Mathematics with Complex Variables: A Visual Approach

Course: Math 3614, Fall 2016

Time & Place: MW 3:30 – 5:00 pm (SC 201)

Instructor: Jorge Basilio (jbasilio@sarahlawrence.edu)

Office Hours:

By appointment

Handouts

Introduction to Calculus and Analysis

Courant & John

Courant and John pages 1-12 only containing sections 1.1a, 1.1b, and 1.1c.

Cultural context: Introduces the subject of analysis with real variables. It is important to read this since it explains what a 'real number,' and surprisingly enough, and will lay the foundation for more advanced topics in Complex Analysis.

What is Mathematics?

Courant & Robbins

This is the introduction to the excellent book of the same title by Courant & Robbins. In these four little pages you will get a great idea of how modern mathematicians view mathematics. As fresh today as it was in 1941 when it was first published.

On Writing Proofs

On Reading Mathematics

Suggestions for Effectively Reading Mathematics

Reaction/Reflection Pieces

Guidelines

Announcements

Final Conference Work Submission Instructions

It’s that exciting time of the semester where we are almost done and we get to compile our efforts into a final document. To guarantee the final submission looks professional, please follow these instructions. I can’t wait to read your work!

Final Due date: Tuesday, December 20, by 5 pm on your Google Drive submitted as a single PDF document (instructions may easily be found on the web).
What to submit:
Cover page
Table of Contents
A paragraph (or two) that explains what the conference project was, what we discussed in meetings, and why you chose this topic.
Main Work (this is your final paper)
References
Cover page: Please choose an appropriate title for your conference work. Please have me approve the title before the final submission. This is the title that I will submit in my evaluation. Please also include: Name, Date, Course Number, Course Name, Instructor
The "Main Work" might have the same or a different title than the Conference Project.
Format: Please write using 12 point font (either Georgia or Arial font), double-spaced, with 1 inch margins. Please include proper citations which you will include in the “References” page but you can follow any of the standard ways to do so (APA, MLA,...).

Other notes:
(i) Make sure that the font is the same for the entire document.
(ii) Typing this up using LaTeX is highly recommended, but I decided not to require it.
If you want info on LaTeX, see: Professor Ethan Bloch's LaTeX website

If you have any questions, please do not hesitate to ask!

Errata for HHW7

There's a few typos spotted on the questions. I'll post them briefly here and when I get back home I'll upload a corrected PDF:
Problem 10(a): the denominator should be

$1+z$ not

$1-z$ Problem 10(c): Should be

$f_1(f_2(f_3(z))))$ . Problem 7 after part (d), before (e): the formulas for the dot product are incorrect! The correct formulas should read:

$|z||proj_z(w)|$ or

$|w||proj_w(z)|$ . Let me know if you find any other typos!

HHW Key

regular=turn-in
( )=practice, don't turn-in
*=optional, prepare to discuss some in conference

Videos & Readings

Videos & Readings for first day of class added to the schedule

Hello!

Welcome to "Mathematics with Complex Variables"!
I'll post important information here.
I posted the course syllabus below--for students who have not taken a course with me please read it carefully.
More info coming soon...

Week	Class	Hand-in homework
1 {9/5 & 9/7}	{M} Class introductions; Read: [Img] Ch 1; {W} Read: [Img] Ch 2 & Ch 3	HHW1 due 9/10 @ 5pm
2 {9/12 & 9/14}	{M} Read [CA] 1.1, 1.2; {W} Read [CA] 1.3, 1.4	HHW2 due 9/17 @ 5pm
3 {9/19 & 9/21}	{M} Read [CA] 1.5, 1.6*, 2.1; {W} Read [CA] 2.2, 2.3	HHW3 due 9/24 @ 5pm § 1.3: (7), 9, 13, 19, (23), 25, (27), 34, 35, 37, 28, 41, 48, 45, 47; § 1.4: 4, 5, 17, (18), 19, 20, 21, 24, 29, 31, 32
4 {9/26 & 9/28}	{M} Read [CA] 2.3, 2.4; {W} Read [CA] 2.5, 2.6*	HHW4 due 10/1 @ 5pm § 1.5: (7), 9, 13, 19, (23), 25, (27), 34, 35, 37, 28, 41, 48, 45, 47; § 1.6: 1, 3, 5, 7, 9, 11, 29, 30; § 2.1: 3, (6, 8, 9), 14, (19), 20, (21), 27, (29c), 30, 32, 33, 36
5 {10/3 & 10/5}	{M} Read [CA] 3.1, 3.2; {W} Read [CA] 3.3, 3.4, 3.5, [VCA] Ch 4*	HHW5 due 10/8 @ 5pm § 2.2: (5, 7, 11, 12, 13), 15, 17, (21), 22, 23, (25), 27, 30, 31; § 2.3: 5, (9, 11), 12, 15, (18, 19), 21, 23, 24, 25, 31, 32, 33, 35, 37; § 2.4: 11, 13, 17, 21, 23, 25, 29, (31, 37,), 41, 47, 49*, 51
6 {10/10 & 10/12}	{M} Read [CA] 4.1, [VCA] 2.IV-2.VII; {W} Read [CA] 4.2, 4.3	HHW6 due 10/16 @ 5pm § 2.5: 3, 4, 9, 13, 15, (16, 17, 19), 21, 22, 25, 27; § 2.6: (1-43 odd); § 3.1: 2, 3, 11, 15, 17, (19), 20, 21, 28, 31, 37, 49, 50, 51, 54
7 {10/17 & 10/19}	{M} October Study Days -- No meetings!; {W} Review [CA] Ch 1 -- 4	HHW7 due 10/21 @ 5pm § 3.2: 5, 6, 8, 15, 19, 21, 27, 35, 37; § 3.3: 1, 3, 5, 11, 15, 17, 18, 23, 24, 25, 27, 29 § 3.4: 4, 6, 7, 11, 13, 14, 17, 18, 23
7	HHW 8 (Take-home Midterm)	10/20 @ 9 pm
8 {10/24 & 10/26}	{M} Read [CA] 5.1; {W} Read [CA] 5.2	HHW9 due 10/31 @ 5pm § 4.1: 3,(4, 5), 7, (8), 11, 15, 17, 19, 23, 31, 34, 35, 42, 43, (44), 45, 50, (51, 52, 55); § 4.2: (3), 4, 9, 10, (11), 13, 17, 22; § 4.3: 2, 9, 29, 37, 43, 44, 53; § 4.5: 4.5: 9, 10, 12*
9 {10/31 & 11/2}	{M} Read [CA] 5.3; {W} Read [CA] 5.4	HHW10 due 11/4 @ 5pm § 5.1: (9,11,17,19),25,27,33; § 5.2: (3,5),7,(9,15),19,20,(23),25,27,29,33
10 {11/7 & 11/9}	{M} Student presentations of Problems from § 5.1 & 5.2; {W} Student presentations of Problems from § 5.3 & 5.4	HHW11 due 11/11 @ 5pm § 5.3: 3,5,9,11,17,21,25,27,29,30,31,34; § 5.4: 10,11,12,15,25,29*
11 {11/14 & 11/16}	{M} Read [CA] 5.5; {W} Proofs from 5.5	HHW12 due 11/20 @ 5pm § 5.5: 5,9,11,17,19,21,23,25, 26, 27(c),28, 29, 33
12 {11/21 & 11/23}	{M} Read [CA] 6.1; {W} Thanksgiving -- No Meetings!	—
13 {11/28 & 11/30}	{M} Read [CA] 6.2, 6.3; {W} Read [CA] 6.4, 6.5	HHW13 due 12/5 @ 5pm § 6.1: (15, 19, 23,) 25, 29, 32, 33, 35, 36, 38, 39, 41, 42, 47, 48; § 6.2: (1), 3, 5, 11, (15,) 21, 24, 25, 40,41,42,43; § 6.3: (5), 7, 8, 9, 10, (13, 14, 15, 16,) 25, 34
14 {12/5 & 12/7}	{M} Read [CA] 6.6; {W} Proof of Cauchy-Goursat's Theorem	HHW14 due 12/10 @ 5pm § 6.4: 1, 3, 5, 7, 9, 11, 15, 27, 31, 32, 33, 34, 35; § 6.5: 1-17 odd, 17, 18, 19, 21, 24, 25, 26, 30
15 {12/12 & 12/14}	{M} Student presentations of Conference Work; {W} Student presentations of Conference Work	—
15	HHW 15	12/18 @ 9 pm

Course Policies

Please consult the Course Syllabus for a more detailed description.

Prerequisites

Successful completion of Calculus II or the equivalent (a score of 4 or 5 on the Calculus BC AP test) is required; completion of an intermediate-level mathematics course (e.g. Multivariable Calculus, Linear Algebra, or Discrete Mathematics) is recommended.

Hand-in Homework

There will be homework assignments consisting of various types of problems or tasks. Some might be straight-forward calculations, others might me short-essays and free-response. You are encouraged to work with other students in solving the homework problems, but you should write your own solutions, and you must acknowledge anyone that you work with. Your solutions should be written clearly and in complete sentences, with enough detail that another student in the class would be able to follow your reasoning.

Required Texts

[CA] Complex Analysis: A First Course with Applications, Third Edition, by Zill and Shanahan.
[WA] WebAssign Access (for online HW).
An Imaginary Tale: The story of $\sqrt{-1}$ , by Paul Nahin.
[VCA] Visual Complex Analysis, by Tristan Needham OPTIONAL!

A few comments about the required texts.

[CA] will be our main text and we'll cover almost all of it. Students are not required to purchase a physical copy of the book as an electronic version is provided at no additional cost with a purchase of access to WebAssign--Note bene: access to the ebook ends after the semester ends! However, I strongly recommend students buy a physical copy of the textbook.
WebAssign access can be purchased directly online (with a credit card) or with the textbook (if bought with the textbook, don't lose this key code''!!!)
We will begin the course by reading the first three chapters from "An Imaginary Tale" in the first week of class so be sure to have a copy before the first day of class! A copy will be place on reserve at the library. After the first three chapters, you can read the other chapters for enjoyment in like a day or two after the semester is done or just keep reading it before jumping into the official textbook by Zill & Shanahan. Chapter 4 is really charming and hopefully will get you hooked on the subject. You can skip Ch 5, 6, & 7. If you want to preview the theory of Complex Analysis early, then read/skim chapter 7. Let anything tricky wash over you ;-)
I will provide selections from [VCA] via MySLC. A copy will be placed on reserve at the library so I recommend you skim through it at the library first. If you fall in love, by all means, buy it and maybe study from it in conference or after the basic course is done. This book has a lot of material. Enough for two semesters, and maybe more. It can be frustrating sometimes so be patient and find comfort in the fact that everyone probably struggles with this book early. Finally, I'll note that there are lots of potential topics for conference work!

Additional Texts

Calculus with Complex Variables is a vast (and fun!) subject. There's lots of ways to teach it and lots of resources out there.
Here's a few:

A First Course in Complex Analysis, Beck et al. Free online book.
Nice lecture notes from a professor at Georgia Tech. Free online book.
A nice book from a professor at UIUC. Free online book.
An Introduction to Complex Analysis for Engineers Michael D. Alder Free online book.

What is this class?

Once described as "that amphibian between existence and nonexistence" by Leibniz, the so-called complex numbers, $z=x+iy$ , where $x$ and $y$ are real numbers and $i=\sqrt{-1}$ is the imaginary unit, were met with suspicion and hostility for almost two and a half centuries. These numbers have since proven to have a profound impact on the whole of mathematics. Once accepted, however, the development of a beautiful new theory of how to do calculus with such numbers was astonishingly rapid going from birth (in 1814) to maturity (in 1851) in less than 40 years! After this intense period of investigation, any lingering suspicions held by the scientific community over the reality of complex numbers were subsequently squashed due to the amazing utility of this new calculus'' in mathematics, physics, engineering, and elsewhere.

We'll learn the meaning of complex multiplication, exponentiation, as in Euler's famous equation $e^{i\pi}= -1$ , and the associated geometry of these numbers. We'll study complex functions and their power series, and learn the many deep properties of M\"{o}bius transformations. We'll explore differentiation of complex functions and learn how to integrate them in the complex plane. We'll see the easiest proof of the Fundamental Theorem of Algebra, which says that every algebraic equation has a solution as long as you allow complex numbers. Numerous applications to other fields of mathematics (such as number theory or non-euclidean geometry), physics, and engineering can also be explored in seminar and in conference work according to the tastes and wishes of the students.

Thematic Outline

Pre-Course

Watch A Whirwind Tour of Complex Analysis (5 short videos) on YouTube:
Watch before semester starts, then re-watch & re-watch & re-watch :-)

Historical Introduction

Chapter 1, 2, & 3 of "An Imaginary Tale" [Img] in the first week.

[CA] Textbook

Chapters 1: Complex Numbers and the Complex Plane
Chapters 2: Complex Functions and Mappings
Chapters 3: Analytic Functions
Chapters 4: Elementary Functions
Midterm on Chapters 1-4
Chapters 5: Integration in the Complex Plane
Chapters 6: Series and Residues
Chapters 7: Conformal Mappings (time permitting)
Final on Chapters 5-7 (take-home)

Postscript: A Brave new World with Complex Numbers!

Student Conference presentations.