{
    "query": "Is this a clarifying question or a clarification statement? Answer Yes if it is a clarifying question or statement, and No in all other cases. Here's the text: ## Werkst\u00fccktypen\n\n| Typ | Gewicht Rohteil | Gewicht Fertigteil | Abma\u00dfe (L\u00e4nge, Breite, H\u00f6he) | Hauptlagerdurchmesser | Pleuellagerdurchmesser |\n|---|---|---|---|---|---|\n| Kurbelwelle | 20 | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert |\n| Zylinderblock | 60 | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert |\n| Zylinderblock | 40 | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert |\n| Geh\u00e4use E-Motor | 25 | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert |\n| Zylinderblock | 20 | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert |\n| Kurbelwelle | 20 | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert |\n| Zylinderblock | 40 | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert |\n| Rotor | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert |\n| Batterie | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert |\n| Batterie | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert |\n| Motorgeh\u00e4use SSP HA L43-4 | nicht spezifiziert | 14.875 kg | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert |\n| Getriebegeh\u00e4use SSP H4 LK34.4 | nicht spezifiziert | 5,840 kg | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert |\n| Motorph\u00e4notyp SSP HA LX34 | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert |\n| Motorblock | 150 kg | nicht spezifiziert | 900 mm, 600 mm, 500 mm | nicht spezifiziert | nicht spezifiziert |\n| Getriebegeh\u00e4use SSP HA LK3.4 | nicht spezifiziert | 6,630 kg | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert | \n",
    "namespace": "9b7a0d3b-f149-4977-879c-cf22ea8d011f",
    "messages": [],
    "stream": false,
    "language_level": "",
    "chat_channel": "",
    "language": "German",
    "tone": "neutral",
    "writing_style": "standard",
    "model": "gemini-1.5-flash",
    "knowledgebase": "ki-dev-large",
    "seed": 0,
    "client_id": 1029,
    "all_context": false,
    "follow_up_for": null,
    "knowledgebase_files_count": 0,
    "override_command": "",
    "disable_clarity_check": false,
    "custom_primer": "",
    "logging": true,
    "query_route": "simple_conversation"
}


INITIALIZATION
Knowledgebase: ki-dev-large
Base Query: Is this a clarifying question or a clarification statement? Answer Yes if it is a clarifying question or statement, and No in all other cases. Here's the text: ## Werkstücktypen

| Typ | Gewicht Rohteil | Gewicht Fertigteil | Abmaße (Länge, Breite, Höhe) | Hauptlagerdurchmesser | Pleuellagerdurchmesser |
|---|---|---|---|---|---|
| Kurbelwelle | 20 | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert |
| Zylinderblock | 60 | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert |
| Zylinderblock | 40 | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert |
| Gehäuse E-Motor | 25 | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert |
| Zylinderblock | 20 | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert |
| Kurbelwelle | 20 | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert |
| Zylinderblock | 40 | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert |
| Rotor | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert |
| Batterie | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert |
| Batterie | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert |
| Motorgehäuse SSP HA L43-4 | nicht spezifiziert | 14.875 kg | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert |
| Getriebegehäuse SSP H4 LK34.4 | nicht spezifiziert | 5,840 kg | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert |
| Motorphänotyp SSP HA LX34 | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert |
| Motorblock | 150 kg | nicht spezifiziert | 900 mm, 600 mm, 500 mm | nicht spezifiziert | nicht spezifiziert |
| Getriebegehäuse SSP HA LK3.4 | nicht spezifiziert | 6,630 kg | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert | 

Model: gemini-1.5-flash
Use Curl?: 
==================================================
**Elapsed Time: 0.00 seconds**
==================================================


ROUTING
Query type: simple_conversation
==================================================
**Elapsed Time: 0.00 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.

        
            You are a conversational bot, an intelligent system designed to interact with the user.
            Respond courteously to the message of the user.
            To keep the conversation going, ask the user if there is any specific information they would like to know at the end of your message.
        
==================================================
**Elapsed Time: 0.00 seconds**
==================================================


FINAL QUERY 
Final Query: Is this a clarifying question or a clarification statement? Answer Yes if it is a clarifying question or statement, and No in all other cases. Here's the text: ## Werkstücktypen

| Typ | Gewicht Rohteil | Gewicht Fertigteil | Abmaße (Länge, Breite, Höhe) | Hauptlagerdurchmesser | Pleuellagerdurchmesser |
|---|---|---|---|---|---|
| Kurbelwelle | 20 | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert |
| Zylinderblock | 60 | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert |
| Zylinderblock | 40 | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert |
| Gehäuse E-Motor | 25 | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert |
| Zylinderblock | 20 | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert |
| Kurbelwelle | 20 | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert |
| Zylinderblock | 40 | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert |
| Rotor | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert |
| Batterie | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert |
| Batterie | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert |
| Motorgehäuse SSP HA L43-4 | nicht spezifiziert | 14.875 kg | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert |
| Getriebegehäuse SSP H4 LK34.4 | nicht spezifiziert | 5,840 kg | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert |
| Motorphänotyp SSP HA LX34 | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert |
| Motorblock | 150 kg | nicht spezifiziert | 900 mm, 600 mm, 500 mm | nicht spezifiziert | nicht spezifiziert |
| Getriebegehäuse SSP HA LK3.4 | nicht spezifiziert | 6,630 kg | nicht spezifiziert | nicht spezifiziert | nicht spezifiziert | 
Important: Take a look at the QUERY and only the QUERY. If this is vague or unclear, please ignore everything and ask a follow-up question instead!


Final Files Sources: 
==================================================
**Elapsed Time: 0.71 seconds**
==================================================


FINAL ANSWER 
Answer: <IPython.core.display.Markdown object>
==================================================
**Elapsed Time: 0.00 seconds**
==================================================


ROUTING
Query type: summary
==================================================
**Elapsed Time: 1.63 seconds**
==================================================


RAG PARAMETERS
Max Context To Include: 100
Lowest Score to Consider: 0.1
==================================================
**Elapsed Time: 0.00 seconds**
==================================================


VECTOR SEARCH ALGORITHM TO USE 
Use MMR search?: True
Use Similarity search?: False
==================================================
**Elapsed Time: 0.11 seconds**
==================================================


VECTOR SEARCH DONE 
==================================================
**Elapsed Time: 8.08 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: 0.24 seconds**
==================================================


FINAL QUERY 
Final Query: CONTEXT: ##########
File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf

Page: 3

Context: # "And what is the use of a book," thought Alice, "without pictures or conversations?"  
Lewis Carroll (Alice in Wonderland)

## About this book  
*A First Course in Complex Analysis* was written for a one-semester undergraduate course developed at Binghamton University (SUNY) and San Francisco State University, and has been adopted at several other institutions. For many of our students, Complex Analysis is their first rigorous analysis (if not mathematics) class they take, and this book reflects this very much. We tried to rely on as few concepts from real analysis as possible. In particular, series and sequences are treated from scratch, which has the consequence that power series are introduced late in the course. The goal our book works toward is the Residue Theorem, including some nontraditional applications from both continuous and discrete mathematics.

A printed paperback version of this open textbook is available from Orthogonal Publishing (www.orthogonalpublishing.com) or your favorite online bookseller.

## About the authors  
**Matthias Beck** is a professor in the Mathematics Department at San Francisco State University. His research interests are in geometric combinatorics and analytic number theory. He is the author of three other books, *Computing the Continuous Discretion: Integer-point Enumeration in Polyhedra* (with Sinal Robins, Springer 2007), *The Art of Proof: Basic Training for Mathematics* (with Ross Geoghegan, Springer 2010), and *Combinatorial Reciprocity Theorem: An Invitation to Enumerative Geometric Combinatorics* (with Raman Sanyal, AMS 2018).

**Gerald M. Green** is a lecturer in the Department of Mathematical Sciences at Binghamton University (SUNY).

**Dennis P. Stinson** is a professor emeritus in the Department of Mathematical Sciences at Binghamton University (SUNY). His research interests are in dynamical systems and formal languages.

**Lucas Sabalka** is an applied mathematician at a technology company in Lincoln, Nebraska. He works on 3-dimensional computer vision applications. He was formerly a professor of mathematics at St. Louis University, after postdoctoral positions at UC Davis and Binghamton University (SUNY). His mathematical research interests are in geometric group theory, low dimensional topology, and computational algebra.
####################
File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf

Page: 5

Context: # Contents

1. **Complex Numbers**  
   1.1. Definitions and Algebraic Properties . . . . . . . . 2  
   1.2. From Algebra to Geometry and Back . . . . . . . . 5  
   1.3. Geometric Properties . . . . . . . . . . . . . . . . 9  
   1.4. Elementary Topology of the Plane . . . . . . . . . 12  
   Optional Lab . . . . . . . . . . . . . . . . . . . . . . 21  

2. **Differentiation**  
   2.1. Limits and Continuity . . . . . . . . . . . . . . . . 23  
   2.2. Differentiability and Holomorphicity . . . . . . . 28  
   2.3. The Cauchy–Riemann Equations . . . . . . . . . . . 32  
   2.4. Constant Functions . . . . . . . . . . . . . . . . . 36  

3. **Examples of Functions**  
   3.1. Möbius Transformations . . . . . . . . . . . . . . . 43  
   3.2. Infinity and the Cross Ratio . . . . . . . . . . . 46  
   3.3. Stereographic Projection . . . . . . . . . . . . . . 48  
   3.4. Exponential and Trigonometric Functions . . . . . 55  
   3.5. Logarithms and Complex Exponentials . . . . . . . 59  

4. **Integration**  
   4.1. Definition and Basic Properties . . . . . . . . . . 71  
   4.2. Antiderivatives . . . . . . . . . . . . . . . . . . 76  
   4.3. Cauchy’s Theorem . . . . . . . . . . . . . . . . . 80  
   4.4. Cauchy’s Integral Formula . . . . . . . . . . . . 85  

5. **Consequences of Cauchy’s Theorem**  
   5.1. Variations of a Theme . . . . . . . . . . . . . . . 97  
   5.2. Antiderivatives Again . . . . . . . . . . . . . . 100  
   5.3. Taking Cauchy's Formulas to the Limit . . . . . . 102
####################
File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf

Page: 6

Context: ```
# Contents

6. Harmonic Functions ................................. 110  
   6.1 Definitions and Basic Properties ................. 114  
   6.2 Mean-Value Principle ............................ 114  

7. Power Series ....................................... 121  
   7.1 Sequences and Completeness ...................... 122  
   7.2 Series .......................................... 125  
   7.3 Sequences and Series of Functions ............... 131  
   7.4 Regions of Convergence .......................... 135  

8. Taylor and Laurent Series .......................... 146  
   8.1 Power Series and Holomorphic Functions .......... 146  
   8.2 Classification of Zeros and the Identity Principle 152  
   8.3 Laurent Series .................................. 156  

9. Isolated Singularities and the Residue Theorem..... 169  
   9.1 Classification of Singularities ................... 169  
   9.2 Residues ........................................ 176  
   9.3 Argument Principle and Rouché’s Theorem ........ 180  

10. Discrete Applications of the Residue Theorem ...... 188  
   10.1 Infinite Sums ................................... 188  
   10.2 Binomial Coefficients ........................... 189  
   10.3 Fibonacci Numbers ............................... 190  
   10.4 The Coin-Exchange Problem ...................... 193  
   10.5 Dedekind Sums .................................. 193  

## Theorems From Calculus

## Solutions to Selected Exercises

## Index
```
####################
File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf

Page: 7

Context: # Chapter 1

## Complex Numbers

Die ganzen Zahlen hat der liebe Gott geschaffen, alles andere ist Menschenwerk.  
(God created the integers, everything else is made by humans.)  
Leopold Kronecker (1823–1891)

The real numbers have many useful properties. There are operations such as addition, subtraction, and multiplication, as well as division by any nonzero number. There are useful laws that govern these operations, such as the commutative and distributive laws. We can take limits and do calculus, differentiating and integrating functions. But you cannot take a square root of \(-1\); that is, you cannot find a real root of the equation

$$
x^2 + 1 = 0. \tag{1.1}
$$

Most of you have heard that there is a "new" number that is a root of (1.1); that is, \(x^2 + 1 = 0\) or \(x^2 = -1\). We will show that when the real numbers are enlarged to a new system called the **complex numbers**, which includes \(i\), not only do we gain numbers with interesting properties, but we do not lose many of the nice properties that we had before.

The complex numbers, like the real numbers, will have the operations of addition, subtraction, multiplication, as well as division by any complex number except zero. These operations will follow all the laws that we are used to, such as the commutative and distributive laws. We will also be able to take limits and do calculus. And there will be a root of (1.1).

As a brief historical aside, complex numbers did not originate with the search for a square root of \(-1\); rather, they were introduced in the context of cubic equations. Scipione del Ferro (1465–1526) and Niccolò Tartaglia (1499–1557) discovered a way to find a root of any cubic polynomial, which was publicized by Gerolamo Cardano (1501–1576) and is often referred to as **Cardano's formula**. For the cubic polynomial \(x^3 + px + q\), Cardano's formula involves the quantity 

$$
\sqrt[3]{-\frac{q}{2} + \sqrt{\left(\frac{q}{2}\right)^2 + \left(\frac{p}{3}\right)^3}} + \sqrt[3]{-\frac{q}{2} - \sqrt{\left(\frac{q}{2}\right)^2 + \left(\frac{p}{3}\right)^3}}.
$$
####################
File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf

Page: 8

Context: ```
# Complex Numbers

It is not hard to come up with examples for \( p \) and \( q \) for which the argument of this square root becomes negative and thus not computable within the real numbers. On the other hand (e.g., by arguing through the graph of a cubic polynomial), every cubic polynomial has at least one real root. This seeming contradiction can be solved using complex numbers, as was probably first exemplified by Rafael Bombelli (1526–1572).

In the next section we show exactly how the complex numbers are set up, and in the rest of this chapter we will explore the properties of the complex numbers. These properties will be of both algebraic (such as the commutative and distributive properties mentioned already) and geometric nature. You will see, for example, that multiplication can be described geometrically. In the rest of the book, the calculus of complex numbers will be built on the properties that we develop in this chapter.

## 1.1 Definitions and Algebraic Properties

There are many equivalent ways to think about a complex number, each of which is useful in its own right. In this section, we begin with a formal definition of a complex number. We then interpret this formal definition in more useful and easier-to-work-with algebraic language. Later we will see several more ways of thinking about complex numbers.

**Definition.** The complex numbers are pairs of real numbers:  
\[
C = \{(x, y) : x, y \in \mathbb{R} \}
\]
equipped with the addition
\[
(x, y) + (a, b) = (x + a, y + b) \quad (1.2)
\]
and the multiplication
\[
(x, y) \cdot (a, b) = (x a - y b, x b + y a) \quad (1.3)
\]

One reason to believe that the definitions of these binary operations are acceptable is that \( C \) is an extension of \( \mathbb{R} \), in the sense that the complex numbers of the form \((x, 0)\) behave just like real numbers:
\[
(x, 0) + (y, 0) = (x + y, 0) \quad \text{and} \quad (x, 0) \cdot (y, 0) = (xy, 0).
\]
```
####################
File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf

Page: 9

Context: So we can think of the real numbers being embedded in \( \mathbb{C} \) as those complex numbers whose second coordinate is zero.

The following result states the algebraic structure that we established with our definitions.

## Proposition 1.1

Let \( (x, y) \) be a field, that is, for all \( (a, b) \in \mathbb{C} \):

1. \( (x, y) + (a, b) \in \mathbb{C} \) (1.4)  
2. \( (x, y) + (a, d) + (b, c) = (x, y) + ((a + b), c) \) (1.5)  
3. \( (x, y) + (0, 0) = (x, y) \) (1.6)  
4. \( (x, y) + (-x, -y) = (0, 0) \) (1.7)  
5. \( (a, b) + (c, d) = (x, y) \cdot (a, b) + (c, d) \) (1.8)  
6. \( (x, y) \in \mathbb{C} \) (1.9)  
7. \( (x, y) \cdot (a, b) = (x \cdot a, y \cdot b) \) (1.10)  
8. \( (x, y) \cdot (b, d) = (x \cdot b, (a \cdot b)(c, d)) \) (1.11)  
9. \( (x, y) \cdot (a, b) = (x, y) \) (1.12)  
10. \( (x, y) \cdot (1, 0) = (x, y) \) (1.13)  
11. For all \( (x, y) \in \mathbb{C} \setminus \{(0, 0)\} \):  
    \( (x, y) \cdot \left( \frac{(y, -x)}{x^2 + y^2} \right) = (1, 0) \) (1.14)  

What we are stating here can be compressed in the language of algebra: equations (1.4)–(1.8) say that \( (\mathbb{C}, +) \) is an Abelian group with identity \( (0, 0) \); equations (1.10)–(1.14) say that \( (\mathbb{C} \setminus \{(0, 0)\}, \cdot) \) is an Abelian group with identity \( (1, 0) \).

The proof of Proposition 1.1 is straightforward but nevertheless makes for good practice (Exercise 1.14). We give one sample:

**Proof (1.18)**. By our definition for complex addition and properties of additive inverse in \( \mathbb{R} \):

\[
(x, y) + (-x, -y) = (x + (-x), y + (-y)) = (0, 0).
\]

The definition of our multiplication implies the innocent-looking statement:

\[
(0, 1) \cdot (1, 0) = (1, 0). \quad (1.15)
\]
####################
File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf

Page: 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: 11

Context: # 1.2 From Algebra to Geometry and Back

Although we just introduced a new way of writing complex numbers, let’s for a moment return to the \((x, y)\)-notation. It suggests that we can think of a complex number as a two-dimensional real vector. When plotting these vectors in the plane \(\mathbb{R}^2\), we will call the \(x\)-axis the real axis and the \(y\)-axis the imaginary axis. The addition that we defined for complex numbers resembles vector addition; see Figure 1.1. The analogy stops at multiplication: there is no “usual” multiplication of two vectors in \(\mathbb{R}^2\) that gives another vector, and certainly not one that agrees with our definition of the product of two complex numbers.

![Figure 1.1: Addition of complex numbers.](#)

Any vector in \(\mathbb{R}^2\) is defined by its two coordinates. On the other hand, it is also determined by its length and the angle it encloses with, say, the positive real axis; let’s define these concepts thoroughly.

### Definition

The absolute value (also called the modulus) of \(z = x + iy\) is

\[
r = |z| = \sqrt{x^2 + y^2},
\]

and an argument of \(z = x + iy\) is a number \(\phi \in \mathbb{R}\) such that

\[
x = r \cos \phi \quad \text{and} \quad y = r \sin \phi.
\]

A given complex number \(z = x + iy\) has infinitely many possible arguments. For instance, the number \(1 = 1 + 0i\) lies on the positive real axis, and has argument \(0\), but we could just as well say it has argument \(2\pi, -2\pi, \text{or } 2k\pi\) for any integer \(k\). The number \(0 = 0 + 0i\) has modulus \(0\), and every real number \(y\) is an argument.
####################
File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf

Page: 13

Context: # FROM ALGEBRA TO GEOMETRY AND BACK

## 7

ϕ₂. This means we can write \(x_1 + iy_1 = (r_1 \cos \phi_1) + i(r_1 \sin \phi_1)\) and \(x_2 + iy_2 = (r_2 \cos \phi_2) + i(r_2 \sin \phi_2)\). To compute the product, we make use of some classic trigonometric identities:

\[
(x_1 + iy_1)(x_2 + iy_2) = (r_1 \cos \phi_1 + ir_1 \sin \phi_1)(r_2 \cos \phi_2 + ir_2 \sin \phi_2)
\]
\[
= r_1 r_2 (\cos \phi_1 \cos \phi_2 - \sin \phi_1 \sin \phi_2) + i(r_1 r_2 \cos \phi_2 \sin \phi_1 + r_1 r_2 \sin \phi_2 \cos \phi_1)
\]
\[
= r_1 r_2 \left( \cos(\phi_1 + \phi_2) + i \sin(\phi_1 + \phi_2) \right).
\]

So the absolute value of the product is \(r_1 r_2\) and one of its arguments is \(ϕ_1 + ϕ_2\). Geometrically, we are multiplying the lengths of the two vectors representing our two complex numbers and adding their angles measured with respect to the positive real axis. 

![Figure 1.3: Multiplication of complex numbers](#)

In view of the above calculation, it should come as no surprise that we will have to deal with quantities of the form \(x + iy\) (where \(y\) is some real number) quite a bit. To save space, bytes, ink, etc., and because "Mathematics is for lazy people" we introduce a shortcut notation and define:

\[
e^{i\phi} = \cos \phi + i \sin \phi.
\] 

^1 You should convince yourself that there is no problem with this fact since there are many possible arguments for complex numbers, as both cosine and sine are periodic functions with period \(2\pi\).  
^2 Peter Halton (unpublished articles, Hudson River Undergraduate Mathematics Conference 2000).
####################
File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf

Page: 14

Context: # COMPLEX NUMBERS

Figure 1.4 shows three examples. At this point, this exponential notation is indeed 

![Figure 1.4: Three sample complex numbers of the form \( e^{x} \).](image-path)

purely a notation. We will later see in Chapter 3 that it has an intimate connection to the complex exponential function. For now, we motivate this maybe strange seeming definition by collecting some of its properties:

## Proposition 1.3. 

For any \( x_1, x_2 \in \mathbb{R} \):

1. \( e^{(x_1 + x_2 i)} = e^{x_1} e^{x_2 i} \)
2. \( e^{0} = 1 \)
3. \( \frac{d}{dx} e^{x} = e^{x} \)
4. \( |e^{x}| = 1 \)
5. \( e^{i\pi} = -1 \)

You are encouraged to prove them (Exercise 1.16); again, we give a sample.

## Proof of \( e^{x} \).

By definition of \( e^{x} \):

\[
\frac{d}{dx} e^{(x + i)} = -\sin{(x)} + \cos{(x)} = i ( \cos{(x)} + i\sin{(x)}) = i e^{(x+i)}
\]
####################
File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf

Page: 15

Context: # Geometric Properties

Proposition 1.3 implies that \( z^{n} = 1 \) for any integers \( n \) and \( r > 0 \). Thus, numbers of the form \( z^{n} \) with \( z \in \mathbb{Q} \) play a pivotal role in solving equations of the form \( z^{n} = 1 \)—plenty of reason to give them a special name.

**Definition.** A root of unity is of the form \( z^{n} \) for some integers \( n \) and \( r > 0 \). A root of unity \( z \) is a complex number \( z \) such that \( z^{n} = 1 \) for some positive integer \( n \). In this case, we call \( z \) an \( n \)th root of unity. If \( n \) is the smallest positive integer with the property \( z^{n} = 1 \), then \( z \) is a primitive \( n \)th root of unity.

**Example 1.4.** The 4th roots of unity are \( 1 \) and \( i \) and \( e^{i\frac{\pi}{2}} \). The latter two are primitive 4th roots of unity. 

With our new notation, the sentence "the complex number \( x + iy \) has absolute value \( r \) and argument \( \theta \) now becomes the identity

\[
x + iy = r e^{i\theta}.
\]

The left-hand side is often called the **rectangular form**, the right-hand side the **polar form** of this complex number.

We now have five different ways of thinking about a complex number: the formal definition, in rectangular form, in polar form, and geometrically, using Cartesian coordinates or polar coordinates. Each of these five ways is useful in different situations, and translating between them is an essential ingredient in complex analysis. The five ways and their corresponding notation are listed in Figure 1.5. This list is not exhaustive; see, e.g., Exercise 1.21.

## 1.3 Geometric Properties

From the chain of basic inequalities 

\[
-\sqrt{x^{2} + y^{2}} \leq -|z| \leq \sqrt{x^{2} + y^{2}} \leq |z| \leq \sqrt{x^{2} + y^{2}},
\]

(or, alternatively, by arguing with basic geometric properties of triangles), we obtain the inequalities

\[
-|z| \leq \text{Re}(z) \leq |z| 
\]
and 
\[
-|z| \leq \text{Im}(z) \leq |z|. \tag{1.16}
\]

The square of the absolute value has the nice property 

\[
|x + iy|^{2} = x^{2} + y^{2} = (x + iy)(x - iy).
\]
####################
File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf

Page: 16

Context: # COMPLEX NUMBERS

|            | Algebraic        | Geometric       | Exponential   |
|------------|------------------|------------------|---------------|
| Formal     | \(x + iy\)       | Cartesian        | \(re^{i\theta}\) |
|            | Rectangular      | \(x\)            | Polar         |
|            |                   | \(y\)            |               |

**Figure 1.5:** Five ways of thinking about a complex number.

This is one of many reasons to give the process of passing from \(x + iy\) to \(x - iy\) a special name.

## Definition
The number \(x - iy\) is the (complex) conjugate of \(x + iy\). We denote the conjugate by

\[
x + iy = x - iy.
\]

Geometrically, conjugating \(z\) means reflecting the vector corresponding to \(z\) with respect to the real axis. The following collects some basic properties of the conjugate.

## Proposition 1.5
For any \(z_1, z_2, z_3 \in \mathbb{C}\):

1. \(z_1 \overline{z_2} = \overline{z_1 z_2}\)
2. \(\overline{z_1 z_2} = \overline{z_2} \overline{z_1}\)
3. \(\overline{\left(\frac{z_1}{z_2}\right)} = \frac{\overline{z_2}}{\overline{z_1}}\)
4. \(\overline{z} = z\)
5. \(|z| = |z|\)

The proofs of these properties are easy (Exercise 1.22); once more we give a sample.

### Proof of (b)
Let \(z_1 = x_1 + iy_1\) and \(z_2 = x_2 + iy_2\). Then

\[
z_1 z_2 = (x_1 + iy_1)(x_2 + iy_2) = (x_1x_2 - y_1y_2) + i(x_1y_2 + y_1x_2) = \overline{z_1} \overline{z_2}.
\]

Image Analysis: 

### 1. Localization and Attribution

- **Page**: This is a page from a document or book, numbered at the top left as "10."
- **Image 1**: An illustrated table at the top of the page labeled as "Figure 1.5." 
- **Text Block**: Multiple paragraphs of text underneath the table, including definitions, propositions, and proofs.

### 2. Object Detection and Classification
- **Image 1:**
  - **Objects**: A table with three rows and three columns.
  - **Categories**: Types of complex number representations.
  - **Key Features**:
    - **Row 1**: "Formal" corresponds to (x, y).
    - **Row 2**: "Algebraic" includes rectangular (x + iy) and exponential (re^iθ).
    - **Row 3**: "Geometric" includes cartesian and polar representations.

### 3. Scene and Activity Analysis
- **Scene**: 
  - The scene depicts a page from a textbook or academic document.
  - The key activity is educational content delivery, specifically discussing complex numbers.

### 4. Text Analysis
- **Detected Text**:
  - Title: "COMPLEX NUMBERS."
  - Sections:
    - "Figure 1.5: Five ways of thinking..."
    - Definitions and propositions related to complex numbers.
    - Proof of a property.
- **Analysis**:
  - The page discusses different representations of complex numbers, highlights their properties, and presents relevant mathematical proofs.

### 5. Diagram and Chart Analysis
- **Image 1:**
  - **Content**: The table categorizes the representations of complex numbers.
  - **Axes/Scales**: Not applicable.
  - **Legends**: Explanatory text within the cells.
  - **Key Insights**: 
    - Complex numbers can be represented in multiple forms: formal, algebraic, geometric.
    - The table is a clear and concise way to present these different perspectives.

### 6. Product Analysis
- **Not applicable**: No products were depicted in the image.

### 7. Anomaly Detection
- **Not applicable**: No anomalies in the visual content.

### 8. Color Analysis
- **Dominant Colors**: Black text on a white background.
- **Impact**: Standard colors for printed or digital educational content, aiding readability and focus.

### 9. Perspective and Composition
- **Perspective**: Straight-on view, typical for document pages.
- **Composition**: 
  - The table (Image 1) is at the top, drawing initial attention.
  - Subsequent text follows a structured format, typical of academic documents.

### 10. Contextual Significance
- **Contribution**:
  - The image and text provide fundamental knowledge about complex numbers.
  - It enriches the educational material by categorizing and defining complex number representations.

### 11. Metadata Analysis
- **Not available**: No metadata provided from the image.

### 12. Graph and Trend Analysis
- **Not applicable**: No graphs present.

### 13. Graph Numbers
- **Not applicable**: No graphs present.

### Additional Aspects
- **Process Flows and Descriptions**:
  - A defined process for understanding the conjugate of a complex number.
  - Steps to prove a proposition about complex numbers.
- **Type Designations**:
  - Designates formal, algebraic, and geometric types of complex number representations.
- **Trend and Interpretation**:
  - Emphasizes the versatility and various intuitions behind complex number representations.
- **Tables**:
  - The table encapsulates the five types of complex number representations succinctly.

This comprehensive examination focuses on the detailed aspects and provides a thorough understanding of the educational content on the described page.
####################
File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf

Page: 17

Context: # GEOMETRIC PROPERTIES

We note that \( f(z) \) yields a neat formula for the inverse of a nonzero complex number, which is implicit already in (1.14):

\[
z^{-1} = \frac{1}{|z|^2} \overline{z}.
\]

A famous geometric inequality (which holds, more generally, for vectors in \( \mathbb{R}^n \)) goes as follows.

## Proposition 1.6 (Triangle Inequality)

For any \( z_1, z_2 \in \mathbb{C} \) we have \( |z_1 + z_2| \leq |z_1| + |z_2| \).

By drawing a picture in the complex plane, you should be able to come up with a geometric proof of the triangle inequality. Here we proceed algebraically:

### Proof

We make extensive use of Proposition 1.5:

\[
|z_1 + z_2|^2 = |(z_1 + z_2)(\overline{z_1 + z_2})| = z_1\overline{z_1} + z_1\overline{z_2} + z_2\overline{z_1} + z_2\overline{z_2} = |z_1|^2 + z_1\overline{z_2} + z_2\overline{z_1} + |z_2|^2
\]

\[
\leq |z_1|^2 + 2\Re(z_1\overline{z_2}) + |z_2|^2
\]

\[
= |z_1|^2 + |z_2|^2 + 2|z_1||z_2| \cdot \frac{\Re(z_1\overline{z_2})}{|z_1||z_2|} \leq (|z_1| + |z_2|)^2,
\]

where the inequality follows from (1.16). Taking square roots on the left- and right-hand sides proves our claim. 

For future reference we list several useful variants of the triangle inequality:

## Corollary 1.7

For \( z_1, z_2, \ldots, z_n \in \mathbb{C} \), we have the following relations:

(a) The triangle inequality: \( |z_1 + z_2| \leq |z_1| + |z_2|. \)

(b) The reverse triangle inequality: \( |z_1 + z_2| \geq ||z_1| - |z_2||. \)

(c) The triangle inequality for sums:

\[
\left| \sum_{k=1}^{n} z_k \right| \leq \sum_{k=1}^{n} |z_k|.
\]

Inequality (a) is just a rewrite of the original triangle inequality, using the fact that \( |z| = |z| \) and follows by induction. The proof of the reverse triangle inequality (b) is left as Exercise 1.25.
####################
File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf

Page: 18

Context: # 1.4 Elementary Topology of the Plane

In Section 1.2 we saw that the complex numbers \( C \), which were initially defined algebraically, can be identified with the points in the Euclidean plane \( \mathbb{R}^2 \). In this section we collect some definitions and results concerning the topology of the plane.

\[
\begin{array}{c}
    y \\
    \text{C}(2+i,2) \\
    \hspace{-30pt} \text{D}(-2, 1) \\
    \hspace{80pt} 1 \\
    \hspace{-30pt} x
\end{array}
\]

**Figure 1.6:** Sample circle and disk.

In Proposition 1.2, we interpreted \( |z - w| \) as the distance between the complex numbers \( z \) and \( w \), viewed as points in the plane. So if we fix a complex number \( a \) and a positive real number \( r \), then all \( z \in C \) satisfying \( |z - a| = r \) form the set of points at distance \( r \) from \( a \); this set is the circle with center \( a \) and radius \( r \), which we denote by 

\[
C(a, r) = \{ z \in C : |z - a| = r \}
\]

The inside of this circle is called the open disk with center \( a \) and radius \( r \); we use the notation 

\[
D(a, r) = \{ z \in C : |z - a| < r \}
\]

Note that \( D(a, r) \) does not include the points on \( C(a, r) \). Figure 1.6 illustrates these definitions.

Next we need some terminology for talking about subsets of \( C \).

**Definition.** Suppose \( G \) is a subset of \( C \).
####################
File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf

Page: 19

Context: # ELEMENTARY TOPOLOGY OF THE PLANE

## 1. Types of Points

(a) A point \( a \in G \) is an **interior point** of \( G \) if some open disk centered at \( a \) is a subset of \( G \).

(b) A point \( b \in E \) is a **boundary point** of \( G \) if every open disk centered at \( b \) contains a point in \( G \) and also a point that is not in \( G \).

(c) A point \( c \in C \) is an **accumulation point** of \( G \) if every open disk centered at \( c \) contains a point of \( G \) different from \( c \).

(d) A point \( d \in G \) is an **isolated point** of \( G \) if some open disk centered at \( d \) contains no point of \( G \) other than \( d \).

The idea is that if you don't move too far from an interior point of \( G \) then you remain in \( G \); but a boundary point you can make an arbitrarily small move and get to a point inside \( G \) and you can also make an arbitrarily small move and get to a point outside \( G \).

## Definition
A set is **open** if all its points are interior points. A set is **closed** if it contains all its boundary points.

## Example 1.18
For \( r > 0 \) and \( a \in \mathbb{C} \), the sets \( \{ z \in \mathbb{C} : |z - a| < r \} \) and \( \{ z \in \mathbb{C} : |z - a| = r \} \) are open. The closed disk 

\[
\overline{D[a, r]} := \{ z \in \mathbb{C} : |z - a| \leq r \}
\]

is an example of a closed set.

A given set might be neither open nor closed. The complex plane \( \mathbb{C} \) and the empty set \( \emptyset \) are the only sets that are both open and closed.

## Definition
The **boundary** \( \partial G \) of a set \( G \) is the set of all boundary points of \( G \). The interior of \( G \) is the set of all interior points of \( G \). The closure of \( G \) is the set \( \overline{G} = G \cup \partial G \).

## Example 1.19
The closure of the open disk \( D[a, r] \) is \( \overline{D[a, r]} \). The boundary of \( D[a, r] \) is the circle \( \partial D[a, r] \).

## Definition
The set \( G \) is **bounded** if \( G \subset D[0, R] \) for some \( R \).

One notion that is somewhat subtle in the complex domain is the idea of **connectedness**. Intuitively, a set is connected if it is "in one piece." In a set \( R \), a set is connected if and only if it is an interval, so there is little reason to discuss the matter. However, in the plane there is a vast variety of connected subsets.
####################
File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf

Page: 20

Context: # Complex Numbers

![Figure 1.7: The intervals [0,1) and (1,2) are separated.](image_link)

## Definition
Two sets \( X, Y \subset \mathbb{C} \) are separated if there are disjoint open sets \( A, B \subset C \) so that \( X \subset A \) and \( Y \subset B \). A set \( G \subset \mathbb{C} \) is connected if it is impossible to find two separated nonempty sets whose union is \( G \). A region is a connected open set.

The idea of separation is that the two open sets \( A \) and \( B \) ensure that \( X \) and \( Y \) cannot just "stick together." It is usually easy to check that a set is not connected. On the other hand, it is hard to use the above definition to show that a set is connected, since we have to rule out any possible separation.

## Example 1.10
The intervals \( X = [0,1) \) and \( Y = (1,2) \) on the real axis are separated: There are infinitely many choices for \( A \) and \( B \) that work; one choice is \( A = D[0,1) \) and \( B = D(1,2) \), depicted in Figure 1.7. Hence \( X \cup Y = [0,1) \cup (1,2) \) is not connected.

One type of connected set that we will use frequently is a path.

## Definition
A path (or curve) in \( \mathbb{C} \) is a continuous function \( \gamma : [a,b] \to \mathbb{C} \), where \([a,b]\) is a closed interval in \( \mathbb{R} \). We may think of \( \gamma \) as a parametrization of the image that is painted by the path and will often write this parametrization as \( \gamma(t) \), for \( a \leq t \leq b \). The path is smooth if it is differentiable and the derivative \( \gamma' \) is continuous and nonzero.
####################
File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf

Page: 21

Context: # ELEMENTARY TOPOLOGY OF THE PLANE

This definition uses the calculus notions of continuity and differentiability; that is, \( \gamma: [a, b] \to C \) being continuous means that for all \( t_0 \in [a, b] \)

\[
\lim_{t \to t_0} \gamma(t) = \gamma(t_0)
\]

and the derivative of \( \gamma \) at \( t_0 \) is defined by

\[
\gamma'(t_0) = \lim_{t \to t_0} \frac{\gamma(t) - \gamma(t_0)}{t - t_0}
\]

\[
\gamma_1(t) = -2 + 2e^{it}, \quad 0 \leq t \leq 2\pi
\]

\[
\gamma_2(t) = 
\begin{cases} 
3 + i(t - 2) & \text{if } 0 \leq t \leq 3 \\ 
6 - i + i(t - 1) & \text{if } 3 < t \leq 5 
\end{cases}
\]

Figure 1.8: Two paths and their parameterizations: \( \gamma_1 \) is smooth and \( \gamma_2 \) is continuous and piecewise smooth.

Figure 1.8 shows two examples. We remark that each path comes with an orientation, i.e., a sense of direction. For example, the path \( \gamma_1 \) in Figure 1.8 is different from 

\[
\gamma_1(t) = -2 + 2e^{it}, \quad 0 \leq t \leq \frac{3\pi}{2}
\]

even though both \( \gamma_1 \) and \( \gamma_3 \) yield the same picture; \( \gamma_1 \) features a counter-clockwise orientation, where as that of \( \gamma_2 \) is clockwise.

It is a customary and practical abuse of notation to use the same letter for the path and its parameterization. We emphasize that a path must have a parameterization, and that the parameterization must be defined and continuous on a closed and bounded interval \([a, b]\). Since topologically we may identify \( C \) with \( \mathbb{R}^2 \), a path can be specified.
####################
File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf

Page: 22

Context: ```
# Complex Numbers

by giving two continuous real-valued functions of a real variable, \( x(t) \) and \( y(t) \), and setting \( y(t) = x(t) + iy(t) \).

## Definition

The path \( \gamma: [a, b] \to C \) is simple if \( \gamma(t_1) = \gamma(t_2) \) is one-to-one, with the possible exception that \( \gamma(a) = \gamma(b) \) in plain English: the path does not cross itself. A path \( \gamma: [a, b] \to C \) is closed if \( \gamma(a) = \gamma(b) \).

## Example 1.11

The unit circle \( C[0,1] \), parametrized, e.g., by \( \gamma(t) = e^{it} \), \( 0 \leq t < 2\pi \), is a simple closed path.

As seems intuitively clear, any path is connected; however, a proof of this fact requires a bit more preparation in topology. The same goes for the following result, which gives a useful property of open connected sets.

## Theorem 1.12

If any two points in \( G \subset C \) can be connected by a path in \( G \), then \( G \) is connected. Conversely, if \( G \subset C \) is open and connected, then any two points of \( G \) can be connected by a path in \( G \); in fact, we can connect any two points of \( G \) by a chain of horizontal and vertical segments lying in \( G \).

Here a chain of segments in \( G \) means the following: there are points \( z_{0}, z_{1}, \ldots, z_{n} \) such that \( z_{0} \) and \( z_{n} \) are the endpoints of a horizontal or vertical segment in \( G \) for all \( k = 0, 1, \ldots, n-1 \) (it is not hard to parametrize such a chain, so it determines a path).

## Example 1.13

Consider the open unit disk \( D[0, 1) \). Any two points in \( D[0, 1) \) can be connected by a chain of at most two segments in \( D[0, 1) \) and so \( D[0, 1) \) is connected. Now let \( G = (D[0, 1) \setminus \{0\}) \). This is the punctured disk obtained by removing the center from \( D[0, 1) \). Then \( G \) is open and it is connected, but now you may need more than two segments to connect points. For example, you need three segments to connect \( z_{0} \to z_{1} \to z_{2} \) since we cannot go through \( 0 \). 

We remark that the second part of Theorem 1.12 is not generally true if \( G \) is not open. For example, circles are connected but there is no way to connect two distinct points of a circle by a chain of segments that are subsets of the circle. A more extreme example, discussed in topology texts, is the "topologist's sine curve," which is a connected set \( S \) that contains points that cannot be connected by a path of any sort within \( S \).
```
####################
File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf

Page: 23

Context: # Exercises

## 1.1. 
Let \( z = 1 + 2i \) and \( w = 2 - i \). Compute the following:

(a) \( z + 3w \)  
(b) \( w - z \)  
(c) \( z^2 \)  
(d) \( \text{Re}(wz + w) \)  
(e) \( z^2 + z + i \)  

## 1.2. 
Find the real and imaginary parts of each of the following:

(a) \( \frac{e^{ia}}{1 + i} \) for any \( a \in \mathbb{R} \)  
(b) \( 1 - i \)  
(c) \( \left( \frac{1}{\sqrt{2}} \right)^3 \)  
(d) \( i^n \) for any \( n \in \mathbb{Z} \)  

## 1.3. 
Find the absolute value and conjugate of each of the following:

(a) \( -2 + i \)  
(b) \( (2 + i\sqrt{3}) \)  
(c) \( \frac{3}{\sqrt{5}} - \sqrt{5}i \)  
(d) \( (1 + i)^{6} \)  

## 1.4. 
Write in polar form:

(a) \( 2i \)  
(b) \( 1 + i \)  
(c) \( 3 + \sqrt{3}i \)  
(d) \( -i \)  
(e) \( (2 - i)^2 \)  
(f) \( [3 - 4i] \)  
(g) \( \sqrt{5} - i \)  
(h) \( \left( \frac{1}{4} \right)^{4} \)  

## 1.5. 
Write in rectangular form:

(a) \( \sqrt{2}e^{i\frac{\pi}{4}} \)  
(b) \( 3e^{i\frac{\pi}{6}} \)  
(c) \( -r^{2}e^{i2\theta} \)  
(d) \( 2e^{i\frac{\pi}{3}} \)  

## 1.6. 
Write in both polar and rectangular form:

(a) \( e^{i\frac{\pi}{6}} \)  
(b) \( \frac{1}{2} e^{i\theta} \)
####################
File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf

Page: 24

Context: # COMPLEX NUMBERS

## 1.7
Show that the quadratic formula works. That is, for \( a, b, c \in \mathbb{R} \) with \( a \neq 0 \), prove that the roots of the equation \( ax^2 + bx + c = 0 \) are 

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Here we define \( \sqrt{b^2 - 4ac} = i \sqrt{-(b^2 - 4ac)} \) if the discriminant \( b^2 - 4ac \) is negative.

## 1.8
Use the quadratic formula to solve the following equations:

- (a) \( x^2 + 25 = 0 \)
- (b) \( 2x^2 + 2x + 5 = 0 \)
- (c) \( 5x^2 + 4x + 1 = 0 \)
- (d) \( x^2 - 2x = 0 \)
- (e) \( x^2 = 2x \)

## 1.9
Find all solutions of the equation \( x^2 + 2x + (1-i) = 0 \).

## 1.10
Fix \( a \in \mathbb{C} \) and \( b \in \mathbb{R} \). Show that the equation \( |z|^2 + \text{Re}(z) + b = 0 \) has a solution if and only if \( |a|^2 \geq 4b \). When solutions exist, show the solution set is a circle.

## 1.11
Find all solutions to the following equations:

- (a) \( z^6 = 1 \)
- (b) \( z^6 = -16 \)
- (c) \( z^6 = e^{-3} \)
- (d) \( z^6 - z^2 = 0 \)

## 1.12
Show that \( |z| = 1 \) if and only if \( z = \bar{z} \).

## 1.13
Show that:

- (a) \( z \) is a real number if and only if \( z = \bar{z} \).
- (b) \( z \) is either real or purely imaginary if and only if \( |z|^2 = z^2 \).

## 1.14
Review Proposition 1.1.

## 1.15
Show that if \( z_1 z_2 = 0 \) then \( z_1 = 0 \) or \( z_2 = 0 \).
####################
File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf

Page: 25

Context: # Elementary Topology of the Plane

## 1.16
Prove Proposition 1.3.

## 1.17
Fix a positive integer \( n \). Prove that the solutions to the equation \( z^n = 1 \) are precisely \( z = e^{2 \pi i k/n} \) where \( k \in \mathbb{Z} \). (Hint: To show that every solution of \( z^n = 1 \) is of this form, first prove that it must be of the form \( z = e^{i \theta} \) for some \( \theta \in \mathbb{R} \), then write \( z = m + ib \) for some integer \( m \) and some real number \( 0 < b < 1 \), and then argue that \( b \) has to be zero.)

## 1.18
Show that 

\[
z^2 - 1 = (z - 1)\left(z^2 + 2 \cos \frac{\pi}{3}(z^2 - 2z \cos \frac{\pi}{3} + 1\right)
\]

and deduce from this closed formula for \( \cos \frac{\pi}{3} \) and \( \cos \frac{2\pi}{3} \).

## 1.19
Fix a positive integer \( n \) and a complex number \( w \). Find all solutions to \( z^n = w \). (Hint: Write \( w \) in terms of polar coordinates.)

## 1.20
Use Proposition 1.3 to derive the triple angle formulas:

(a) \(\cos(3\phi) = 4\cos^3\phi - 3\cos\phi \sin^2\phi\)

(b) \(\sin(3\phi) = 3\cos^2\phi \sin\phi - \sin^3\phi\)

## 1.21
Given \( x, y \in \mathbb{R} \), define the matrix \( M(x, y) = \begin{bmatrix} x & -y \\ y & x \end{bmatrix} \). Show that 

\[
M(x, y) + M(a, b) = M(x + a, y + b)
\]

and 

\[
M(x, y)M(a, b) = M(xa - yb, xb + ya)
\]

(This means that the set \( \{ M(x, y) : x, y \in \mathbb{R} \} \) equipped with the usual addition and multiplication of matrices, behaves exactly like \( C = \{ (x, y) : x, y \in \mathbb{R} \} \).)

## 1.22
Prove Proposition 1.5.
####################
File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf

Page: 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: 38

Context: # 2 Differentiation

Now define \( z_0 = g(x_0) \) and set

\[
\phi(w) = 
\begin{cases}
\frac{f(w) - f(w_0)}{w - w_0} & \text{if } w \neq w_0 \\
f'(w_0) & \text{if } w = w_0 
\end{cases}
\]

This is continuous at \( w_0 \) and \(\lim_{w \to w_0} \phi(w) = f'(w_0)\), so we can apply Proposition 2.6:

\[
g'(x_0) = \lim_{h \to 0} \frac{1}{\phi(g(x_0 + h))} \cdot \frac{1}{f'(w_0)}
\]

## 2.3 The Cauchy–Riemann Equations

When considering a real-valued function \( f: \mathbb{R}^2 \to \mathbb{R} \) of two variables, there is no notion of the derivative of a function. For such a function, we instead only have partial derivatives \( \frac{\partial f}{\partial x}(x_0, y_0) \) and \( \frac{\partial f}{\partial y}(x_0, y_0) \) (and also directional derivatives) which depend on the way in which we approach a point \( (x_0, y_0) \in \mathbb{R}^2 \). For a complex-valued function \( f(z) \), we now have a new concept of the derivative \( f'(z_0) \), which definition cannot depend on the way in which we approach a point \( z_0 = (x_0, y_0) \in \mathbb{C} \). It is logical, then, that there should be a relationship between the complex derivative \( f'(z_0) \) and the partial derivatives

\[
\frac{\partial f}{\partial x}(z_0) = \lim_{h \to 0} \frac{f(x_0 + h) - f(x_0, y_0)}{h}
\]

and

\[
\frac{\partial f}{\partial y}(z_0) = \lim_{k \to 0} \frac{f(x_0, y_0 + k) - f(x_0, y_0)}{k}
\]

(this definition is exactly as in the real-valued case). This relationship between the complex derivative and partial derivatives is very strong, and it is a powerful computational tool. It is described by the Cauchy–Riemann equations, named after Augustin Louis Cauchy (1789–1857) and Georg Friedrich Bernhard Riemann (1826–1866), even though the equations appeared already in the works of Jean le Rond d'Alembert (1717–1783) and Leonhard Euler (1707–1783).
####################
File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf

Page: 41

Context: In the second case, \( \Delta z = i \Delta y \) and 

\[
f'(z_0) = \lim_{y_0 \to y_0} \frac{f(x_0 + i y) - f(x_0)}{i y} 
= \lim_{y_0 \to 0} \frac{f(x_0 + i y) - f(x_0)}{y}
\]

Thus, we have shown that \( f'(z_0) = f_x(z_0) - i f_y(z_0) \).

(b) Suppose the Cauchy-Riemann equation (2.2) holds and the partial derivatives \( f_x \) and \( f_y \) are continuous in an open disk centered at \( z_0 \). Our goal is to prove that \( f'(z_0) = f_x(z_0) + i f_y(z_0) \). By (2.2),

\[
f(z_0) = x_0 + i y_0 \quad f(z_0 + \Delta z) = f(z_0 + \Delta x + i \Delta y) 
\]

On the other hand, we can rewrite the difference quotient for \( f'(z_0) \) as

\[
f(z_0 + \Delta z) - f(z_0) = \frac{f(z_0 + \Delta z) - f(z_0)}{\Delta z} 
= \frac{f(z_0 + \Delta x + i \Delta y) - f(z_0)}{\Delta z} 
= \frac{f(z_0 + \Delta x) - f(z_0)}{\Delta z} + \frac{f(z_0 + \Delta y) - f(z_0)}{\Delta z}
\]

Thus, 

\[
\lim_{\Delta z \to 0} \frac{f(z_0 + \Delta z) - f(z_0)}{\Delta z} = \lim_{\Delta y \to 0} \frac{f(z_0 + i \Delta y) - f(z_0)}{\Delta y}
= \lim_{\Delta x \to 0} \left( \frac{f(z_0 + \Delta x + i \Delta y) - f(z_0)}{\Delta z} \right).
\]

We claim that both limits on the right-hand side are 0, so we have achieved our set goal. The fractions \( \frac{\Delta y}{\Delta z} \) and \( \frac{\Delta x}{\Delta z} \) are bounded in absolute value by 1, so we just need to see that the differences in parentheses are 0. The second term on the right-hand side of (2.5) has a limit of 0 since, by definition,

\[
f(z) = \lim_{\Delta z \to 0} \frac{f(z + \Delta z) - f(z)}{\Delta z}
\]

and taking the limit here as \( \Delta z \to 0 \) is the same as taking the limit as \( \Delta x \to 0 \).
####################
File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf

Page: 42

Context: ```markdown
36
# Differentiation

We cannot do something equivalent for the first term in (2.5), since now both \( \Delta x \) and \( \Delta y \) are involved, and both change as \( \Delta x \to 0 \). Instead, we apply the Mean-Value Theorem A.2 for real functions\(^1\) to the real and imaginary parts \( u(x, y) \) and \( v(x, y) \) of \( f(z) \). Theorem A.2 gives real numbers \( \xi \) and \( \eta \) such that

\[
\frac{u(x_0 + \Delta x, y_0 + \Delta y) - u(x_0, y_0)}{\Delta y} = u_y(x_0, y_0 + \eta)
\]

and

\[
\frac{v(x_0 + \Delta x, y_0 + \Delta y) - v(x_0, y_0)}{\Delta y} = v_y(x_0, y_0 + \xi).
\]

Thus,

\[
\frac{f(x_0 + \Delta x + i \Delta y) - f(x_0 + i y)}{\Delta y} = \frac{u(x_0 + \Delta x, y_0 + \Delta y) - u(x_0, y_0)}{\Delta y} - iv_x(x_0, y_0)
\]

\[
= \left( \frac{u(x_0 + \Delta x, y_0 + \Delta y) - u(x_0, y_0)}{\Delta y} \right) - i v_y(x_0, y_0)
+ \left( \frac{u(x_0 + \Delta x, y_0 + \Delta y) - u(x_0, y_0)}{\Delta y} - iv_y(x_0, y_0) \right) = u_y(x_0, y_0) - iv_y(x_0, y_0). \tag{2.6}
\]

Because \( u \) and \( v \) are continuous at \( (x_0, y_0) \),

\[
\lim_{\Delta y \to 0} u(x_0 + \Delta x, y_0 + \Delta y) = u(x_0, y_0)
\]

and

\[
\lim_{\Delta y \to 0} v(x_0 + \Delta x, y_0 + \Delta y) = v(x_0, y_0),
\]

and so (2.6) goes to \( \Delta z \to 0 \), which we set out to prove. 

## 2.4 Constant Functions

As a simple application of the definition of the derivative of a complex function, we consider functions that have a derivative of 0. In a typical calculus course, one of the first applications of the Mean-Value Theorem for real-valued functions (Theorem A.1) is to constant functions.
```
####################
File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf

Page: 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: 46

Context: ```
## 2.11. 
Show that the two definitions of continuity in Section 2.1 are equivalent. Consider separately the cases where \(x_0\) is an accumulation point of \(G\) and where \(x_0\) is an isolated point of \(G\).

## 2.12. 
Consider the function \(f : C \setminus \{0\} \to G\) given by \(f(z) = \frac{1}{z}\). Apply the definition of the derivative to give a direct proof that \(f'(z) = -\frac{1}{z^2}\).

## 2.13. 
Prove Proposition 2.6.

## 2.14. 
Prove Proposition 2.10.

## 2.15. 
Find the derivative of the function \(T(z) = \frac{a z^2 + b z + c}{e^{z}}\), where \(a, b, c \in C\) with \(ad - bc = 0\). When is \(T'(z) = 0\)?

## 2.16. 
Prove that if \(f(z)\) is given by a polynomial in \(z\) then \(f\) is entire. What can you say if \(f(z)\) is given by a polynomial in \(z = x + iy\) and \(y = \Im z\)?

## 2.17. 
Prove or find a counterexample. If \(x\) and \(y\) are real valued and continuous, then \(f(z) = u(x,y) + iv(x,y)\) is continuous; if \(u\) and \(v\) are (real) differentiable then \(f\) is (complex) differentiable.

## 2.18. 
Where are the following functions differentiable? Where are they holomorphic? Determine their derivatives at points where they are differentiable.

1. \(f(z) = e^{z} \cdot e^{c+iy}\)
2. \(f(z) = 2x + iy\)
3. \(f(z) = x^3 + iy\)
4. \(f(z) = e^{c-iy}\)
5. \(f(z) = \cosh x - i \sin y\)
6. \(f(z) = \Im z\)
7. \(f(z) = |z|^2 = x^2 + y^2\)
8. \(f(z) = z^2\)
9. \(f(z) = \Im z\)
10. \(f(z) = \frac{4}{Re z} - i(Re z)^2\)
```
####################
File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf

Page: 48

Context: ```
42
# Differentiation

## 2.27
Consider the general real homogeneous quadratic function \( u(x,y) = ax^2 + bxy + cy^2 \), where \( a, b, \) and \( c \) are real constants.

(a) Show that \( u \) is harmonic if and only if \( a + c = 0 \).

(b) If \( u \) is harmonic then show that \( u \) is the real part of a function of the form \( f(z) = A z^2 \) for some \( A \in \mathbb{C} \). Give a formula for \( A \) in terms of \( a \) and \( c \).

## 2.28
Re-prove Proposition 2.10 by using the formula for \( f' \) given in Theorem 2.13.

## 2.29
Prove that, if \( G \subset \mathbb{C} \) is a region and \( f : G \to \mathbb{C} \) is a complex-valued function with \( f''(z) \) defined and equal to 0 for all \( z \in G \), then \( f(z) = az + b \) for some \( a, b \in \mathbb{C} \). (Hint: Use Theorem 2.17 to show that \( f'(z) = a \), and then use Theorem 2.17 again for the function \( f(z) - az \).)
```
####################
File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf

Page: 49

Context: # Chapter 3

## Examples of Functions

_To many, mathematics is a collection of theorems. For us, mathematics is a collection of examples; a theorem is a statement about a collection of examples and the purpose of proving theorems is to classify and explain the examples..._  
— John B. Conway

In this chapter we develop a toolkit of complex functions. Our ingredients are familiar from calculus: linear functions, exponentials and logarithms, and trigonometric functions. Yet, when we move these functions into the complex world, they take on—at times drastically different—new features.

### 3.1 Möbius Transformations

The first class of functions that we will discuss in some detail are built from linear polynomials.

**Definition.** A linear fractional transformation is a function of the form 

\[ f(z) = \frac{az + b}{cz + d} \]

where \(a, b, c, d \in \mathbb{C}\) and \(ad - bc \neq 0\). Then \(f\) is called a Möbius transformation.

Exercise 2.16 said that any polynomial \(p\) is an entire function, and so the linear fractional transformation \(f(z) = \frac{az + b}{cz + d}\) is holomorphic on \(\mathbb{C} \setminus \{-\frac{d}{c}\}\), unless \(c = 0\) (in which case \(c\) is entire). If \(c \neq 0\) then \(f(z) = \frac{az + b}{cz + d}\) implies \(ad - bc \neq 0\), which means that a Möbius transformation \(f(z) = \frac{az + b}{cz + d}\) will never take on the value \(-\frac{b}{d}\). Our first proposition in this chapter says that with these small observations about the domain and image of a Möbius transformation, we obtain a class of bijections, which are quite special among complex functions.
####################
File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf

Page: 52

Context: # Examples of Functions

For real numbers \( a, b, \gamma, y, r \) and \( s \) that satisfy \( \beta^2 + \gamma^2 > 4\alpha \) (Exercise 3.3). The form (3.1) is more convenient for us, because it includes the possibility that the equation describes a line (precisely when \( a = 0 \)).

Suppose \( x + iy \) satisfies (3.1); we need to prove that \( x + iy = \frac{x - iy}{x^2 + y^2} \) satisfies a similar equation.

\[
u + iv = \frac{x - iy}{x^2 + y^2}.
\]

We can rewrite (3.1) as:

\[
0 = \alpha + \frac{y}{x^2 + y^2} + \frac{8}{x^2 + y^2}.
\]

This leads to:

\[
0 = \alpha + \beta u + \gamma (a + b + r^2). \tag{3.2}
\]

But this equation, in conjunction with Exercise 3.3, says that \( x + iy \) lies on a circle or line. 

## 3.2 Infinity and the Cross Ratio

In the context of Möbius transformations, it is useful to introduce a formal way of saying that a function \( f \) "blows up" in absolute value, and this gives rise to a notion of infinity.

**Definition.** Suppose \( f : G \to \mathbb{C} \).

1. \( \lim_{x \to a} f(x) = \infty \) means that for every \( M > 0 \) we can find \( \delta > 0 \) so that, for all \( x \in G \) satisfying \( 0 < |x - a| < \delta \), we have \( |f(x)| > M \).

2. \( \lim_{x \to b} f(x) = L \) means that for every \( \epsilon > 0 \) we can find \( N > 0 \) so that, for all \( x \in G \) satisfying \( |x| > N \), we have \( |f(x) - L| < \epsilon \).

3. \( \lim_{x \to \infty} f(x) = \infty \) means that for every \( M > 0 \) we can find \( N > 0 \) so that, for all \( x \in G \) satisfying \( |x| > N \), we have \( |f(x)| > M \).

In the first definition we require that \( a \) be an accumulation point of \( G \), while in the second and third we require that \( b \) be an "extended accumulation point" of \( G \), in the sense that for every \( B > 0 \) there is some \( z \in G \) with \( |z| > B \).

As in Section 2.1, the limit, in any of these senses, is unique if it exists.
####################
File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf

Page: 54

Context: # EXAMPLES OF FUNCTIONS

is not defined, but if we take the limit of \( z + (-z) = 0 \) as \( z \to \infty \) we will get \( 0 \), even though the individual limits of \( z \) and \( -z \) are both \( \infty \).

Now we reconsider Möbius transformations with \( c \) in mind. For example, \( f(z) = \frac{az + b}{cz + d} \) is now defined for \( z = 0 \) and \( z = \infty \), with \( f(0) = \frac{b}{d} \) and \( f(\infty) = \frac{a}{c} \), so we might argue the proper domain for \( f(z) \) is actually \( \mathbb{C} \). Let’s consider the other basic types of Möbius transformations. A translation \( f(z) = z + b \) is now defined for \( z = \infty \), with \( f(\infty) = \infty \) and a dilation \( f(z) = az \) (with \( a \neq 0 \)) is also defined for \( z = \infty \), with \( f(\infty) = \infty \). Since every Möbius transformation can be expressed as a composition of translations, dilations, and inversions (Proposition 3.3), we see that every Möbius transformation may be interpreted as a transformation of \( \mathbb{C} \) onto \( \mathbb{C} \). This general case is summarized in the following extension of Proposition 3.1.

## Corollary 3.8
Suppose \( a, b, c, d \) are \( \neq 0 \) and consider \( f: \mathbb{C} \to \mathbb{C} \) defined through

\[
f(z) =
\begin{cases}
\frac{az + b}{cz + d} & \text{if } z \in \mathbb{C} \setminus \{-\frac{d}{c}\} \\
\infty & \text{if } z = -\frac{d}{c} \\
\end{cases}
\]

Then \( f \) is a bijection.

This corollary also holds for \( c = 0 \) if we define \( f(\infty) = 0 \).

## Example 3.9
Continuing Examples 3.2 and 3.5, consider once more the Möbius transformation \( f(z) = \frac{1}{z} \). With the definitions \( f(1) = 1 \) and \( f(\infty) = 0 \), we can extend \( f \) to a function \( \mathbb{C} \).

With \( 0 \) on our mind, we can also add some insight to Theorem 3.4. We recall that in Example 3.5, we proved that \( f(z) = \frac{1}{z} \) maps the unit circle to the real line. Essentially, the same proof shows that, more generally, any circle passing through \( \infty \) gets mapped to a line (see Exercise 3.4). The original domain of \( f \) was \( \mathbb{C} \setminus \{0\} \), so the point \( z = 1 \) must be excluded from these circles. However, by thinking of \( f \) as a function from \( \mathbb{C} \) to \( \mathbb{C} \), we can put \( z = 1 \) back into the picture, and \( f \) transforms the circle defined by \( |z| = 1 \) into a line plus a point. If we make this a definition, then Theorem 3.4 can be expressed as: any Möbius transformation \( f(z) \) transforms circles to lines.
####################
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: 65

Context: # Logarithms and Complex Exponentials

The complex logarithm is the first function we'll encounter that is of somewhat tricky nature. It is motivated as an inverse to the exponential function, that is, we're looking for a function \( \Log \) such that

\[
\exp(\Log(z)) = z = \Log(\exp z).
\]

But because \( \exp \) is not one-to-one, this is too much to hope for. In fact, given a function \( \Log \) that satisfies the first equation, the function \( f(z) = \Log(z) + 2\pi i k \) does as well, and so there cannot be an inverse of \( \exp \) (which would have to be unique). On the other hand, \( \exp \) becomes one-to-one if we restrict its domain, so there is hope for a logarithm if we're careful about its construction and about its domain.

## Definition

Given a region \( G \), any continuous function \( \Log: G \rightarrow \mathbb{C} \) that satisfies \( \exp(\Log(z)) = z \) is a branch of the logarithm (on \( G \)).

To make sure this definition is not vacuous, let's write, as usual, \( z = re^{i \theta} \), and suppose that \( \Log(z) = \log |z| + i \arg(z) \). Then for the first equation to hold, we need

\[
\exp(\Log(z)) = e^{\log |z| + i \arg(z)} = z,
\]

that is, \( e^{\log r} = r \). The latter equation is equivalent to \( r = e^{\log r} \) for some \( k \in \mathbb{Z} \), and denoting the natural logarithm of the positive real number \( r \) by \( \ln(r) \), the former equation is equivalent to \( u = \ln |z| \). A reasonable definition of a logarithm function \( \Log \) would hence be \( \Log(z) = \ln |z| + i \arg(z) \) where \( \arg(z) \) gives the argument for the complex number \( z \) according to some convention—here is an example:

## Definition

Let \( \Arg \) denote the unique argument \( \theta \) of \( z \) that lies in \( (-\pi, \pi) \) (the principal argument of \( z \)). Then the principal logarithm is the function \( \Log: \mathbb{C} \setminus \{0\} \rightarrow \mathbb{C} \) defined through

\[
\Log(z) = \ln |z| + i \Arg(z).
\]
####################
File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf

Page: 71

Context: # LOGARITHMS AND COMPLEX EXPONENTIALS

### 3.18
Find a Möbius transformation that maps the unit disk to \(\{x + iy \in \mathbb{C} : x + y > 0\}\).

### 3.19
The Jacobian of a transformation \(z = u(x, y)\), \(w = v(x, y)\) is the determinant of the matrix 

\[
\begin{bmatrix}
\frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} \\
\frac{\partial v}{\partial x} & \frac{\partial v}{\partial y}
\end{bmatrix}.
\]

Show that if \(f = u + iv\) is holomorphic then the Jacobian equals \(|f'(z)|^2\).

### 3.20
Find the fixed points in \(C\) of \(f(z) = \frac{az + b}{cz + d}\).

### 3.21
Find each Möbius transformation \(f\):

(a) \(f\) maps \(0 \mapsto 1\), \(1 \mapsto \infty\), \(\infty \mapsto 0\).

(b) \(f\) maps \(1 \mapsto -1\), \(-1 \mapsto i\), and \(i \mapsto -1\).

(c) \(f\) maps the \(x\)-axis to \(y = x\), the \(y\)-axis to \(y = -x\), and the unit circle to itself.

### 3.22
(a) Find a Möbius transformation that maps the unit circle to \(\{x + iy \in \mathbb{C} : x + y = 0\}\).

(b) Find two Möbius transformations that map the unit disk 

\[
\{z \in \mathbb{C} : |z| < 1\}
\]

to 

\(\{x + iy \in \mathbb{C} : x + y > 0\}\) and \(\{x + iy \in \mathbb{C} : x + y < 0\}\), respectively.

### 3.23
Given \(a \in \mathbb{R} \setminus \{0\}\), show that the image of the line \(y = a\) under inversion is the circle with center \(\frac{1}{a}\) and radius \(\frac{1}{|a|}\).

### 3.24
Suppose \(z_1, z_2\), and \(z_3\) are distinct points in \(\mathbb{C}\). Show that \(z\) is on the circle passing through \(z_1, z_2\), and \(z_3\) if and only if \([z, z_1, z_2, z_3]\) is real or \(\infty\).
####################
File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf

Page: 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: 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: 94

Context: # Integration

This all smells like good coffee, except ... we might be just dreaming. The argument may be intuitively clear, but intuition doesn’t prove anything. We'll look at it carefully, fill in the gaps, and then we'll see what we have proved.

First, we need a notion of the interior of a simple closed path. The fact that any such path \( \gamma \) divides the complex plane into two connected open sets \( \gamma \) (the bounded one of which we call the inside or interior of \( \gamma \)) is one of the first substantial theorems ever proved in topology, the Jordan Curve Theorem, due to Camille Jordan (1838–1922). In this book, we shall assume the validity of the Jordan Curve Theorem.

Second, we need to specify the orientation of \( \gamma \), since if the formula gives \( F(w) \) for one orientation then it will give \( -F(w) \) for the other orientation.

**Definition.** A piecewise smooth closed path \( \gamma \) is positively oriented if it is parameterized so that its inside is on the left as our parameterization traverses \( \gamma \). An example is a counter-clockwise oriented circle.

Third, if \( \gamma \) is positively oriented and \( D[\gamma, R] \) is a closed disk inside \( \gamma \), then we need a homotopy from \( \gamma \) to the counterclockwise circle \( C[\gamma, R] \) that stays inside \( \gamma \) and away from \( D[\gamma, R] \). This is provided directly by another substantial theorem of topology, the Arzelà–Ascoli Theorem, although there are other methods. Again, in this book we shall assume the existence of this homotopy.

These results of topology seem intuitively obvious but are surprisingly difficult to prove. If you'd like to see a proof, we recommend that you take a course in topology.

There is still a subtle problem with our proof. We assumed that \( \gamma \) is in \( G \), but we also need the interior of \( \gamma \) to be contained in \( G \). Since we need to apply Cauchy’s Theorem to the homotopy between \( \gamma \) and \( C[\gamma, R] \), we could just add this as an assumption to our theorem, but the following formulation will be more convenient later.

**Theorem 4.27 (Cauchy’s Integral Formula).** Suppose \( f \) is holomorphic in the region \( G \) and \( \gamma \) is a positively oriented, simple, closed, piecewise smooth path, such that \( w \) is inside \( \gamma \) and \( r > 0 \). Then

\[
f(w) = \frac{1}{2\pi i} \int_{\gamma} \frac{f(z)}{z - w} \, dz.
\]

\[
\text{This is the Jordan of Jordan normal form fame, but not the one of Gauss–Jordan elimination.}
\]
####################
File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf

Page: 104

Context: we rewrite using Theorem 4.27.

$$
\frac{f(z + \Delta z) - f(z)}{\Delta z} = \frac{1}{2\pi i} \int_{\gamma} \frac{f(t)}{t - z} \, dt - \frac{1}{2\pi i} \int_{\gamma} \frac{f(t)}{(t - z)^2} \, dz
$$

$$
= \frac{1}{2\pi i} \int_{\gamma} \frac{f(t)}{(t - z)^2} \, dt. \tag{5.1}
$$

Theorem 5.1 will follow if we can show that the following expression gets arbitrarily small as \( \Delta z \to 0 \):

$$
\frac{f(z + \Delta z) - f(z)}{\Delta z} = \frac{1}{2\pi i} \int_{\gamma} \frac{f(t)}{(t - z)^2} \, dt - \frac{f(z)}{(z - w)^2} \, dz
$$

This can be made arbitrarily small if we can show that the integral on the right-hand side stays bounded as \( \Delta z \to 0 \). In fact, by Proposition 4.6, it suffices to show that the integrand stays bounded as \( \Delta z \to 0 \) (because \( z \) and curve length \( \gamma \) are fixed).

Let \( M = \max_{t \in \gamma} |f(t)| \) (whose existence is guaranteed by Theorem A1). Choose \( \delta > 0 \) such that \( |z - w| < \delta \): that is, \( |z - w| \geq \epsilon \) for all \( y \in \gamma \). By the reverse triangle inequality (Corollary 1.7(b)), for all \( y \in \gamma \):

$$
\frac{|f(t)|}{|t - z|^2} \leq \frac{|f(z)|}{|z - w|^2} \leq \frac{M}{(a - |z - w|)^2},
$$

which certainly stays bounded as \( \Delta z \to 0 \). This proves (5.1) and thus the Cauchy Integral Formula for \( f' \).

The proof of the formula for \( f'' \) is very similar and will be left to Exercise 5.2. 

Theorem 5.1 suggests that there are similar formulas for the higher derivatives of \( f \). This is in fact true, and theoretically we could obtain them one by one with the methods of the proof of Theorem 5.1. However, once we start studying power series for holomorphic functions, we will obtain such a result much more easily; so we save the derivation of integral formulas for higher derivatives of \( f \) for later (Corollary 8.11).
####################
File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf

Page: 105

Context: # VARIATIONS OF A THEME

Theorem 5.1 has several important consequences. For starters, it can be used to compute certain integrals.

## Example 5.2
\[
\int_{C(0,1)} \frac{\sin(z)}{z^2} \, dz = \frac{2\pi i}{2\pi} \left. \frac{d}{dx} \sin(z) \right|_{z=0} = 2\pi i \cos(0) = 2\pi i.
\]

![Figure 5.1: The integration paths in Example 5.3.](path/to/figure5.1.png)

## Example 5.3
To compute the integral
\[
\int_{C(0,2)} \frac{dz}{z^2(z-1)},
\]
we could employ a partial fractions expansion similar to the one in Example 4.23, or moving the integration path similar to the one in Exercise 4.29. To exhibit an alternative, we split up the integration path as illustrated in Figure 5.1: we introduce an additional path that separates 0 and 1. If we integrate on these two new closed paths (\(Y_1\) and \(Y_2\)) counterclockwise, the two contributions along the new path will cancel each other. The effect is that we transformed an integral for which two singularities were inside the integration path into a sum of two integrals, each of which has only one singularity inside the integration path; these new integrals will...
####################
File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf

Page: 107

Context: Proof: Theorem 4.15 yields an antiderivative \( F \) for \( f \) in \( G \). Because \( F \) is holomorphic in \( G \), Corollary 5.5 implies that \( f \) is also holomorphic in \( G \). 

Just like there are several variations of Theorem 4.15, we have variations of Corollary 5.6. For example, by Corollary 4.16, we can replace the condition for all piecewise smooth closed paths \( \gamma \) in the statement of Corollary 5.6 by the condition for every closed path in \( G \) (which, in fact, gives a stronger version of this result). 

A special case of Theorem 4.15 applies to regions in which every closed path is contractible.

### Definition
A region \( G \subset \mathbb{C} \) is **simply connected** if \( \gamma \) is contractible for every closed path \( \gamma \) in \( G \).

### Example 5.7
Any disk \( D_r \) is simply connected, as is \( \mathbb{C} \setminus \{0\} \). (You should draw a few closed paths in \( \mathbb{C} \setminus \{0\} \) to convince yourself that they are all contractible.) The region \( \{0\} \) is not simply connected, as e.g., the unit circle is not \( \{(0)\} \)-contractible.

If \( f \) is holomorphic in a simply-connected region then Corollary 4.20 implies that \( f \) satisfies the conditions of Theorem 4.15, hence we conclude:

### Corollary 5.8
Every holomorphic function on a simply-connected region \( G \subset \mathbb{C} \) has an antiderivative on \( G \).

Note that this corollary gives us no indication of how to compute an antiderivative. For example, it says that the (entire) function \( f : \mathbb{C} \to \mathbb{C} \) given by \( f(z) = \exp(z^2) \) has an antiderivative \( F \) in \( C \); it is an entirely different matter to derive a formula for \( F \).

Corollary 5.8 also illustrates the role played by two of the regions in Example 5.7, in connection with the function \( f(z) = \frac{1}{z} \). This function has no antiderivative on \( \mathbb{C} \setminus \{0\} \), as was proved in Example 4.14. Consequently, since \( \mathbb{C} \setminus \{0\} \) is not simply connected, however, the function \( f(z) = \frac{1}{z} \) does not illustrate the instance implied by Corollary 5.8.

Finally, Corollary 5.8 implies that, if we have two paths in a simply-connected region with the same endpoints, we can concatenate them—changing direction on one—to form a closed path, which proves:
####################
File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf

Page: 108

Context: # Consequences of Cauchy’s Theorem

## Corollary 5.9
If \( f \) is holomorphic in a simply-connected region \( G \) then \( f \) is independent of the piecewise smooth path \( \gamma \) between \( y(a) \) and \( y(b) \).

When an integral depends only on the endpoints of the path, the integral is called path independent. Example 41 shows that this situation is quite special; it also says that the function \( z^2 \) does not have an antiderivative in, for example, the region \( \{ z \in \mathbb{C} : |z| < 2 \} \). (Actually, the function \( z^2 \) does not have an antiderivative in any nonempty region — see Exercise 5.7.)

## 5.3 Taking Cauchy’s Formula to the Limit

Many beautiful applications of Cauchy’s Integral Formulas (such as Theorems 4.27 and 5.1) arise from considerations of the limiting behavior of the integral as the path gets arbitrarily large. The first and most famous application concerns the roots of polynomials. As a preparation we prove the following inequality, which is generally quite useful. It says that for \( | z | \) large enough, a polynomial \( p(z) \) of degree \( a_n \) tends to behave almost like a constant times \( z^{a_n} \).

### Proposition 5.10
Suppose \( p(z) \) is a polynomial of degree \( a_n \) with leading coefficient \( a_n \). Then there is a real number \( R \) such that

\[
\frac{1}{2} | z |^{a_n} \leq | p(z) | \leq 2 | a_n | | z |^{a_n}
\]

for all \( z \) satisfying \( | z | \geq R \).

**Proof:** Since \( p(z) \) has degree \( a_n \), its leading coefficient \( a_n \) is not zero, and we can factor out \( a_n z^{a_n} \):

\[
| p(z) | = | a_n z^{a_n} + a_{n-1} z^{a_n-1} + \ldots + a_0 | 
= | a_n | \cdot \left| z^{a_n} \left(1 + \frac{a_{n-1}}{a_n} \frac{1}{z} + \ldots + \frac{a_0}{a_n} \frac{1}{z^{a_n}} \right) \right|.
\]

Then the term inside the last factor has limit as \( z \to \infty \) (by Exercise 3.12), and so its modulus is between \( \frac{1}{2} \) and \( 2 \) as long as \( | z | \) is large enough.

## Theorem 5.11 (Fundamental Theorem of Algebra)
Every nonconstant polynomial has a root in \( \mathbb{C} \).
####################
File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf

Page: 109

Context: ```markdown
# Taking Cauchy’s Formulas to the Limit

## Proof

Suppose (by way of contradiction) that \( p(z) \) does not have any roots, that is, \( p(z) \neq 0 \) for all \( z \in \mathbb{C} \). Then \( \frac{1}{p(z)} \) is entire, and so Cauchy’s Integral Formula (Theorem 4.24) gives

\[
\frac{1}{p(0)} = \frac{1}{2\pi i} \int_{\mathcal{C}(0, R)} \frac{p'(z)}{p(z)} \, dz, \quad \text{for any } R > 0.
\]

Let \( d \) be the degree of \( p(z) \) and \( a_k \) its leading coefficient. Propositions 4.6(d) and 5.10 allow us to estimate, for sufficiently large \( R \):

\[
\left| \frac{1}{p(0)} \right| = \frac{1}{2\pi} \int_{\mathcal{C}(0, R)} \frac{1}{|p(z)|} \, |dz| \leq \frac{1}{2\pi} \cdot \frac{1}{\text{ext } |a_k| R^{d}} \cdot |2\pi R| = \frac{2 |a_k|}{R^{d}}.
\]

The left-hand side is independent of \( R \), while the right-hand side can be made arbitrarily small (by choosing \( R \) sufficiently large), and so we conclude that \( \frac{1}{p(0)} = 0 \), which is impossible.

## Theorem 5.11

Theorem 5.11 implies that any polynomial \( p \) can be factored into linear terms of the form \( z - a \) where \( a \) is a root of \( p \) as we can apply the corollary, after getting a root to \( z^n = 0 \) (which is again a polynomial by the division algorithm), etc. (see also Exercise 5.11).

A compact reformulation of the Fundamental Theorem of Algebra (Theorem 5.11) is to say that \( \mathbb{C} \) is algebraically closed in contrast, \( \mathbb{R} \) is not algebraically closed.

### Example 5.12

The polynomial \( p(z) = z^4 + 5z^2 + 3 \) has no roots in \( \mathbb{R} \). The Fundamental Theorem of Algebra (Theorem 5.11) states that \( p \) must have a root (in fact, four roots) in \( \mathbb{C} \):

\[
p(z) = (z^2 + 1)(z^2 + 3) = (z + i)(z - i)(\sqrt{3} + z)(\sqrt{3} - z) \quad \Box
\]

Another powerful consequence of Theorem 5.1 is the following result, which again has no counterpart in real analysis (consider, for example, the ratio test).
```
####################
File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf

Page: 110

Context: # Corollary 5.13 (Liouville's Theorem)  

Any bounded entire function is constant.

## Proof:  
Suppose \( |f(z)| \leq M \) for all \( z \in \mathbb{C} \). Given any \( \epsilon > 0 \), we apply Theorem 5.1 with the circle \( C(R) \); note that we can choose any \( R > 0 \) because \( f \) is entire. By Proposition 4.16(1),

\[
\left| \frac{f(w)}{2\pi i \int_{C(R)} (z - w)^{-1} dz} \right| = \frac{1}{2\pi i} \int_{C(R)} \frac{f(z)}{(z - w)^2} dz \leq \frac{1}{2\pi} \max_{z \in C(R)} |f(z)| \cdot \frac{1}{2\pi R}.
\]

The right-hand side can be made arbitrarily small, as we are allowed to choose \( R \) as large as we want. This implies that \( f' = 0 \), and hence, by Theorem 2.17, \( f \) is constant.

As an example of the usefulness of Liouville's theorem (Corollary 5.13), we give another proof of the Fundamental Theorem of Algebra, close to Gauss's original proof.

## Second proof of Theorem 5.11 (Fundamental Theorem of Algebra):  
Suppose (by way of contradiction) that \( f \) does not have any roots, that is, \( f(z) \neq 0 \) for all \( z \in \mathbb{C} \). Then the function \( f(z) = \frac{1}{f(z)} \) is entire. But \( f \to 0 \) as \( |z| \to \infty \), by Proposition 5.10; consequently, by Exercise 5.10, \( f \) is bounded. Now we apply Corollary 5.13 to deduce that \( f \) is constant. Hence \( f \) is constant, which contradicts our assumptions.

As one more example of the theme of getting results from Cauchy's Integral Formulas by taking the limit as a path “goes to infinity,” we compute an improper integral.

## Example 5.14:  
We will compute the (real) integral

\[
\int_0^{\infty} \frac{dx}{x^2 + 1} = \pi.
\]

Let \( C_R \) be the counterclockwise semicircle formed by the segment \([-R, R]\) of the real axis from \(-R\) to \(R\), followed by the circular arc \( \gamma_R \) of radius \( R \) in the upper half plane from \( R \to -R \), see Figure 5.2. 

> *This theorem (the historical reason commonly attributed to Joseph Liouville (1809–1883). It was published earlier by Cauchy in fact, Gauss may well have known about it before Cauchy.)*
####################
File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf

Page: 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: 124

Context: where \( P_\nu(r) \) is the Poisson kernel which we introduced in Exercise 4.31. Then \( u \) is the desired extension: it is continuous on the closed unit disk, harmonic in the open unit disk, and agrees with \( u \) on the unit circle. In simple cases this solution can be converted to solutions in other regions, using a conformal map to the unit disk. All of this is beyond the scope of this book, though Exercise 6.13 gives some indication why the above formula does the trick. At any rate, we remark that Corollary 6.14 says that the solution to the Dirichlet problem is unique.

## Exercises

6.1. Show that all partial derivatives of a harmonic function are harmonic.

6.2. Suppose \( u(x, y) \) and \( v(x, y) \) are harmonic in \( G \), and \( \varepsilon \in \mathbb{R} \). Prove that \( u(x, y) + \varepsilon v(x, y) \) is also harmonic in \( G \).

6.3. Give an example that shows that the product of two harmonic functions is not necessarily harmonic.

6.4. Let \( u(x, y) = e^y \sin y \).

(a) Show that \( u \) is harmonic on \( \mathbb{C} \).

(b) Find an entire function \( f \) such that \( \text{Re}(f) = u \).

6.5. Consider \( u(x, y) = \ln(x^2 + y^2) \).

(a) Show that \( u \) is harmonic on \( \mathbb{C} \setminus \{(0, 0)\} \).

(b) Prove that \( u \) is not the real part of a function that is holomorphic in \( \mathbb{C} \setminus \{(0, 0)\} \).

6.6. Show that if \( f \) is holomorphic and nonzero in \( G \), then \( \ln|f(x, y)| \) is harmonic in \( G \).

6.7. Suppose \( u(x, y) \) is a function \( \mathbb{R}^2 \rightarrow \mathbb{R} \) that depends only on \( x \). When is \( u \) harmonic?
####################
File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf

Page: 127

Context: # Chapter 7

## Power Series

*It is a pain to think about convergence but sometimes you really have to.*  
**Sinai Robbins**

Looking back to what machinery we have established so far for integrating complex functions, there are several useful theorems we developed in Chapters 4 and 5. But there are some simple-looking integrals, such as:

\[
\int_{C} \frac{\exp(z)}{\sin(z)} \, dz, \tag{7.1}
\]

that we cannot compute with this machinery. The problems, naturally, come from the singularities at \(0\) and \(x\) inside the integration path, which in turn stem from the roots of the sine function. We might try to simplify this problem a bit by writing the integral as the sum of integrals over the two "D" shaped paths shown in Figure 5.1.

![Figure 7.1: Modifying the integration path for (7.1).](path/to/figure5.1)
####################
File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf

Page: 129

Context: To prove that a sequence \((a_n)\) is divergent, we have to show the negation of the statement that defines convergence; that is, given any \(L \in \mathbb{C}\), there exists \(\epsilon > 0\) such that, given any integer \(N\), there exists an integer \(n > N\) such that \(|a_n - L| > \epsilon\). (If you have not negated any mathematical statements, this is worth meditating about.)

### Example 2.7:
The sequence \((a_n = i^n)\) diverges: Given \(L \in \mathbb{C}\), choose \(L = \frac{1}{2}\). We consider two cases: If \(\text{Re}(L) > 0\), then for any \(N\), choose \(n > N\) such that \(a_n = -1\). (This is always possible since \(a_{2k} = i^{2k} = -1\) for any \(k \geq 0\).) Then

\[
|a_n - L| = |1 + L| > \frac{1}{2}.
\]

If \(\text{Re}(L) < 0\), then for any \(N\), choose \(n > N\) such that \(a_n = 1\). (This is always possible since \(a_{4k} = i^{4k} = 1\) for any \(k \geq 0\).) Then

\[
|a_n - L| = |1 - L| > \frac{1}{2}.
\]

This proves that \((a_n = i^n)\) diverges.

The following limit laws are the cousins of the identities in Propositions 2.4 and 2.6, with one little twist.

### Proposition 7.3.
Let \((a_k)\) and \((b_k)\) be convergent sequences and \(c \in \mathbb{C}\). Then

1. \(\lim_{k \to \infty} (a_k + b_k) = \lim_{k \to \infty} a_k + \lim_{k \to \infty} b_k.\)
2. \(\lim_{k \to \infty} (a_k - b_k) = \lim_{k \to \infty} a_k - \lim_{k \to \infty} b_k.\)
3. \(\lim_{k \to \infty} (c a_k) = c \cdot \lim_{k \to \infty} a_k.\)
4. \(\lim_{k \to \infty} (a_k b_k) = \lim_{k \to \infty} a_k \cdot \lim_{k \to \infty} b_k.\)
5. \(\lim_{k \to \infty} \frac{a_k}{b_k} = \frac{\lim_{k \to \infty} a_k}{\lim_{k \to \infty} b_k}\) where \(b_k \neq 0\) for all sufficiently large \(k\), assuming \(\lim_{k \to \infty} b_k \neq 0.\)
6. \(\lim_{k \to \infty} k = \infty.\)

Again, the proof of this proposition is essentially a repeat from arguments we have given in Chapters 2 and 3, as you should convince yourself in Exercise 7.4.

We will assume, as an axiom, that it is complete. To phrase this precisely, we need the following.
####################
File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf

Page: 130

Context: # POWER SERIES

## Definition
The sequence \((a_n)\) is monotonic if it is either nondecreasing \((a_n \geq a_{n-1}\) for all \(n)\) or nonincreasing \((a_n \leq a_{n-1}\) for all \(n)\).

There are many equivalent ways of formulating the completeness property for the reals. Here is what we'll go by:

## Axiom (Monotone Sequence Property)
Any bounded monotonic sequence converges.

This axiom (or one of its many equivalent statements) gives arguably the most important property of the real number system: namely, that we can, in many cases, determine that a given sequence converges without knowing the value of the limit. In this sense we can use the sequence to define a real number.

## Example 7.4
Consider the sequence \((a_n)\) defined by

\[
a_n = 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n}.
\]

This sequence is increasing (by definition) and each \(a_n \leq 3\) by Exercise 7.9. By the Monotone Sequence Property, \((a_n)\) converges, which allows us to define one of the most famous numbers in all of mathematics,

\[
\epsilon = 1 + \lim_{n \to \infty} a_n.
\]

## Example 7.5
Fix \(0 < r < 1\). We claim that \(\lim_{n \to \infty} r^n = 0\). First, the sequence \((a_n = r^n)\) converges because it is decreasing and bounded below by 0. Let \(L = \lim_{n \to \infty} r^n\). By Proposition 7.3,

\[
L = \lim_{n \to \infty} r^n = r^1 = \lim_{n \to \infty} r^n = 0.
\]

Thus \((1 - r)L = 0\) and (since \(1 - r \neq 0\)) we conclude that \(L = 0\).

We remark that the Monotone Sequence Property implies the Least Upper Bound Property: every nonempty set of real numbers with an upper bound has a least upper bound. The Least Upper Bound Property, in turn, implies the following theorem, which is often stated as a separate axiom:

**Footnote:**
Both the Archimedean Property and the Least Upper Bound Property can be used in (different) axiomatic developments of \(\mathbb{R}\).
####################
File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf

Page: 131

Context: # Theorem 7.6. [Archimedean Property] 
If \( x \) is any real number then there is an integer \( N \) that is greater than \( x \).

For a proof see Exercise 7.10. Theorem 7.6 essentially says that infinity is not part of the real numbers. Note that we already used Theorem 7.6 in Example 7.1. The Archimedean Property underlies the construction of an infinite decimal expansion for any real number, while the Monotone Sequence Property shows that such infinite decimal expansion actually converges to a real number.

We close this discussion of limits with a pair of standard limits. The first of these can be established by calculus methods (such as L'Hôpital's rule (Theorem A.11), by treating \( x \) as the variable); both of them can be proved by more elementary considerations. Either way, we leave the proof of the following to Exercise 7.11.

## Proposition 7.7.

(a) Exponential best polynomials: For any polynomial \( p(x) \) (with complex coefficients) and any \( x \in \mathbb{C} \) with \( |x| > 1 \),

\[
\lim_{x \to \infty} \frac{p(x)}{e^x} = 0.
\]

(b) Factorials best exponentials: For any \( z \in \mathbb{C} \),

\[
\lim_{n \to \infty} \frac{z^n}{n!} = 0.
\]

## 7.2 Series

**Definition.** A series is a sequence \( (a_n) \) whose members are of the form \( a_n = \sum_{k=1}^{n} b_k \) (or \( a_n = \sum_{k=1}^{\infty} b_k \)); we call \( (b_k) \) the sequence of terms of the series. The \( a_n = \sum_{k=1}^{n} b_k \) (or \( a = \sum_{k=1}^{\infty} b_k \)) are the partial sums of the series.

If we wanted to be lazy we would define convergence of a series simply by referring to convergence of the partial sums of the series—after all, we just defined series through sums. However, there are some convergence features that hinge on special appearances for series, so we mention them here explicitly. For starters, a 

*Archimedes of Syracuse (287–212 BCE) attributes this property to Eudoxus of Cnidus (408–355 BCE).
####################
File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf

Page: 134

Context: ```markdown
# Power Series

## Proof of Proposition 7.13

Suppose \(\sum_{k=1}^{\infty} b_k\) converges. Then, using Proposition 7.3(a) & (d),

\[
0 = \lim_{n \to \infty} \sum_{k=1}^{n} b_k = \lim_{n \to \infty} \left( \sum_{k=0}^{n} b_k - \sum_{k=1}^{n} b_k \right) = \lim_{n \to \infty} b_n.
\]

A common mistake is to try to use the converse of Proposition 7.13, but the converse is false:

### Example 7.16

The harmonic series \(\sum_{k=1}^{\infty} \frac{1}{k}\) diverges (even though the terms go to 0). If we assume the series converges to \(L\), then

\[
L = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots
\]

\[
> \frac{1}{2} + \frac{1}{4} + \frac{1}{6} + \cdots = \frac{1}{2} \left( 1 + \frac{1}{2} + \frac{1}{3} + \cdots \right) = \frac{1}{2}L.
\]

Thus,

\[
L > \frac{L}{2}
\]

which leads to

\[
L = 1 + \frac{1}{2} + \frac{1}{3} + \cdots = L,
\]

a contradiction.

![Figure 7.2: The integral test.](link_to_image)
```
####################
File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf

Page: 138

Context: Let's digest these two notions of convergence of a function sequence by describing them using quantifiers: as usual, \( \forall \) denotes for all and \( \exists \) means there exists. Pointwise convergence on \( G \) translates into 

\[
V \to \forall z \in G \, \exists N \in \mathbb{N} \, |f_n(z) - f(z)| < \epsilon,
\]

whereas uniform convergence on \( G \) translates into 

\[
V \to \exists N \in \mathbb{N} \, \forall z \in G \, \forall n \in \mathbb{N} \, |f_n(z) - f(z)| < \epsilon.
\]

No big deal — we only exchanged two of the quantifiers. In the first case, \( N \) may well depend on \( z \); in the second case, we need to find an \( N \) that works for all \( z \in G \). And this can make all the difference ...

### Example 7.23

Let \( f: [0, 1] \to \mathbb{R} \) be defined by \( f_n(x) = x^n \). We claim that this sequence of functions converges pointwise to \( f: [0, 1] \to \mathbb{R} \) given by \( f(x) = 0 \). This is immediate for the point \( z = 0 \). Now given any \( \epsilon > 0 \) and \( 0 < |x| < 1 \), choose \( N > \frac{1}{\epsilon} \). Then for all \( n \geq N \),

\[
|f_n(x) - f(x)| = |x^n - 0| = |x|^n \leq |x|^N < \epsilon.
\]

(You ought to check carefully that all our inequalities work the way we claim they do.)

### Example 7.24

Let \( f: [0, 1] \to \mathbb{R} \) be defined by \( f_n(x) = x^n \). We claim that this sequence of functions converges uniformly to \( f: [0, 1] \to \mathbb{R} \) given by \( f(x) = 0 \). Given any \( \epsilon > 0 \) and \( |x| < \frac{1}{n} \), choose \( N \geq \frac{1}{\epsilon} \). Then for all \( n \geq N \),

\[
|f_n(x) - f(x)| = |x^n - 0| \leq |x|^n \left( \frac{1}{n} \right)^N.
\]

(Again, you should carefully check our inequalities.) 

The differences between Example 7.23 and Example 7.24 are subtle, and we suggest you meditate over them for a while with a good cup of coffee. You might already suspect that the function sequence in Example 7.23 does not converge uniformly, as we will see in a moment.

The first application illustrating the difference between pointwise and uniform convergence says, in essence, that if we have a sequence of functions \( f_n \) that
####################
File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf

Page: 140

Context: # POWER SERIES

uniform either, though this needs a separate proof, as the domain of the functions in Example 7.23 is the unit disk (Exercise 7.20(b)).  

Now that we have established Proposition 7.25 about continuity, we can ask about integration of sequences or series of functions. The next theorem should come as no surprise; however, its consequences (which we will see shortly) are wide ranging.

## Proposition 7.27
Suppose \( f_n : G \to \mathbb{C} \) is continuous, for \( n \geq 1 \). Then \( (f_n) \) converges uniformly to \( f : G \to \mathbb{C} \) and \( G \) is a piecewise smooth path. Then

\[
\lim_{n \to \infty} \int_{C} f_n(z) \, dz = \int_{C} f(z) \, dz.
\]

**Proof:** We may assume that \( z \) is not just a point, in which case the proposition holds trivially. Given \( \epsilon > 0 \), there exists \( N \) such that for all \( z \in G \) and all \( n \in \mathbb{N}, \)

\[
|f_n(z) - f(z)| < \frac{\epsilon}{\text{length}(C)}.
\]

With Proposition 4.6(d) we can thus estimate

\[
\left| \int_{C} f_n(z) \, dz - \int_{C} f(z) \, dz \right| \leq \max_{z \in C} |f_n(z) - f(z)| \cdot \text{length}(C) < \epsilon.
\]

All of these notions for sequences of functions hold verbatim for series of functions. For example, if \( \sum_{n} f_n(z) \) converges uniformly on \( G \) and \( G \) is a piecewise smooth path, then

\[
\int_{C} \sum_{n} f_n(z) \, dz = \sum_{n} \int_{C} f_n(z) \, dz.
\]

In some sense, the above identity is why we care about uniform convergence.

There are several criteria for uniform convergence; see, e.g., Exercises 7.19 and 7.20, and the following result, sometimes called the Weierstrass M-test.

## Proposition 7.28
Suppose \( f_n : G \to \mathbb{C} \) for \( n \geq 1 \), and \( |f_n(z)| \leq M_n \) for all \( z \in G \), where \( \sum_{n} M_n \) converges. Then \( \sum_{n} f_n(z) \) converges absolutely and uniformly in \( G \). (We say the series \( \sum_{n} f_n(z) \) converges absolutely and uniformly.)

**Proof:** For each fixed \( z \), the series \( \sum_{n} |f_n(z)| \) converges absolutely by Corollary 7.12. To show that the convergence is uniform, let \( \epsilon > 0 \). Then there exists \( N \) such that
####################
File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf

Page: 141

Context: for all \( n \in \mathbb{N} \).

\[
\sum_{n=1}^N M_n - \sum_{n=1}^N M_n < \epsilon.
\]

Thus for all \( z \in G \) and \( n \in \mathbb{N} \):

\[
\sum_{k=1}^{\infty} \left( f_k(z) - \sum_{k=1}^N f_k(z) \right) = \left| \sum_{k=1}^{\infty} f_k(z) \right| \leq \sum_{k=1}^N \left| f_k(z) \right| < \epsilon,
\]

which proves uniform convergence. Replace \( f_k \) with \( |f_k| \) in this argument to see that \( \sum_{k=1}^{\infty} |f_k| \) also converges uniformly. 

Example 7.29. We revisit Example 7.8 and consider the geometric series \( \sum_{k=1}^{\infty} z^k \) as a series of functions in \( z \). We know from Example 7.8 that this function series converges pointwise for \( |z| < 1 \):

\[
\sum_{k=0}^{\infty} z^k = \frac{1}{1-z}.
\]

To study uniform convergence, we apply Proposition 7.28 with \( f(k) = z^k \). We need a series of upper bounds that converges, so fix a real number \( 0 < r < 1 \) and let \( M_k = M_r^k \). Then 

\[
|f_k(z)| = |z|^k < r^k \quad \text{for } |z| < r.
\]

And \( \sum_{k=1}^{\infty} r^k \) converges by Example 7.8. Thus, Proposition 7.28 says that \( \sum_{k=1}^{\infty} z^k \) converges uniformly for \( |z| < r \).

We note the subtle distinction of domains for pointwise/uniform convergence: 

\[
\sum_{k=1}^{\infty} z^k \text{ converges (absolutely) for } |z| < 1, \text{ but to force uniform convergence, we need to shrink the domain to } |z| < r \text{ (arbitrary but fixed)} < 1.
\]

### 7.4 Regions of Convergence

For the remainder of this chapter (indeed, this book) we concentrate on some very special series of functions.

**Definition.** A power series centered at \( z_0 \) is a series of the form 

\[
\sum_{k=0}^{\infty} c_k (z-z_0)^k
\]

where \( c_0, c_1, \ldots \in \mathbb{C} \).
####################
File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf

Page: 143

Context: Now if \( |z - z_0| < r - |a| \), then  
\[
\sum_{k=0}^{\infty} |(z - z_0)^k|^p < \sum_{k=0}^{\infty} \frac{|(z - z_0)|^k}{r^k} \leq \frac{1}{1 - \frac{|z - z_0|}{r}}.
\]

The sum on the right-hand side is a convergent geometric sequence, since \( |z - z_0| < r \), and so \( \sum_{k=0}^{\infty} (z - z_0)^k \) converges absolutely by Corollary 7.12. 

**Proof of Theorem 7.31.** Consider the set  
\[
S = \left\{ x \in \mathbb{R}^2 : \sum_{k \neq 0} c_k (x - z_0)^k \text{ converges} \right\}.
\]

(This set is nonempty since \( 0 \in S \).) 

If \( R \) is unbounded, then \( \sum_{k \neq 0} c_k (z - z_0)^{-k} \) converges absolutely and uniformly for \( |z - z_0| < r \), for any \( r \) and so this gives the \( R = 0 \) case of Theorem 7.31: choose \( S \) with \( x \) such that \( R < R \). Then Proposition 7.32 says that \( \sum_{k} c_k (z - z_0)^k \) converges absolutely. 

Since \( |(z - z_0)|^k \leq |z - z_0|^k \), we can now use Proposition 7.28. 

If \( R \) is bounded, let \( R \) be its least upper bound. If \( R < \infty \), then \( \sum_{k \neq 0} c_k (z - z_0)^{-k} \) converges only for \( z \in R \), which establishes Theorem 7.31 in this case. 

Now assume \( R > 0 \). If \( |z - z_0| < R \) for some \( r < R \), again we can find \( z \) such that \( r < x < R \). Then \( \sum_{k} c_k |(z - z_0)|^k \) converges absolutely and uniformly for \( |z - z_0| < |z| \) by Proposition 7.13. This proves (b). 

Finally, if \( |z - z_0| > R \) then there exists \( r < R \) such that  
\[
|z - z_0| < R - |a|. 
\]
But \( \sum_{k} c_k (z - z_0)^{-k} \) diverges, so by the contrapositive of Theorem 7.20, \( \sum_{k} |c_k| |(z - z_0)|^k \) diverges, and so by the contrapositive of Proposition 7.32, \( \sum_{k} c_k (z - z_0)^{-k} \) diverges, which finishes the proof.
####################
File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf

Page: 144

Context: Corollary 7.33. If \( \lim_{n \to \infty} \sqrt{|z|} \) exists, then the radius of convergence of the series 

\[
\sum_{n=0}^{\infty} c_n (z-a)^n
\]

equals 

\[
R = \begin{cases} 
\frac{\infty}{\lim_{n \to \infty} \sqrt{|c_n|}} & \text{if } \lim_{n \to \infty} \sqrt{|c_n|} = 0, \\ 
\text{otherwise}. 
\end{cases}
\]

**Proof:** We treat the case that \( R \) is finite and leave the case \( R = \infty \) to Exercise 7.31. Given \( R \) as in the statement of the corollary, it suffices (by Theorem 7.31) to show that \( \sum_{n=0}^{\infty} c_n (z-a)^n \) converges for \( |z-a| < R \) and diverges for \( |z-a| > R \). 

Suppose \( r = |z-a| < R \). Since \( \lim_{n \to \infty} \sqrt{|c_n|} = \frac{1}{R} \) and \( \sqrt{R} < r \), there exists \( N \) such that \( \sqrt{|c_n|} < \frac{R}{2} \) for those \( n \geq N \). For those \( k \):

\[
|c_n (z-a)^n| < |c_n| \left( \sqrt{|z-a|} \right)^n < \left( \sqrt{|c_n|} \right)^n \left( \frac{2R}{R+r} \right)^k
\]

and so \( \sum_{n=0}^{\infty} (c_n (z-a)^n) \) converges (absolutely) by Proposition 7.28, because \( |z-a| < 1 \) and thus \( \sum \frac{2R}{R+r} \) converges as a geometric series. Thus \( \sum c_n (z-a)^n \) converges.

Now suppose \( r = |z-a| > R \). Again, using \( \sqrt{|c_n|} = \frac{1}{R} \) and now \( \sqrt{R} < r \), there exists \( N \) such that \( \sqrt{|c_n|} < \frac{R}{k} \) for \( n \geq N \). For those \( k \):

\[
|c_n (z-a)^n| > \left( \sqrt{|c_n|} \right)^n \left( \frac{2R}{R+r} \right)^k > 1,
\]

and so the sequence \( \sum c_n (z-a)^n \) cannot converge to 0. Subsequently (by Corollary 7.14), \( \sum_{n=0}^{\infty} c_n (z-a)^n \) diverges. 

You might remember this corollary from calculus, where it goes by the name root test. Its twin sister, the ratio test, is the subject of Exercise 7.32.

**Example 7.34.** For the power series \( \sum_{k=0}^{\infty} k z^k \) we compute

\[
\lim_{|z| \to 1} \sqrt[k]{k} = \lim_{k \to \infty} \sqrt{k} = e^{\lim_{k \to \infty} \frac{1}{k} \ln(k)} = 1,
\]

and Corollary 7.33 gives the radius of convergence \( R = 1 \). (Alternatively, we can argue by differentiating the geometric series.)
####################
File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf

Page: 148

Context: # Power Series

![Functions](https://example.com/path/to/image)  
**Figure 7.3:** The functions \( f_n(x) = \sin\left(\frac{x}{n}\right) \) in Exercise 7.21.

## 7.18
1. Show that  
   \[
   \sum_{k=1}^{\infty} \frac{k}{k+1} \text{ diverges.}
   \]

2. Show that  
   \[
   \sum_{k=1}^{\infty} \frac{k}{k^2 + 1} \text{ converges.}
   \]

## 7.19
1. Suppose \( G \subset \mathbb{C} \) and \( f_n : G \to \mathbb{C} \) for \( n \geq 1 \). Suppose \( (a_n) \) is a sequence in \( \mathbb{R} \) with \( \lim_{n \to \infty} a_n = 0 \) and, for each \( n \geq 1 \),
   \[
   |f_n(a_n)| \leq a_n \quad \text{for all } a \in G.
   \]

   Show that \( f_n \) converges uniformly to the zero function in \( G \).

2. Re-prove the statement of Example 7.24 using part (a).

## 7.20
1. Suppose \( G \subset \mathbb{C}, f_n : G \to \mathbb{C} \) for \( n \geq 1 \), and \( f_n \) converges uniformly to the zero function in \( G \). Show that if \( (a_n) \) is any sequence in \( G \), then  
   \[
   \lim_{n \to \infty} f_n(a_n) = 0.
   \]

2. Apply (a) to the function sequence given in Example 7.23, together with the sequence \( (b_n = \frac{1}{n}) \), to prove that the convergence given in Example 7.23 is not uniform.

Image Analysis: 

### Comprehensive Examination

#### 1. Localization and Attribution
- **Image Positioning and Numbering**:
  - **Image 1 (Top)**: Positioned at the upper part of the page.
  - **Image 2 (Middle)**: Text content starting from "7.18" to "7.20" located beneath the Image 1.

#### 2. Object Detection and Classification
- **Image 1**:
  - **Objects**: Graph/Chart depicting a series of curves.
  - **Classification**: Mathematical function graphs.

#### 3. Scene and Activity Analysis
- **Image 1**:
  - **Scene Description**: Graphical representation of functions \( f_n(x) = \sin^n(x) \) for different values of \( n \).
  - **Activity**: Visualization of function behavior as \( n \) increases, showing how the functions change shape.

#### 4. Text Analysis
- **Image 2**:
  - **Text Content**:
    - **Figure Caption**: "Figure 7.3: The functions \( f_n(x) = \sin^n(x) \) in Exercise 7.21."
    - **7.18**: Two parts (a) and (b) discussing convergence and divergence of series.
    - **7.19**: Two parts (a) and (b), with (a) detailing properties of a function sequence and convergence criteria; (b) asks to re-prove part (a).
    - **7.20**: Two parts (a) and (b), with (a) expanding on function convergence criteria and (b) applying these criteria to a given example.
  - **Significance**: The text provides mathematical exercises and explanations related to the visual content (graph of sin functions).

#### 5. Diagram and Chart Analysis
- **Image 1**:
  - **Diagram**: Graph of \( f_n(x) = \sin^n(x) \)
  - **Axes**: 
    - **X-Axis**: Represents the variable \( x \).
    - **Y-Axis**: Represents the value of \( f_n(x) \).
  - **Scales**: Standard numerical scales for both x and y axes.
  - **Key Insights**: Demonstrates how the function \( f_n(x) \) narrows and approaches a delta-like function centered at \( \pi \) and other periodic points as \( n \) increases.

#### 8. Color Analysis
- **Image 1**:
  - **Dominant Colors**: Blue curves on a white background.
  - **Impact on Perception**: The color choice helps emphasize the variation among different curves, making it easy to distinguish between different values of \( n \).

#### 9. Perspective and Composition
- **Image 1**:
  - **Perspective**: Straight-on view of the graph.
  - **Composition**: Centralized graph with labeled axes. The curves are symmetric around the origin, following the properties of sine functions.

#### 10. Contextual Significance
- **Image 1**:
  - **Context**: Related to exercises and explanations in the study of power series and function convergence in mathematics.
  - **Contribution**: Provides visual understanding and concrete examples to complement the theoretical exercises below.

#### 12. Graph Numbers
- **Image 1**:
  - **Data Points**: Not explicitly provided, but the curves can be understood as sampled points from the functions \( \sin^n(x) \).
  - **Trends**: As \( n \) increases, the curve sharply peaks at specific points (multiples of \( \pi \)), demonstrating convergence behavior.

#### 13. Graph and Trend Analysis
- **Image 1**:
  - **Trends**: The sequence of functions \( f_n(x) = \sin^n(x) \) becomes more peaked at multiples of \( \pi \) as \( n \) increases, indicating how the function evolves with higher powers.
  - **Interpretation**: This trend is significant in understanding pointwise convergence and properties of Fourier series in mathematical analysis.

#### Additional Aspects
- **Prozessbeschreibungen (Process Descriptions)**:
  - **Series Convergence** (7.18): Analysis of divergence and convergence criteria.
  - **Function Sequence Convergence** (7.19, 7.20): Conditions and proof of uniform convergence of function sequences on given domains.
####################
File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf

Page: 153

Context: # POWER SERIES AND HOLOMORPHIC FUNCTIONS

A special case of this result concerns power series with infinite radius of convergence: those represent entire functions.

Now that we know that power series are differentiable in their regions of convergence, we can also know how to find their derivatives. The next result says that we can similarly differentiate the series term by term.

## Theorem 8.2

Suppose \( f(z) = \sum_{k=0}^{\infty} c_k (z - z_0)^k \) has radius of convergence \( R > 0 \). Then

\[
f'(z) = \sum_{k=1}^{\infty} k c_k (z - z_0)^{k-1} \quad \text{for any } z \in D_{[z_0,R]}.
\]

The radius of convergence of this power series is also \( R \).

**Proof.** If \( z \in D_{[z_0,R]} \) then \( |z - z_0| < R \), so we can choose \( R' \) such that \( |z - z_0| < R' < R \). Then the circle \( \gamma = C(z_0,R') \) lies in \( D_{[z_0,R]} \) and is inside \( S \). Since \( f \) is holomorphic in \( D_{[z_0,R]} \), we can use Cauchy’s Integral Formula for \( f \) (Theorem 5.1), as well as Corollary 3.7:

\[
f'(z) = \frac{1}{2 \pi i} \int_{\gamma} \frac{f(w)}{(w - z)^2} \, dw = \frac{1}{2 \pi i} \int_{\gamma} \frac{(w - z)^k}{(w - z)^2} \, dw = \sum_{k=0}^{\infty} k c_k \int_{\gamma} (w - z)^{k-1} \, dw
\]

Note that we used Theorem 5.1 again in the penultimate step, but now applied to the function \( f(w) \).

The last statement of the theorem is easy to show: the radius of convergence of \( f'(z) \) is at least \( R \) since we have shown that the series for \( f' \) converges whenever \( |z - z_0| < R \), and it cannot be larger than \( R \) by comparison to the series for \( f(z) \), since the coefficients for \( (z - z_0)^{k-1} \) are larger than the corresponding ones for \( f(z) \).

## Example 8.3

Let 

\[
f(z) = \sum_{k=0}^{\infty} \frac{z^k}{k!}.
\]
####################
File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf

Page: 158

Context: # TAYLOR AND LAURENT SERIES

**Proof:** Let \( r < R \). By Corollary 8.11 and Proposition 4.6(d),

\[
\left| \phi'(x) \right| = \frac{k}{2 \pi i} \int_{C(r)} \frac{f(z)}{(z - x)^{2}} \, dz 
\leq \frac{k}{r^{2}} \max_{z \in C(r)} |f(z)| \cdot \text{length}(C(r)) 
\leq \frac{k}{r^{2}} \cdot \frac{m}{r} \cdot \frac{2\pi r}{1}
\]

The statement now follows since \( r \) can be chosen arbitrarily close to \( R \).

A key aspect of this section is worth emphasizing: namely, we have developed an alternative characterization of what it means for a function to be holomorphic. In Chapter 2, we defined a function to be holomorphic in a region \( G \) if it is differentiable at each point \( x_0 \in G \). We now define what it means for a function to be **analytic** in \( G \).

**Definition:** Let \( f : G \to \mathbb{C} \) and \( x_0 \in G \). If there exist \( R > 0 \) and \( c_0, c_1, \ldots, c_n \in \mathbb{C} \) such that the power series 

\[
\sum_{k=0}^{\infty} c_k (x - x_0)^{k}
\]

converges in \( D(x_0, R) \) and agrees with \( f(x) \) in \( D(x_0, R) \), then \( f \) is analytic at \( x_0 \). We call \( f \) analytic in \( G \) if it is analytic at each point in \( G \).

What we have proved in this section can be summed up as follows:

**Theorem 8.13.** For any region \( G \), the class of all analytic functions in \( G \) coincides with the class of all holomorphic functions in \( G \).

While the terms **holomorphic** and **analytic** do not always mean the same thing, the study of complex analysis they do and are frequently used interchangeably.

## 8.2 Classification of Zeros and the Identity Principle

When we proved the Fundamental Theorem of Algebra (Theorem 5.11; see also Exercise 5.11), we remarked that, if a polynomial \( p(z) \) of degree \( d > 0 \) has a zero at \( a \) (that is, \( p(a) = 0 \)), then \( p(z) \) has \( z - a \) as a factor. That is, we can write 

\[
p(z) = (z - a) q(z)
\]

where \( q(z) \) is a polynomial of degree \( d - 1 \). We can then ask whether \( q(z) \) itself has a zero at \( a \), and if so, we can factor out another \( z - a \).
####################
File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf

Page: 159

Context: # Classification of Zeros and the Identity Principle

Continuing in this way, we see that we can factor \( p(z) \) as 

\( p(z) = (z-a)^m g(z) \)

where \( m \) is a positive integer and \( g(z) \) is a polynomial that does not have a zero at \( a \). The integer \( m \) is called the multiplicity of the zero of \( p(z) \). Almost exactly the same thing happens for holomorphic functions.

## Theorem 8.14 (Classification of Zeros)

Suppose \( f : G \to \mathbb{C} \) is holomorphic and \( f \) has a zero at \( a \in G \). Then either:

1. \( f \) is identically zero on some disk \( D \) centered at \( a \) (that is, \( f(z) = 0 \) for all \( z \in D \)); or
2. There exists a positive integer \( m \) and a holomorphic function \( g : G \to \mathbb{C} \), such that 

   \( f(z) = (z-a)^m g(z) \) for all \( z \in G \).

In this case, the zero is isolated: there is a disk \( D[a,r] \) which contains no other zero of \( f \).

The integer \( m \) in the second case is uniquely determined by \( f \) and is called the multiplicity of the zero at \( a \).

### Proof

By Theorem 8.8, there exists \( R > 0 \) such that we can expand 

\[
f(z) = \sum_{k=0}^{\infty} c_k (z-a)^k \text{ for } z \in D[a,R].
\]

and \( c_0 = f(a) = 0 \). There are now exactly two possibilities: either

1. \( c_0 = 0 \) for all \( k \geq 0 \); or
2. There is some positive integer \( m \) so that \( c_k = 0 \) for all \( k < m \) but \( c_m \neq 0 \).

The first case gives \( f(z) = 0 \) for all \( z \in D[a,R] \). So now consider the second case. We note that for \( z \in D[a,R] \):

\[
f(z) = (z-a)^m \left( c_m + c_{m+1} (z-a) + \ldots \right) = (z-a)^m \sum_{n=0}^{\infty} c_{m+n} (z-a)^n.
\]
####################
File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf

Page: 160

Context: # TAYLOR AND LAURENT SERIES

Thus we can define a function \( g : G \rightarrow \mathbb{C} \) through

\( g(z) = \begin{cases} 
\sum_{n=0}^{\infty} a_n (z - a)^n & \text{if } z \in D(a, R), \\ 
f(z) & \text{if } z \in G \setminus \{ a \}. 
\end{cases} \)

(According to our calculations above, the two definitions give the same value when \( z \in D(a, R) \). The function \( g \) is holomorphic in \( D(a, R) \) by the first definition, and \( g \) is holomorphic in \( G \setminus \{ a \} \) by the second definition. Note that \( g(a) = c \neq 0 \) and, by construction,

\( f(z) = (z - a)^n s(z) \quad \text{for all } z \in G. \)

Since \( s(a) \neq 0 \) by continuity, \( r > 0 \) so that \( s(z) \neq 0 \) for all \( z \in D(a, r) \), so \( D(a, r) \) contains no other zero of \( f \). The integer \( r \) is unique, since it is defined in terms of the power series expansion of \( f \) at \( a \), which is unique by Corollary 8.6.

Theorem 8.14 gives rise to the following result, which is sometimes called the identity principle or the uniqueness theorem.

## Theorem 8.15
Suppose \( G \) is a region, \( f : G \rightarrow \mathbb{C} \) is holomorphic, and \( f(a) = 0 \) where \( (a_k) \) is a sequence of distinct numbers that converges in \( G \). Then \( f \) is the zero function on \( G \).

Applying this theorem to the difference of two functions immediately gives the following variant.

## Corollary 8.16
Suppose \( f \) and \( g \) are holomorphic in a region \( G \) and \( f(a) = g(a) = 0 \) at a sequence that converges to \( a \in G \) with \( a_k \neq a \) for all \( k \). Then \( f(z) = g(z) \) for all \( z \in G \).

### Proof of Theorem 8.15
Consider the following two subsets of \( G \):

- \( X := \{ z \in G : \text{ there exists } z \text{ such that } F(z) = 0 \text{ for all } z \in D(a, r) \} \)
- \( Y := \{ z \in G : \text{ there exists } z \text{ such that } F(z) \neq 0 \text{ for all } z \in D(a, r) \setminus \{ a \} \} \)

If \( f(a) \neq 0 \) then, by continuity of \( f \), there exists a disk centered at \( a \) in which \( f \) is nonzero, and so \( a \in Y \).
####################
File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf

Page: 163

Context: Here \( a_k \in \mathbb{C} \) are terms indexed by the integers. The double series above converges if and only if the two series on the right-hand side do. Absolute and uniform convergence are defined analogously. Equipped with this, we can now introduce the following central concept.

## Definition
A Laurent series centered at \( z_0 \) is a double series of the form

\[
\sum_{n \in \mathbb{Z}} c_n (z - z_0)^{n}.
\]

### Example 8.21
The series that started this section is the Laurent series of \(\exp\left(\frac{1}{z}\right)\) centered at 0.

### Example 8.22
Any power series is a Laurent series (with \( a_k = 0 \) for \( k < 0 \)).

We should pause for a minute and ask for which \( z \) a general Laurent series can possibly converge. By definition,

\[
\sum_{n \in \mathbb{Z}} c_n (z - z_0)^n = \sum_{n \geq 0} c_n (z - z_0)^n + \sum_{n < 0} c_n (z - z_0)^n.
\]

The first series on the right-hand side is a power series with some radius of convergence \( R_z \), that is, with Theorem 7.31, it converges in \( \{ z \in \mathbb{C} : |z - z_0| < R_z \} \), and the convergence is uniform in \( \{ z \in \mathbb{C} : |z - z_0| \leq r \} \) for any fixed \( r < R_z \). For the second series, we invite you (in Exercise 8.30) to revise our proof of Theorem 7.31 to show that this series converges for 

\[
\frac{1}{|z - z_0|} < \frac{1}{R_1}
\]

for some \( R_1 \), and that the convergence is uniform in \( \{ z \in \mathbb{C} : |z - z_0| \leq r \} \) for any \( r > R_1 \). Thus the Laurent series converges in the annulus

\[
A = \{ z \in \mathbb{C} : R_1 < |z - z_0| < R_2 \}
\]

(assuming this set is not empty, i.e., \( R_1 < R_2 \)), and the convergence is uniform on any set of the form

\[
\{ z \in \mathbb{C} : R_1 < |z - z_0| < r \}
\]

for \( R_1 < r < R_2 \).

*Jean-Pierre Antoine Laurent (1813–1894).*
####################
File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf

Page: 165

Context: # LAURENT SERIES

## Theorem 8.24
Suppose \( f \) is a function that is holomorphic in  
\( A = \{ z \in \mathbb{C} : R_1 < |z - z_0| < R_2 \} \).

Then \( f \) can be represented in \( A \) as a Laurent series centered at \( z_0 \):

\[
f(z) = \sum_{n=-\infty}^{\infty} c_n (z - z_0)^n 
\]

with 

\[
c_n = \frac{1}{2 \pi i} \int_{C_{r_1}} \frac{f(w)}{(w - z_0)^{n+1}} \, dw,
\]

where \( R_1 < r < R_2 \).

By Cauchy's Theorem 4.18, we can replace the circle \( C_{r_1} \) in the formula for the Laurent coefficients by any path \( \gamma \subset A [C_{r_1}, r_2] \).

![Figure 8.1: The path γ in our proof of Theorem 8.24.](path_to_figure)

## Proof
Let \( g(z) = f(z + z_0) \), so \( g \) is a function holomorphic in  
\( \{ z \in \mathbb{C} : R_1 < |z| < R_2 \} \).

Fix \( R_1 < r_1 < r_2 < R_2 \), and let \( \gamma \) be the path in Figure 8.1, where \( \gamma_1 = C[r_1, r_2] \) and \( \gamma_2 = C[0, r_2] \). By Cauchy's Integral Formula (Theorem 4.27),

\[
g(z) = \frac{1}{2 \pi i} \int_{\gamma} \frac{g(w)}{w - z} \, dw. \tag{8.1}
\]

Complete any missing details from the original image as necessary while ensuring the accuracy of the markup.
####################
File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf

Page: 170

Context: 8.11. Suppose \( G \) is a region and \( f : G \to \mathbb{C} \) is holomorphic. Prove that the sets
- \( X = \{ a \in G : \text{there exists } r \text{ such that } f(a) = 0 \text{ for all } z \in D(a, r) \} \)
- \( Y = \{ a \in G : \text{there exists } r \text{ such that } f(z) \neq 0 \text{ for all } z \in D(a, r) \} \)

in our proof of Theorem 8.15 are open.

8.12. Prove the Minimum-Modulus Theorem (Corollary 8.19): Suppose \( f \) is holomorphic and nonconstant in a region \( G \). Then \( |f| \) does not attain a weak relative minimum at a point \( a \in G \) unless \( f(a) = 0 \).

8.13. Prove Corollary 8.20: Assume that \( u \) is harmonic in a region \( G \) and has a weak local maximum at \( a \in G \).

(a) If \( G \) is simply connected then apply Theorem 8.17 to \( u = \operatorname{Re}(f) \) where \( f \) is a harmonic conjugate of \( u \). Conclude that \( u \) is constant on \( G \).

(b) If \( G \) is not simply connected, then the above argument applies to \( u \) on any disk \( D(a, r) \cap G \). Conclude that the partials \( u_x \) and \( u_y \) are zero on \( G \), and adapt the argument of Theorem 2.17 to show that \( u \) is constant.

8.14. Let \( C : \mathbb{C} \to \mathbb{C} \) be given by \( C(z) = z^2 - 2 \). Find the maximum and minimum of \( |f(z)| \) on the closed unit disk.

8.15. Give another proof of the Fundamental Theorem of Algebra (Theorem 5.11), using the Minimum-Modulus Theorem (Corollary 8.19). (Hint: Use Proposition 5.10 to show that a polynomial does not achieve its minimum modulus on a large circle; then use the Minimum-Modulus Theorem to deduce that the polynomial has a zero.)

8.16. Give another proof of (a variant of) the Maximum-Modulus Theorem 8.17 via Corollary 8.11, as follows: Suppose \( f \) is holomorphic in a region containing \( \overline{D(a, r)} \) and \( |f(z)| \leq M \) for each point \( z \in D(a, r) \). Show (e.g., by Corollary 8.11) that there is a constant \( c \in \mathbb{C} \ such that 

\[
|f(z)| \leq c < M.
\]

Conclude that \( |f(a)| \leq M \).
####################
File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf

Page: 173

Context: 8.31. Show that  
\[
\lim_{z \to 0} \left( \frac{1}{\sin(z)} - \frac{1}{z} \right) = 0 
\]  
and  
\[
\lim_{z \to \infty} \left( \frac{1}{z} - \frac{1}{\sin(z)} \right) = \frac{1}{6}
\]  
(These are the limits we referred to in Example 8.23.)

8.32. Find the three Laurent series of  
\[
f(z) = \frac{3}{(1 - z)(z + 2)}
\]  
centered at 0, defined on the three regions \( |z| < 1, \, 1 < |z| < 2, \, \text{and} \, 2 < |z| \), respectively. (Hint: Use a partial fraction decomposition.)

8.33. Suppose that \( f(z) \) has exactly one zero, at \( a \), inside the circle \( C \), and that it has multiplicity 1. Show that  
\[
a = \frac{1}{2 \pi i} \int_C \frac{f'(z)}{f(z)} \, dz.
\]

8.34. Recall that a function \( f : G \to G \) is even if \( f(-z) = f(z) \) for all \( z \in G \), and \( f \) is odd if \( f(-z) = -f(z) \) for all \( z \in G \). Prove that if \( f \) is even (resp., odd), then the Laurent series of \( f \) has only even (resp., odd) powers.

8.35. Suppose \( f \) is holomorphic and not identically zero on an open disk \( D \) centered at \( a \), and suppose \( f(a) = 0 \). Use the following outline to show that \( \text{Re} \, f(z) > 0 \) for some \( z \in D \).  
(a) Why can you write \( f(z) = (z - a)^m g(z) \) where \( m > 0 \), \( g \) is holomorphic, and \( g(a) \neq 0 \)?  
(b) Write \( g(a) \) in polar coordinates as \( |g(a)| e^{i \theta} \) and define \( G(z) = |g(a)| e^{i \theta} (z - a)^m \). Why is \( \text{Re} \, G(z) > 0 \)?  
(c) Write \( z = a + r e^{i \phi} \) for \( 0 < r < \epsilon \).  
(d) Find a value of \( r \) so that \( \text{Re} \, f(z) \) has positive real part.
####################
File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf

Page: 179

Context: # CLASSIFICATION OF SINGULARITIES

on \( D[\alpha, R] \) and non-zero at \( \alpha_n > 0 \), and  
\( f(z) = \frac{g(z)}{z - \alpha} \) for all \( z \in D[\alpha, R] \).

But then \( \alpha \) is a pole of order \( m \) since

\[
\lim_{z \to \alpha} |f(z)| = \lim_{z \to \alpha} \left| \frac{g(z)}{(z - \alpha)^m} \right| = \lim_{z \to \alpha} \frac{|g(z)|}{|z - \alpha|^m} = \infty.
\]

We underline one feature of the last part of our proof:

## Corollary 9.6

Suppose \( f \) is holomorphic in \( D[\alpha, R] \). Then \( f \) has a pole at \( \alpha \) if and only if there exists a positive integer \( m \) and a holomorphic function \( g : D[\alpha, R] \to \mathbb{C} \) such that \( g(\alpha) \neq 0 \) and

\[
f(z) = \frac{g(z)}{(z - \alpha)^m}
\]

for all \( z \in D[\alpha, R] \).

If \( \alpha \) is a pole then \( m \) is unique.

### Proof

The only part not covered in the proof of Theorem 9.5 is uniqueness of \( m \). Suppose \( f(z) = \frac{(z - \alpha)^{m}}{g(z)} \) and \( f(z) = \frac{(z - \alpha)^{n}}{h(z)} \) both work, with \( m > n \). Then \( g(z) = (z - \alpha)^{m - n} h(z) \), and plugging in \( z = \alpha \) yields \( g(\alpha) = 0 \), violating \( g(\alpha) \neq 0 \).

### Definition

The integer \( m \) in Corollary 9.6 is the order of the pole \( \alpha \).

This definition, naturally coming out of Corollary 9.6, parallels that of the multiplicity of a zero, which naturally came out of Theorem 8.14. The two results also show that if \( f \) has a zero at \( \alpha \) of multiplicity \( m \) and only if \( j \) has a pole of order \( m \), we will make use of the notions of zeros and poles quite extensively in this chapter.

You might have noticed that the Proposition 9.5 did not include any results on essential singularities. Not only does the next theorem make up for this but it also nicely illustrates the strangeness of essential singularities. To appreciate the following result, we suggest meditating about its statement over a good cup of coffee.
####################
File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf

Page: 180

Context: ```markdown
# Theorem 9.7 (Casorati–Weierstrass)

If \( f \) is an essential singularity of \( r \) and \( r \) is any positive real number, then every \( u \in C \) is arbitrarily close to a point in \( f(B(z_0, r)) \). That is, for any \( x \in C \) and \( \epsilon > 0 \) there exists \( z \in B(z_0, r) \) such that \( |u - f(z)| < \epsilon \).

In the language of topology, Theorem 9.7 says that the image of any punctured disk centered at an essential singularity is dense in \( C \).

There is a stronger theorem, beyond the scope of this book, which implies the Casorati–Weierstrass Theorem 9.7. It is due to Charles Émile Picard (1856–1941) and says that the image of any punctured disk centered at an essential singularity misses at most one point of \( C \). (It is worth coming up with examples of functions that do not miss any point in \( C \) and functions that miss exactly one point. Try it!)

## Proof 

Suppose (by way of contradiction) that there exists \( w \in C \) and \( \epsilon > 0 \) such that for all \( z \in B(z_0, r) \)

\[
|u - f(z)| \geq \epsilon.
\]

Then the function \( g(z) = \frac{f(z) - w}{z - z_0} \) stays bounded as \( z \to z_0 \) and so 

\[
\lim_{z \to z_0} (z - z_0) g(z) = \lim_{z \to z_0} \frac{f(z) - w}{z - z_0} = 0.
\]

(Proposition 9.5(a) tells us that \( f \) has a removable singularity at \( z_0 \).) Hence

\[
\lim_{z \to z_0} \left| \frac{f(z) - w}{z - z_0} \right| = \infty
\]

and so the function \( f(z) - w \) has a pole at \( z_0 \). By Proposition 9.5(b), there is a positive integer \( n \) so that 

\[
\lim_{z \to z_0} (z - z_0)^{n+1} f(z) = \lim_{z \to z_0} (z - z_0)^n g(f(z)) = 0.
\]

Invoking Proposition 9.5 again, we conclude that the function \( f(z) \) has a pole or removable singularity at \( z_0 \), which implies the same holds for \( f(z) \), a contradiction.
```
####################
File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf

Page: 184

Context: ```markdown
## ISOLATED SINGULARITIES AND THE RESIDUE THEOREM

(b) We know by Proposition 9.8(b) that the Laurent series at \( z_0 \) looks like 

\[
f(z) = \sum_{k \geq 0} c_k (z - z_0)^k.
\]

But then 

\[
(z - z_0)^j f(z) = \sum_{k \geq j} c_k (z - z_0)^{k - j}
\]

is a power series, and we can use Taylor's formula (Corollary 8.5) to compute \( c_{-l} \).

It is worth noting that we are really coming full circle here; compare this proposition to Cauchy's Integral Formulas (Theorems 4.27 & 4.1 and Corollary 8.11).

### Example 9.12

The integral 

\[
\frac{\operatorname{exp}(z)}{\sin(z)}
\]

in Example 8.28 has poles of order 1 at 0 and \( \pi \). We thus compute 

\[
\operatorname{Res}_{z = 0} \left( \frac{\operatorname{exp}(z)}{\sin(z)} \right) = \lim_{z \to 0} \left( \frac{\operatorname{exp}(z)}{\sin(z)} \right) = \operatorname{exp}(0) \lim_{z \to 0} \frac{z}{\sin(z)} = 1
\]

and 

\[
\operatorname{Res}_{z = \pi} \left( \frac{\operatorname{exp}(z)}{\sin(z)} \right) = \lim_{z \to \pi} \left( (z - \pi) \frac{\operatorname{exp}(z)}{\sin(z)} \right) = \operatorname{exp}(\pi) \lim_{z \to \pi} \frac{z - \pi}{\sin(z)} = -e^{-\pi},
\]

confirming our computations in Example 8.28.

### Example 9.13

Revisiting Example 9.9, the function \( f(z) = \frac{1}{z^2} \) has a double pole at \( 0 \) with 

\[
\operatorname{Res}_{z=0} \left( \frac{\sin(z)}{z^2} \right) = \lim_{z \to 0} \frac{d}{dz} \left( \frac{2 \sin(z)}{z^2} \right) = \lim_{z \to 0} \left( \frac{2 \cos(z) - 2 \sin(z)}{z} \right) = 0,
\]

after a few iterations of 1'Hôpital's Rule. (In this case, it is simpler to read the residue off the Laurent series in Example 9.9.)

### Proposition 9.14

Suppose \( f \) and \( g \) are holomorphic at \( z_0 \), which is a simple zero of \( g \) (i.e., a zero of multiplicity 1). Then 

\[
\operatorname{Res}_{z \to z_0} \left( \frac{f(z)}{g(z)} \right) = \frac{f(z_0)}{g'(z_0)}.
\]
```
####################
File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf

Page: 189

Context: # ARGUMENT PRINCIPLE AND ROUCHÉ'S THEOREM

This theorem is of surprising practicality. It allows us to locate the zeros of a function fairly precisely. Here is an illustration.

## Example 2.19

All the roots of the polynomial \( p(z) = z^5 + z^4 + z^3 + z^2 + z + 1 \) have modulus less than two. To see this, let \( f(z) = z^5 + z^4 + z^3 + z^2 + z + 1 \).

For \( z \in [0, 2] \):

\[
|f(z)| \leq |z^5| + |z^4| + |z^3| + |z^2| + |z| + 1 = 16 + 8 + 4 + 2 + 1 = 31 < 32 = |z^5| \text{ for } z \in [\sqrt{2}, 2].
\]

So \( g \) and \( f \) satisfy the condition of Theorem 9.18. But \( f \) has just one root, of multiplicity 5 at the origin, whence

\[
Z(f) \cap (C[0, 2]) = Z(f + g, C[0, 2]) = Z(f, C[0, 2]) = 5.
\]

### Proof of Theorem 9.18

By (9.1) and the Argument Principle (Theorem 9.17):

\[
Z(f + g, r) = \frac{1}{2\pi i} \int_{\gamma} \frac{f + g}{f + g} \, \frac{(f + g)'}{f + g} \, dz = \frac{1}{2\pi i} \int_{\gamma} \frac{(f + g)'}{f + g} \, dz,
\]

where \( \gamma \) is a contour around \( r \).

We are assuming that \( |g| < |f| \) on \( \gamma \), which means that the function \( \frac{1}{f + g} \) evaluated on \( \gamma \) stays away from \( R_+ \). But then \( \log\left(\frac{1}{f + g}\right) \) is a well-defined holomorphic function on \( \gamma \). Its derivative is 

\[
\frac{(f + g)'}{f + g}.
\]

---

> *The Fundamental Theorem of Algebra (Theorem 5.1) asserts that \( p \) has \( n \) roots in \( C \). What's special about the theorem of Example 2.19 is that they all have modulus < 2. Note also that there is no general formula for composing roots of a polynomial of degree \( n \). Although for this \( n \) it’s not hard to find one root—and therefore all of them.*
####################
File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf

Page: 193

Context: # Argument Principle and Rouche's Theorem

## 9.21. Find the number of zeros of

1. \( 3z^2 - z \) in \( D[0, 1] \)
2. \( \{ z \} \in D[0, 1] \)
3. \( e^{-z} - 5 + 1 \) in \( \{ z \in \mathbb{C} : |z| \leq 2 \} \)

## 9.22. Give another proof of the Fundamental Theorem of Algebra (Theorem 5.11), using Rouche's Theorem 9.18.

*(Hint: If \( p(z) = z^n + a_{n-1}z^{n-1} + \ldots + a_0 \), let \( f(z) = a_n z^n \) and \( g(z) = a_{n-1}z^{n-1} + a_{n-2}z^{n-2} + \ldots + a_0 \), and choose \( r \) a circle that is large enough to make the condition of Rouche's theorem work. You might want to first apply Proposition 5.10 to \( g(z) \).)*

## 9.23. Suppose \( S \subset C \) is closed and bounded and all points of \( S \) are isolated points of \( S \). Show that \( S \) is finite, as follows:

(a) For each \( x \in S \), we can choose \( \varepsilon > 0 \) so that \( D[x, \varepsilon] \) contains no points of \( S \) except \( x \). Show that \( f \) is continuous. *(Hint: This is really easy if you use the first definition of continuity in Section 2.1.)*

(b) Assume \( S \) is non-empty. By the Extreme Value Theorem A.1, \( S \) has a minimum value, \( r_0 > 0 \). Let \( r = r_0/2 \). Since \( S \) is bounded, it lies in a disk \( D[0, M] \) for some \( M > 0 \). Show that the small disks \( D[x, r] \) for \( x \in S \) are disjoint and lie in \( D[0, M+r] \).

(c) Find a bound on the number of such small disks. *(Hint: Compare the areas of \( D[x, r] \) and \( D[0, M + r] \).)*
####################
File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf

Page: 195

Context: # Binomial Coefficients

(4) Evaluate \(\sum_{n=1}^{\infty} \frac{1}{n^2}\).

(5) Repeat the exercise with the function \(f(z) = \frac{z^{-1}}{e^{z} - 1}\) to arrive at an evaluation of 

\[
\sum_{n=1}^{\infty} \frac{(-1)^{n}}{n^2}.
\]

(Hint: To bound this function, you may use the fact that \(\frac{1}{e^z - 1} = 1 + \cos(z_n)\).)

(6) Evaluate \(\sum_{n=1}^{\infty} n\) and \(\sum_{n=1}^{\infty} n^{-1}\).

We remark that, in the language of Example 7.21, you have computed the evaluations \(\zeta(2)\) and \(\zeta(4)\) of the Riemann zeta function. The function \(\zeta'(s) := \sum_{n=1}^{\infty} \frac{1}{n^s}\) is called the alternating zeta function.

## 10.2 Binomial Coefficients

The binomial coefficient \(\binom{n}{k}\) is a natural candidate for being explored analytically, as the binomial theorem states:

\[
(x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{k} y^{n-k}
\]

(for \(x, y \in \mathbb{C}\) and \(n \in \mathbb{Z}_{\geq 0}\)) tells us that \(\binom{n}{k}\) is the coefficient of \(x^{k}\) in \((x + y)^{n}\). You will derive two simple identities in the course of the exercises below.

1. Convince yourself that 

\[
\binom{n}{k} = \frac{1}{2\pi i} \oint_{C} \frac{(x + 1)^{n}}{z^{k + 1}} dz,
\]

where \(C\) is any simple closed piecewise smooth path such that \(0\) is inside \(r\).

2. Derive a recurrence relation for binomial coefficients from the fact that 

\[
\frac{1}{z^{n}} = \frac{1}{2^{n}} \cdot \left( \text{Multiply both sides by } \frac{1}{z^{n}}\right).
\]

3. Now suppose \(z\) is such that \(|z| < 1/4\). Find a simple path \(P\) surrounding the origin such that 

\[
\sum_{n=0}^{\infty} \left( \frac{(z + 1)^{k}}{k} \right) 
\]

converges uniformly on \(y\) as a function of \(\epsilon\). Evaluate this sum.
####################
File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf

Page: 196

Context: ```
Keeping \( x \) and \( y \) from (3), convince yourself that

\[
\sum_{k=0}^{\infty} \left( \frac{2k}{k} \right) k^k = \frac{1}{2\pi i} \int \left( \frac{x^{n+1}}{z^{n+1}} \right) dz,
\]

use (3) to interchange summation and integral, and use the Residue Theorem 9.10 to evaluate the integral, giving an identity for \( \sum_{k=0}^{\infty} \left( \frac{1}{2} \right)^k \).

### 10.3 Fibonacci Numbers

The Fibonacci\(^1\) numbers are a sequence of integers defined recursively through

\[
f_0 = 0,
\]
\[
f_1 = 1,
\]
\[
f_n = f_{n-1} + f_{n-2} \quad \text{for } n \geq 2.
\]

Let \( F(z) = \sum_{n=0}^{\infty} f_n z^n \).

1. Show that \( F \) has a positive radius of convergence.

2. Show that the recurrence relation among the \( f_n \) implies that \( F(z) = \frac{z}{1 - z - z^2} \).  
   (Hint: Write down the power series for \( F(z) \) and \( z^2 F(z) \) and rearrange both so that you can easily add.)

3. Verify that

\[
\text{Res}_{z=0} \left( \frac{1}{z(1 - z - z^2)} \right) = f_0.
\]

4. Use the Residue Theorem 9.10 to derive an identity for \( f_n \). (Hint: Integrate

\[
\frac{1}{z(1 - z - z^2)}
\]

around \( C[0, R] \) and show that this integral vanishes as \( R \to \infty \).

\(^1\) Named after Leonardo Pisano Fibonacci (1170–1250).
```
####################
File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf

Page: 197

Context: # The Coin-Exchange Problem

(5) Generalize to other sequences defined by recurrence relations, e.g., the Fibonacci numbers

- \( t_0 = 0 \)
- \( t_1 = 0 \)
- \( t_2 = 1 \)
- \( t_n = t_{n-1} + t_{n-2} + t_{n-3} \) for \( n \geq 3 \).

## 10.4 The Coin-Exchange Problem

In this exercise, we will solve and extend a classical problem of Ferdinand Georg Frobenius (1849–1917). Suppose \( a \) and \( b \) are relatively prime positive integers, and suppose \( r \) is a positive integer. Consider the function

\[ f(c) = \frac{1}{(1 - z)(1 - z^2)z^r} \]

1. Compute the residues at all nonzero poles of \( f \).

2. Verify that \( Res_{z = 0}(f) = N(r) \), where
   \[ N(r) = \{(m, n) \in \mathbb{Z}^2 : m \geq 0, n \geq 0, m + nb = r\} \].

3. Use the Residue Theorem, Theorem 9.10, to derive an identity for \( N(r) \). (Hint: Integrate \( f \) around \( [0, R] \) and show that this integral vanishes as \( R \to \infty \).)

4. Use the following three steps to simplify this identity to

\[ N(t) = \frac{1}{ab} \left( \frac{b-1}{a} - \left( \frac{a-1}{b} \right) \right) + 1. \]

*This means that the integers do not have any common factor.*
####################
File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf

Page: 198

Context: # Discrete Applications of the Residue Theorem

Here, \(\{x_n\}\) denotes the fractional part of \(x\): \(a_n \equiv 1 \,(\text{mod } b)\), and \(b \cdot 6 \equiv 1 \,(\text{mod } d)\).

## (a) 

Verify that for \(b = 1\),

\[
N(n) = \left| \{(m,n) \in \mathbb{Z}^2 : m \geq 0, n + n \leq s \} \right| = \left| \{ m \in \mathbb{Z} : m \geq 0, m \cdot s \leq 1 \} \right| = \left[ \frac{0}{d} \right] n^2 = \left\lfloor \frac{1}{d} \right\rfloor.
\]

## (b)

Use this together with the identity found in (3) to obtain:

\[
-\frac{1}{2} \sum_{k=1}^{\infty} \frac{1}{(1 - 2e^{2\pi i k})\cdot 2k} = \left( -\frac{1}{2} + \frac{1}{2} \right) + \frac{1}{z} \,
\]

## (c)

Verify that

\[
\sum_{k=1}^{\infty} \frac{1}{(1 - 2e^{2\pi i k})} = \sum_{k=1}^{\infty} \frac{1}{(1 - 2e^{2\pi i k})}
\]

## (5)

Prove that \(N(a - b) = 0\), and \(N(n) > 0\) for all \(x > a\) and \(x < b\).

### Historical Remark

Given relatively prime positive integers \(a_1, a_2, \ldots\), let’s call an integer \(t\) representable if there exist nonnegative integers \(m_1, m_2, \ldots, m_k\) such that 

\[
t = m_1 a_1 + m_2 a_2 + \cdots + m_k a_k.
\]

(There are many scenarios in which you may ask whether or not \(t\) is representable, given fixed \(a_1, a_2, \ldots, a_k\); for example, if the \(a_i\) are co-prime, this question asks whether you can give exact change for \(t\). In the late 19th century, Frobenius raised the problem of finding the largest integer that is not representable. We will call this largest integer the Frobenius number \(g(a_1, a_2, \ldots, a_k)\). It is well known (probably at least since the 1880s, when James Joseph Sylvester (1814–1897) studied the Frobenius problem) that \(g(a_1, a_2) = a_1 a_2 - a_1 - a_2\). You verified this result in (5). For \(x > 2\), there is no closed formula for \(g(a_1, a_2, \ldots, a_k)\). The formula in (4) states that

The fractional part of a real number \(x\) is loosely speaking the part after the decimal point. More thoroughly, the principal integer function of \(x\), denoted by \(\lfloor x \rfloor\), is the largest integer not exceeding \(x\). The fractional part is defined as \( \{x\} = x - \lfloor x \rfloor \).

**"This means that \(c\) is an integer such that \(c = a + 1 + b + k\) for some \(k \in \mathbb{Z}\)."**
####################
File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf

Page: 199

Context: # Dedekind Sums

This exercise outlines one more nontraditional application of the Residue Theorem 9.10. Given two positive, relatively prime integers \( a \) and \( b \), let 

\[ 
f(z) = \cot(\pi z) \cot(b \pi z). 
\]

1. Choose an \( R > 0 \) such that the rectangular path \( \gamma_R \) from \( -1 - iR \) to \( -1 + iR \) to \( 1 + iR \) to \( 1 - iR \) back to \( -1 - iR \) does not pass through any of the poles of \( f \).

   (a) Compute the residues for the poles of \( f \) inside \( \gamma \). Hint: Use the periodicity of the cotangent and the fact that 

   \[
   \cot z = \frac{1}{z} - \frac{1}{3} z + \text{higher-order terms}. 
   \]

   (b) Prove that \( \lim_{R \to \infty} \int_{\gamma_R} f \, dz = -2i \) and deduce that for any \( R > 0 \)

   \[
   \int_{-1}^{1} f \, dz = -2i. 
   \]

2. Define 

\[
t(a, b) = \frac{1}{48} \sum_{k=1}^{b-1} \cot \left( \frac{\pi k a}{b} \right) \left( \frac{k}{b} \right). 
\]
   
   Use the Residue Theorem 9.10 to show that 

\[
t(a, b) + t(b, a) = -\frac{1}{4} + \frac{1}{12} \left( \frac{a}{b} + \frac{b}{a} \right). 
\]

3. Generalize (10.1) and (10.2).
####################
File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf

Page: 200

Context: # Discrete Applications of the Residue Theorem

## Historical Remark

The sum in (10.1) is called a Dedekind* sum. It first appeared in the study of the Dedekind \( \eta \)-function

\[
\eta(z) = \exp\left(\frac{\pi i}{12} z\right) \prod_{n=1}^{\infty} \left(1 - \exp(2 \pi i n z)\right)
\]

in the 1870s and has since intrigued mathematicians from such different areas as topology, number theory, and discrete geometry. The reciprocity law (10.2) is the most important and famous identity of the Dedekind sum. The proof that is outlined here is due to Hans Rademacher (1892–1969).

---

*Named after Julius Wilhelm Richard Dedekind (1831–1916).
####################
File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf

Page: 201

Context: # Appendix: Theorems From Calculus

First, it is necessary to study the facts, to multiply the number of observations, and then later to search for formulas that connect them so as to discern the particular law governing a certain class of phenomena. In general, it is not until after these particular laws have been established that one can expect to discover and articulate the more general law that connects them by bringing a multitude of apparently very diverse phenomena together under a single governing principle.  
_Augustin Louis Cauchy (1789–1857)_

Here we collect a few theorems from real calculus that we make use of in the course of the text.

## Theorem A.1 (Extreme-Value Theorem)

Suppose \( K \subset \mathbb{R}^n \) is closed and bounded and \( f: K \to \mathbb{R} \) is continuous. Then \( f \) has a minimum and maximum value, i.e.,

\[
\min_{x \in K} f(x) \quad \text{and} \quad \max_{x \in K} f(x)
\]

exist in \( \mathbb{R} \).

## Theorem A.2 (Mean-Value Theorem)

Suppose \( I \subset \mathbb{R} \) is an interval. \( f: I \to \mathbb{R} \) is differentiable, and \( x, x + h \in I \). Then there exists \( 0 < c < h \) such that

\[
f(x + h) - f(x) = f'(c)(x + h - x).
\]

Many of the most important results of analysis concern combinations of limit operations. The most important of all calculus theorems combines differentiation and integration (in two ways):
####################
File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf

Page: 204

Context: # Solutions to Selected Exercises

Well here's another clue for you all.  
John Lennon & Paul McCartney ("Glass Onion," *The White Album*)

## 1.1 
(a) \( 7 - i \)  
(b) \( 1 - i \)  
(c) \( -11 - 2i \)  
(d) \( -2 + 3i \)  

## 1.2 
(b) \( \frac{1}{3} - \frac{1}{2} i \)  

## 1.3 
(a) \( \sqrt{5}, -2 \)  
(b) \( 5\sqrt{5}, 5 - 10i \)  
(c) \( \frac{\sqrt{2 - 1} + \sqrt{(2 + \sqrt{9})}}{5} \)  
(d) \( 8 \)  

## 1.4 
(a) \( 2^{1/2} i \)  
(b) \( 2\sqrt{5}i^2 e^{2} \)  
(c) \( e^{5} \)  

## 1.5 
(a) \( -1 + i \)  
(b) \( 34i \)  
(c) \( -1 \)  
(d) \( 2 \)  

## 1.9 
\( e^{i \pi} = -1 \)  

## 1.11 
(a) \( x = e^{i k}, k = 0, 1, \ldots, 5 \)  
(b) \( x = 2 e^{i t^{5/4}}, k = 0, 1, 2, 3 \)  

## 1.18 
\( \cos \frac{\pi}{5} \) and \( \sin \frac{3\pi}{5} \)  

## 1.33 
For instance:  
(a) \( r(t) = (1 + i)t + e^{it}, 0 \leq t < 2\pi \).  
(b) \( r(t) = (2 - i)(1 - i)(2i) + (2)(1 + i + 3i), 0 \leq t \leq 1 \).  
(c) \( r(t) = 34e^{-i \pi} = 34e^{-i(t)}, 0 \leq t \leq \pi \).  

## 2.2 
(a) \( 1 + i \)  

## 2.18 
(a) differentiable and holomorphic in \( C \) with derivative \( e^{-z} e^{z} \)  
(b) nowhere differentiable or holomorphic  
(c) differentiable only on \( x + iy \in C : x = y \) with derivative \( 2x \), nowhere
####################
File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf

Page: 207

Context: ```
7.26 
(a) \(\sum_{k=0}^{\infty} (-1)^k \frac{z^k}{(2k)!}\)  
(b) \(\sum_{k=0}^{\infty} (-1)^k \frac{z^{2k}}{(2k)!}\)  
(c) \(\sum_{k=0}^{\infty} (-1)^k \frac{2^{k-3}}{2k^2 + 1} z^{2k}\)  
(d) \(\sum_{k=0}^{\infty} (-1)^k \frac{(2^k)(k!)}{(2k)!} z^k\)  

7.28 
\(\sum_{k=0}^{\infty} (-1)^k (z - 1)^k \)  
\(\sum_{k=0}^{\infty} \frac{(-1)^k}{(k+1)} \)  

7.33 \(\forall \, r < 1, \, |f| = 1\) if \(|z| = 1\), and \(|f| \leq 1\) if \(|z| < 1\)  
\(\text{(b)} \, |f| \leq 1 \, (d) \, 1\)  

7.34 
(a) \(e^{z^2} \quad (b) \quad \frac{1}{(z-2)} \quad (c) \quad \frac{2z}{(2-z)}\)  

8.1 
(a) \(C, \, z \in \mathbb{C} : \, |z| < r\) for any \(r\)  
(b) \(\{z \in \mathbb{C}: |z-3| > 1\}\), \(\{z \in \mathbb{C} : |z-3| < R\}\) for any \(1 < r < R\)  

8.2 \(\sin(z)\)  
\(z = -3\)  
\(\implies k = 4\)  

8.14 The maximum is \(3\) (attained at \(z = \pm 1\)), and the minimum is \(1\) (attained at \(z = 1\)).  

8.17 One Laurent series is \(\sum_{n=0}^{\infty} (-2)^{n}(z-1)^{-2-n}\), converging for \(|z| > 1\).  
8.18 The Laurent series is \(\sum_{n=0}^{\infty} (-2)^{n}(z-2)^{-3-n}\), converging for \(|z| > 2\).  
8.19 One Laurent series is \(3-(z-1)^n\), converging for \(z \neq 1\).  

8.25 
\(\sum_{k=0}^{\infty} (-1)^k \frac{z^k}{(2k)!}\)  

8.36 
(a) One Laurent series is \(\sum_{n=0}^{\infty} \frac{(-1)^k}{(1 + (-2)^{n})}\), converging for \(0 < |z| < 4\).  

9.5 
\(\frac{2\pi}{3}\) \quad \(\frac{27\pi}{4} - \frac{2\pi}{17} - \frac{\pi}{3} \) \quad \(6\) \quad \(2\pi i\) \quad \(f(0)\)  

9.21 (a) \(0\) \quad (b) \(1\) \quad (c) \(4\)
```
####################
File: A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf

Page: 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  
```
##########

"""QUERY: in 500 words can you tell me what is algebra?"""

Consider the chat history for relevant information. Use all information included. Use as much tokens as needed. Important: If you find information separated by a | in the context, it is a table formatted in Markdown.Important: Take a look at the QUERY and only the QUERY. If this is vague or unclear, please ignore everything and ask a follow-up question instead!


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 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 8, 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 11, 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 17, 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 19, 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 22, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 23, 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 38, 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 - Page 42, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 44, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 46, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 48, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 49, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 52, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 54, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 57, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 60, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 62, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 63, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 65, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 71, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 74, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 76, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 80, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 81, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 89, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 94, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 104, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 105, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 107, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 108, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 109, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 110, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 112, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 113, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 115, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 118, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 122, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 124, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 127, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 129, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 130, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 131, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 134, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 138, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 140, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 141, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 143, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 144, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 148, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 153, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 158, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 159, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 160, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 163, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 165, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 170, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 173, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 179, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 180, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 184, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 189, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 193, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 195, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 196, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 197, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 198, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 199, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 200, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 201, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 204, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 207, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 208, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 209, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 210, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 211, A%20First%20Course%20in%20Complex%20Analysis%20-%20Matthias%20Beck%2C%20Gerald%20Marchesi%2C%20Dennis%20Pixton%2C%20Lucas%20Sabalka%20%28PDF%29.pdf - Page 212
==================================================
**Elapsed Time: 5.86 seconds**
==================================================


FINAL ANSWER 
Answer: Algebra is a fundamental branch of mathematics that focuses on the study of symbols and the rules for manipulating these symbols to solve equations and express mathematical relationships. At its core, algebra involves the use of variables, which are symbols that represent unknown values, and constants, which are fixed values. The primary operations in algebra include addition, subtraction, multiplication, and division, which can be applied to both numbers and variables. 

One of the key concepts in algebra is the formation of expressions, which are combinations of variables and constants connected by mathematical operations. For example, the expression \(2x + 3\) consists of the variable \(x\), the constant \(2\), and the constant \(3\). Algebra allows for the manipulation of these expressions to simplify them or to solve for the unknown variable. This is often done through the process of solving equations, which are mathematical statements asserting the equality of two expressions. For instance, in the equation \(2x + 3 = 7\), the goal is to find the value of \(x\) that makes the equation true.

Algebra also introduces the concept of functions, which are relationships between sets of values. A function takes an input (or variable) and produces an output based on a specific rule. For example, the function \(f(x) = 2x + 3\) takes an input \(x\), multiplies it by \(2\), and then adds \(3\) to produce the output. Functions can be represented graphically, allowing for visual interpretation of their behavior and properties.

In addition to basic operations and equations, algebra encompasses more advanced topics such as polynomials, which are expressions that involve variables raised to whole number powers, and rational expressions, which are ratios of polynomials. Algebraic structures, such as groups, rings, and fields, provide a framework for understanding the properties and relationships of various algebraic systems. For example, the set of real numbers forms a field, where addition and multiplication are defined, and every non-zero element has a multiplicative inverse.

Algebra is not only a theoretical discipline but also has practical applications in various fields, including science, engineering, economics, and computer science. It serves as a foundational tool for solving real-world problems, modeling relationships, and analyzing data. For instance, algebraic equations can be used to calculate trajectories in physics, optimize functions in economics, and develop algorithms in computer programming.

Moreover, algebra is essential for higher-level mathematics, including calculus and linear algebra, which build upon algebraic principles to explore more complex mathematical concepts. The study of algebra begins in early education and continues to evolve through advanced studies, making it a crucial component of mathematical literacy and critical thinking skills.

In summary, algebra is a versatile and powerful branch of mathematics that provides the tools for expressing and solving problems involving numbers and variables. Its principles are foundational to many areas of mathematics and its applications extend far beyond the classroom, influencing various aspects of everyday life and professional practice.
==================================================
**Elapsed Time: 16.15 seconds**
==================================================