) + g(x)) \, dx = \int_a^b f(x) \, dx + \int_a^b g(x) \, dx. \] ## Theorem A.5. If \( f, g : [a, b] \to \mathbb{R} \) are continuous functions then \[ \int_a^b f(x)g(x) \, dx \leq \int_a^b |f(x)| \, dx \cdot \left( \max_{x \in [a, b]} |g(x)| \right). \] ## Theorem A.6. If \( g : [a, b] \to \mathbb{R} \) is differentiable, \( g' \) is continuous, and \( f : [g(a), g(b)] \to \mathbb{R} \) is continuous then \[ \int_{g(a)}^{g(b)} f(t) \, dt = \int_a^b f(g(t)) g'(t) \, dt. \] For functions of several variables we can perform differentiation/integration operations in any order, if we have sufficient continuity: ## Theorem A.7. If the mixed partials \( \frac{\partial^2 f}{\partial x \partial y} \) and \( \frac{\partial^2 f}{\partial y \partial x} \) are defined on an open set \( G \subset \mathbb{R}^2 \) and are continuous at a point \( (x_0, y_0) \in G \), then they are equal at \( (x_0, y_0) \). ## Theorem A.8. If \( f \) is continuous on \( [a, b] \times [c, d] \subset \mathbb{R}^2 \) then \[ \int_c^d f(x, y) \, dy = \int_a^b \int_c^d f(x, y) \, dy \, dx. \] We can apply differentiation and integration with respect to different variables in either order: #################### 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: 208 Context: # Index - **absolute convergence**, 129 - **absolute value**, 5 - **accumulation point**, 13, 24 - **addition**, 2 - **algebraically closed**, 103 - **alternating harmonic series**, 131 - **alternating zeta function**, 189 - **analytic**, 152 - **analytic continuation**, 158 - **antiderivative**, 76, 100, 196 - **Arg**, 59 - **arg**, 60 - **argument**, 5 - **axis** - imaginary, 5 - real, 5 - **bijection**, 31, 43 - **binary operation**, 2 - **binomial coefficient**, 189 - **boundary**, 13, 117 - **boundary point**, 13 - **bounded**, 13 - **branch of the logarithm**, 59 - **calculus**, 1, 195 - **Casorati–Weierstrass theorem**, 173 - **Cauchy's estimate**, 151 - **Cauchy's integral formula**, 85 - **extensions of**, 97, 151 - **Cauchy's theorem**, 81 - **Cauchy–Goursat theorem**, 82 - **Cauchy–Riemann equations**, 32 - **chain of segments**, 16 - **circle**, 12 - **closed** - disk, 13 - path, 16 - **set**, 13 - **coffee**, 88, 132, 173 - **comparison test**, 127 - **complete**, 123 - **complex number**, 2 - **complex plane**, 5 - extended, 47 - **complex projective line**, 47 - **composition**, 27 - **concatenation**, 74 - **conformal**, 30, 44, 118 - **conjugate**, 10 - **connected**, 14 - **continuous**, 26 - **contractible**, 83 - **convergence**, 122 - **pointwise**, 131 - **uniform**, 131 #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 209 Context: # Table of Contents 1. **Convergence** - Sequence - Series - Cosine - Cotangent - Cross Ratio - Curve - Cycloid 2. **Numerical Concepts** - Dedekind Sum - Dense - Derivative - Partial - Difference Quotient - Differentiable - Dilation - Discriminant 3. **Geometric Concepts** - Disk - Closed - Open - Punctured - Unit - Distance of Complex Numbers - Divergent - Domain - Double Series 4. **Mathematical Constants** - e - Embedding of R into C - Empty Set - Entire Functions - Essential Singularity - Euclidean Plane - Euler’s Formula 5. **Functions and Theorems** - Even - Exponential Function - Exponential Rules - Extended Complex Plane - Fibonacci Numbers - Field - Fixed Point - Frobenius Problem - Functions - Conformal - Even - Exponential - Logarithmic - Odd - Trigonometric - Fundamental Theorem - Of Algebra - Of Calculus 6. **Applications** - Geogebra - Geometric Interpretation of Multiplication - Geometric Series - Group 7. **Harmonic Analysis** - Harmonic - Harmonic Conjugate - Holomorphic - Homotopy - Hyperbolic Trig Functions #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 210 Context: ``` # Index - i, 4 - identity, 3 - identity map, 23 - identity principle, 154 - image - of a function, 27 - of a point, 23 - imaginary axis, 5 - imaginary part, 4 - improper integral, 104, 186 - infinity, 46 - inside, 88 - integral, 71 - path independent, 102 - integral test, 129 - integration by parts, 92 - interior point, 13 - inverse function, 31 - of a Möbius transformation, 43 - inverse parametrization, 74 - inversion, 45 - isolated point, 13 - isolated singularity, 170 - Jacobian, 65 - Jordan curve theorem, 88 - L'Hôpital's rule, 197 - Laplace, 110 - Laurent series, 157 - least upper bound, 124, 137 - Leibniz's rule, 83, 196 - length, 73 - limit - infinity, 46 - of a function, 24 - of a sequence, 122 - of a series, 126 - linear fractional transformation, 43 - Liouville's theorem, 104 - Log, 59 - log, 61 - logarithm, 59 - logarithmic derivative, 180 - max/min property for harmonic functions, 115, 155 - maximum - strong relative, 115 - weak relative, 116, 155 - mean-value theorem - for harmonic functions, 114 - for holomorphic functions, 86 - for real functions, 195 - meromorphic, 181 - minimum - strong relative, 115 - weak relative, 155 - Möbius transformation, 43 - modulus, 5 - monotone, 124 - monotone sequence property, 124 - Morera's theorem, 100 - multiplication, 2 - north pole, 51 ``` #################### File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf Page: 211 Context: ``` obvious, 23 odd, 167 one-to-one, 31 onto, 31 open disk, 12 set, 13 order of a pole, 173 orientation, 15 partial derivative, 32 path, 14 closed, 16 inside of, 88 interior of, 88 polygonal, 79 positively oriented, 88 path independent, 102 periodic, 55, 193 Picard's theorem, 174 piecewise smooth, 72 plane, 12 pointwise convergence, 131 Poisson integral formula, 119 Poisson kernel, 95, 118 polar form, 9 pole, 170 polynomial, 0, 20, 40, 102 positive orientation, 88 power series, 135 differentiation of, 147 integration of, 139 primitive, 76 primitive root of unity, 9 principal argument, 59 principal logarithm, 59 principal value of z, 61 punctured disk, 169 real axis, 5 real number, 2 real part, 4 rectangular form, 9 region, 14 of convergence, 136 simply-connected, 101, 111 removable singularity, 170 reparameterization, 73 residue theorem, 176 reverse triangle inequality, 11, 20 Riemann hypothesis, 130 Riemann sphere, 47 Riemann zeta function, 130 root, 4 root of unity, 9 root test, 138 Rouché's theorem, 182 separated, 14 sequence, 122 convergent, 122 divergent, 122 limit, 122 monotone, 124 series, 125 ``` #################### 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 ``` #################### File: TEST%20REPORT%20-%20Lois%20Layne%20Ramos.pdf Page: 1 Context: # TESTED BY: Lois Layne Ramos ## TEST EMAIL: loislayneramos@gmail.com ## WEB APP URL: [https://tester-task.vercel.app/](https://tester-task.vercel.app/) ## BROWSER: Chrome, Safari, Edge, Firefox, Opera ## OS: Mac OS, Windows ## DATE OF TESTING: 02-13-24 # BUGS/ISSUES FOUND: 1. **[FUNCTIONAL]** INPUT FIELD: Accepts empty values 2. **[FUNCTIONAL]** KEYBOARD FUNCTION: Click 'ENTER' KEY with/without input values create double entries. 3. **[USER INTERFACE GLITCHES]** DELETE BUTTON: Delete all data in one click - no confirmation asked. 4. **[USER INTERFACE GLITCHES]** BUTTON LABEL: Function is unclear. 5. **[USER INTERFACE GLITCHES]** DUPLICATE VALUES: 2 entries added with the same input values. 6. **[SECURITY VULNERABILITIES]** LOGIN: No login page - data is not secure. # PROBLEM DESCRIPTION The user is still able to create/add tasks even without any input. Allowing this might result in a waste of space and an overwhelming number of empty entries. (Refer to bug ticket no. 1) #################### File: TEST%20REPORT%20-%20Lois%20Layne%20Ramos.pdf Page: 3 Context: # PROBLEM DESCRIPTION Pressing the Enter key (regardless of whether there's data entered) results in 2 entries being created: an empty entry and the user's input value. (Refer to bug ticket no. 2) ## Screenshot: ![TODO APP](https://via.placeholder.com/300) ## Steps to replicate 1. Go to [tester-task.vercel.app](https://tester-task.vercel.app/) 2. Click input field 3. Leave textarea blank/ with input value 4. Press 'Enter' in keyboard Image Analysis: ### Comprehensive Examination #### 1. **Localization and Attribution:** - **Image 1:** - Positioned below the text "Screenshot:" and above the "Steps to replicate." - Assigned number: **Image 1**. #### 2. **Object Detection and Classification:** - **Image 1:** - **Objects Detected:** - A digital interface of a "TODO APP." - Text elements such as "Tasks," "Done Tasks," and "TASKS." - Input areas labeled with placeholder text like "New task." - Icons including checkmarks, a plus icon, and a red delete icon. - **Key Features:** - The input fields with checkboxes, indicating they may be tasks or items to be marked as complete. - A distinct blue "Tasks" tab with a number (10) indicating possibly the number of ongoing tasks. - A red "TASKS" tab likely for accessing task functionalities. #### 3. **Scene and Activity Analysis:** - **Image 1:** - **Scene Description:** - The scene showcases a task management interface of a TODO app. - The background is blue, providing a contrast to the primarily white input areas. - **Activities Taking Place:** - Users are presented with a list of tasks ("test 9," "24") and are able to add new tasks. - There are functional components to mark tasks as complete or delete them. #### 4. **Text Analysis:** - **Extracted Text:** - "TODO APP," "Tasks," "Done Tasks," "TASKS," "test 9," "24," "New task." - **Text Content Analysis:** - The text explains the structure and purpose of the app. - "Tasks" tab indicates the total number of tasks (10). - "Done Tasks" likely denotes completed tasks, although the exact number is not visible. - Placeholder and task names give a quick insight into the app's functionality. #### 7. **Anomaly Detection:** - **Image 1:** - **Possible Anomalies:** - The text description highlights a bug where pressing the Enter key creates two entries (an empty entry and the user’s input). - **Impact:** - This bug impacts the usability of the app, leading to unwanted empty entries and potentially cluttered task lists. #### 8. **Color Analysis:** - **Image 1:** - **Dominant Colors:** - Blue: Background, conveying a calm and professional feel. - White: Main input areas, enhancing readability. - Red: Delete icons, drawing attention to functions for removing tasks. - **Impact on Perception:** - The color scheme ensures that key elements like tasks and interactive icons stand out clearly. #### 9. **Perspective and Composition:** - **Perspective:** - Front-facing view of a user interface on a digital screen. - **Composition:** - The interface is composed centrally with task lists and interactive elements balanced within the screen boundaries. - Clear segregation of task categories with respective tabs. #### 10. **Contextual Significance:** - **Context:** - This image is within a bug report or a similar documentation context. - **Contribution:** - The image visually supports the written problem description by illustrating the issue within the TODO app. #### 12. **Graph and Trend Analysis:** - **Process Description (Depicted through steps):** - Specifies steps to replicate the described bug: 1. Visit the given URL. 2. Click the input field. 3. Leave it blank or enter a value. 4. Press 'Enter'. - **Significance:** - Ensures precise replication for developers or testers to observe and address the issue. #### 13. **Ablaufprozesse (Process Flows):** - The described steps detail a straightforward process flow to replicate a bug, critical for identifying and resolving the issue reported. This examination spans various aspects, helping in understanding the content, functionality, and issues of the depicted TODO app. #################### File: TEST%20REPORT%20-%20Lois%20Layne%20Ramos.pdf Page: 5 Context: # Steps to replicate 1. Go to [https://tester-task.vercel.app/](https://tester-task.vercel.app/) 2. Create at least 20 entries. 3. Click the delete task button. ## Actual Behavior - Clicking the "Delete Task" button results in the immediate deletion of all tasks without requiring confirmation. ## Expected Behavior - Clicking the "Delete Task" button should prompt the user for confirmation before permanently removing any data. ## Problem Description The design of the "Done Tasks" button is confusing due to a conflicting element. Upon clicking a specific entry inside the table, a second button with an "x" appears, suggesting task removal, despite the original button label implying task completion. (Refer to bug ticket no. 4) ## Screenshot ![Screenshot](screenshot_link) Image Analysis: ### Examination of the Attached Visual Content #### Image Localization and Attribution - **Image 1**: The text content and instructional steps. - **Image 2**: The screenshot of the "TODO APP". #### Object Detection and Classification - **Image 1**: - Objects: Text blocks. - **Image 2**: - Objects: User interface of a TODO application, including buttons, text fields, and a table. #### Scene and Activity Analysis - **Image 1**: - Scene: An instructional and descriptive document detailing a procedure for a "TODO APP". - Activities: Steps to replicate a task deletion issue, description of actual and expected behavior, problem description. - **Image 2**: - Scene: A screenshot of the "TODO APP" interface. - Activities: Interaction with tasks within the app such as adding or completing tasks. #### Text Analysis - **Image 1**: - **Steps to replicate**: 1. Go to https://tester-task.vercel.app/ 2. Create at least 20 entries. 3. Click delete task button - **Actual Behavior**: - Clicking the "Delete Task" button results in the immediate deletion of all tasks without requiring confirmation. - **Expected Behavior**: - Clicking the "Delete Task" button should prompt the user for confirmation before permanently removing any data. - **Problem Description**: - The design of the "Done Tasks" button is confusing due to a conflicting element. Upon clicking a specific entry inside the table, a second button with an "x" appears, suggesting task removal, despite the original button label implying task completion. - Reference: Bug ticket no. 4. - **Screenshot**: - Caption indicating a visual representation of the problem described. - **Image 2**: - Contains labels like "Tasks", "Done Tasks", "T+STS", "New task", and an "x" button indicating deletion. #### Color Analysis - **Image 2**: - Dominant Colors: Blue background. - Other Colors: White for text fields and buttons, red for the delete button, and gray for the text entries. - Impact: The color composition highlights the interactive elements, particularly the red delete button, drawing attention to it. #### Perspective and Composition - **Image 2**: - Perspective: Frontal view of the application interface. - Composition: The interface is centrally aligned with clearly delineated sections for tasks, done tasks, and task management buttons, providing a clean and organized visual structure. ### Summary The document contains detailed steps to reproduce a bug in a TODO application, describes the actual and expected behaviors, and outlines a design issue related to the "Done Tasks" button that may confuse users. The included screenshot (Image 2) visually supports the problem description by highlighting the TODO APP interface and illustrating the confusing elements. The contextual and functional analysis intends to guide developers or testers in identifying and resolving the issues. The color, perspective, and composition analyses provide insight into how the interface design might impact user interaction. #################### File: TEST%20REPORT%20-%20Lois%20Layne%20Ramos.pdf Page: 8 Context: # Expected Behavior - When a user tries to add an entry with duplicate values, the system should identify the duplicate: recognize that the input matches an existing entry and notify the user. # PROBLEM DESCRIPTION 6. The absence of a login page creates a security vulnerability. When a user opens the web application URL in a browser, they are directly transferred to the main page without authentication. This exposes sensitive user data to unauthorized access, as anyone with the URL can potentially view it (refer to bug ticket no. 6). # Screenshot ![Screenshot](https://tester-task.vercel.app) Image Analysis: ### Analysis of the Attached Visual Content #### 1. **Localization and Attribution** - **Text Content:** - Located at the top and mid sections of the page. - **Image:** - Located at the bottom section of the page. #### 2. **Object Detection and Classification** ##### Image 1: Screenshot of TODO App - **Objects Detected:** - Browser bar - Web application interface containing: - Header section - Task input field - Task list - Buttons for "Tasks," "Done Tasks," and "TASKS" - **Classified Categories:** - User Interface Elements - Text Fields - Buttons #### 3. **Scene and Activity Analysis** ##### Image 1: Screenshot of TODO App - **Scene Description:** - The image portrays a web application named "TODO APP". - **Activities Taking Place:** - A task management interface is shown, listing tasks named "test1" and "test 1" multiple times. - **Main Actors and Actions:** - The user seems to be managing tasks, with the interface displaying a list of tasks alongside options to add a new task and toggle between different views of tasks. #### 4. **Text Analysis** ##### Text Content: - **Expected Behavior:** - Description of how the system should behave when encountering duplicate values in input. - Text: "When a user tries to add an entry with duplicate values, the system should Identify the duplicate: Recognize that the input matches an existing entry - notify the user." - **Problem Description:** - Identifies security issue due to absence of a login page in a web application. - Text: "The absence of a login page creates a security vulnerability. When a user opens the web application URL in a browser, they are directly transferred to the main page without authentication. This exposes sensitive user data to unauthorized access, as anyone with the URL can potentially view it." - **Screenshot Text:** - "TODO APP" - Task entries: "test1", "test1", "test 1" - Buttons labeled: "Tasks," "Done Tasks," "TASKS" #### 6. **Product Analysis** ##### Image 1: Screenshot of TODO App - **Product Details:** - **Main Features:** - Interactive task management application. - Functional buttons for task filtering ("Tasks," "Done Tasks," "TASKS"). - Task input field for adding new tasks. - List of current tasks displayed. - **Materials:** - Digital interface with a clean and simple design. - **Colors:** - Dominant colors are white for the main app interface and blue for the background. #### 7. **Anomaly Detection** ##### Image 1: Screenshot of TODO App - **Anomalies:** - The security concern mentioned highlights a significant anomaly in web application design, as the lack of a login feature poses a security risk. #### 8. **Color Analysis** ##### Image 1: Screenshot of TODO App - **Color Composition:** - Dominant use of white for the application interface. - Blue gradient background for contrast. - Red button indicating active selection (likely for deletion or high priority). #### 9. **Perspective and Composition** ##### Image 1: Screenshot of TODO App - **Perspective:** - Standard front-view interface screenshot. - **Composition:** - The user interface is centered, with the browser bar at the top and main functional areas (task list, input field, and buttons) in the middle. #### 10. **Contextual Significance** - The image and the text combined highlight a critical issue and its expected behavior in managing tasks within a web application. It underscores the importance of secure authentication mechanisms in web applications to protect sensitive user data. #################### File: TEST%20REPORT%20-%20Lois%20Layne%20Ramos.pdf Page: 9 Context: # Steps to replicate 1. Go to [https://tester-task.vercel.app/](https://tester-task.vercel.app/) 2. Redirect to main page ## Actual Behavior Accessing the web app URL directly redirects to the main page, bypassing the intended landing page or login process. ## Expected Behavior Upon accessing the web application, users should be prompted to log in before being granted access to the main page and their data. This ensures that only authorized users can view and interact with sensitive information. # Performance and Browser Compatibility Testing ## Results The web application demonstrated strong performance and consistent functionality across various browsers. No significant issues were identified during testing. ## Details - Data loading remained quick and responsive even when reaching 100 entries. - No performance degradation was observed when reloading the browser. - The application functioned identically on different browsers, including Opera, Safari, Chrome, Microsoft Edge, and Firefox. ## Conclusion These findings indicate a well-optimized application that delivers a reliable and consistent user experience across various performance scenarios and browser platforms. #################### File: TEST%20REPORT%20-%20Lois%20Layne%20Ramos.pdf Page: 10 Context: # Additional Recommendations While no major issues were found, consider conducting further testing under simulated peak loads or with larger datasets to ensure continued performance stability. Monitor user feedback and performance metrics in real-world usage to identify any potential issues that may not have been captured during controlled testing. # Suggestions for Improvements 1. **Implement a hover text or tooltip for the "Delete All Tasks" button** that clearly explains its function. Introduce a confirmation dialog before deleting all tasks, requiring user confirmation and providing an option to cancel. (ref. Bug ticket no. 3) 2. **Implement a dedicated column** within the panel of task entries that clearly displays the total number of tasks created by the user. This would enhance user awareness and facilitate informed decision-making regarding their tasks. Furthermore, providing this data point would significantly simplify automated testing scenarios, enabling direct verification of task creation functionalities without resorting to workarounds. 3. **Modify the behavior** so adding a new entry preserves the user's current scroll position within the table. This would significantly improve the user experience and streamline task management. 4. **Clarify the intended purpose of the "Done Tasks" button** by implementing one of the expected behaviors below. This will provide users with clear functionality and avoid confusion. Clicking the "Done Tasks" button should trigger a specific action, such as: - Displaying a list of all tasks marked as done/completed by the user. - Taking users to a dedicated "Completed Tasks" section within the application. #################### File: TEST%20REPORT%20-%20Lois%20Layne%20Ramos.pdf Page: 11 Context: Additionally, consider providing a tooltip or hover text explaining the button's function on mouseover, further enhancing user experience. 5. The **"Add/Submit"** button should be disabled or visually grayed out when the input field is empty, clearly indicating that no action can be performed until data is entered. Optionally, a tooltip or error message could be displayed upon hovering over the disabled button, explaining the requirement for user input. ########## """QUERY: can you tell me the contents of your context in 250 words?""" Consider the chat history for relevant information. Use all information included. If query is already asked in the history double check the correctness of your answer and maybe correct your previous mistake. Use as much tokens as needed but at the same time be as efficient as possible. If you find information separated by a | in the context, it is a table formatted in Markdown - the whole context is formatted as md structure. Important: Take a look at the QUERY and only the QUERY. Please try always to answer the query question. If the client ask for a formatting structure follow his advise.But if the question is vague or unclear ask a follow-up question based on the context. If too many questions were asked already based on the history, just answer the question Final Files Sources: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 3, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 4, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 5, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 6, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 7, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 9, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 10, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 12, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 13, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 14, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 15, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 16, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 18, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 20, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 21, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 24, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 25, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 26, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 27, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 28, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 29, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 30, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 32, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 33, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 37, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 41, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - 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Page 209, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 210, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 211, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 212, TEST%20REPORT%20-%20Lois%20Layne%20Ramos.pdf - Page 1, TEST%20REPORT%20-%20Lois%20Layne%20Ramos.pdf - Page 3, TEST%20REPORT%20-%20Lois%20Layne%20Ramos.pdf - Page 5, TEST%20REPORT%20-%20Lois%20Layne%20Ramos.pdf - Page 8, TEST%20REPORT%20-%20Lois%20Layne%20Ramos.pdf - Page 9, TEST%20REPORT%20-%20Lois%20Layne%20Ramos.pdf - Page 10, TEST%20REPORT%20-%20Lois%20Layne%20Ramos.pdf - Page 11 ================================================== **Elapsed Time: 0.44 seconds** ================================================== FINAL ANSWER Answer: [Answer was streamed to client] ================================================== **Elapsed Time: 3.24 seconds** ==================================================