Syllabus
… imaginäre Wurzeln als feine und wunderbare Zuflucht des göttlichen Geistes, beinahe ein Zwitterwesen zwischen Sein und Nichtsein.
… imaginary numbers are a fine and wonderful refuge of the divine spirit, almost an amphibian between being and non-being.
—Gottfried Wilhelm Leibniz (1702)
Overview
Content
This course is an introduction to complex analysis, the branch of mathematics which studies complex numbers and their functions. In spite of the mysticism with which complex numbers were once viewed (eg, by Leibniz in the quote above), they have important applications to and connections with a wide variety of other disciplines, including algebraic geometry, number theory, quantum mechanics, electrical engineering, and more.
Topics we will cover include: the geometry and arithmetic of the complex numbers, holomorphic functions and the Cauchy-Riemann equations, Möbius transformations, power series and analyticity, the complex exponential and logarithms, path integrals, Cauchy’s theorem, Cauchy’s integral formula, and residues.
Prerequisites
The course will assume that you are able to compose formal mathematical proofs (MA251), and that you have a working knowledge of calculus and real analysis (MA126–127, MA204, MA375). On a few occasions, some facility with linear algebra (MA220) may also be useful. If you are uncertain about whether or not you meet these prerequisites, feel free to get in touch!
Course Structure Philsophy
In the long run, more important than learning any particular piece of math is learning how to learn math independently (in pedagogical jargon, that’s “how to be a self-regulated learner of math”). Research shows that the following three things are key aspects of this:
Active reading. Reading math is very different from other kinds of reading. You cannot read a math textbook the same way you’d read a novel for pleasure if you want to get anything out of it. You have to stop constantly as you’re reading math. Try to work out examples yourself, instead of just reading through them. Doodle pictures to make sure you have some kind of a picture in your head of what’s going on. Formulate precise questions about things you don’t understand.
Reciprocal teaching. Talking to your peers about math is incredibly important. If you don’t understand a particular concept and ask your peers, you’re much more likely to get an explanation that you actually find helpful. If you think you do understand a particular concept and help a peer who’s struggling, you’ll almost certainly find that the process of explaining the concept to your peer will solidify your own understanding of it.
Metacognition. A key part of learning how to learn is reflecting on your learning and taking the time to ask yourself questions about your learning. What parts of your study habits are working well for you? What parts aren’t working and how can you change these parts?
All three of these are built into the way the course is structured, and it would benefit you to bear these things in mind as you go through the course.
Course Materials
Textbook
The required textbook for the course is freely available: A First Course in Complex Analysis, Version 1.54, by Beck, Marchesi, Pixton, and Sabalka.
Optional, but highly recommended, is Needham’s Visual Complex Analysis. It contains some immensely useful discussions to help develop geometric intuition alongside interesting historical vignettes.
Discord
There is a Discord server for our class. This will be a (virtual) space for you to ask and answer questions about course content, to socialize, and whatever else seems appropriate to you. I will also use Discord to make class-related announcements. You can install Discord on your computer or phone, or you can use it in a web browser.
Course Mechanics
Assignments
There will be a few different types of assignments in the course. All assignments are due at 8am on the day indicated on the calendar. This will likely mean that you have to complete the assignment the day before the deadline indicated on the calendar. Barring extreme circumstances, deadlines will be strict and late assignments will not be accepted.
Reading Assignments: These daily assignments will involve reading some sections from the textbook, attempting some exercises, and participating the class Discord channel.
Reflection Assignments: There will be roughly 1–2 of these a week and they will ask you to reflect on various aspects of your relationship with mathematics, your mathematical learning, etc.
Quizzes: See below.
Project: There will be a final project, due at the end of the block.
In-Class Structure
I’ll begin class with a very brief discussion of the latest reading. This is not intended to be a substitute for having done the reading assignment! Instead, my goal will just be to briefly refresh your memory about the reading. We’ll spend most of class time working on problems.
Quizzes
There will be three take-home, but time-restricted, quizzes.
You may use the textbook and your own notes. You are allocated 2 hours to work on the quiz. You will access the quiz through Gradescope, and the quiz will be due either by 8am on Tuesday, or within 4 hours of when you first access the quiz, whichever comes first. This deadline will be strictly enforced, so please make sure you get quizzes done and submitted on time.
I entrust to the Honor Code that: (a) You will only spend the allocated 2 hours working on the quiz. Gradescope’s 4 hour limit is set only so that you have time to print and/or scan materials, or in case you have arranged for extra time accommodations with Accessibility Resources. (b) You will not use any unauthorized resources (friends, the internet, other textbooks, etc). (c) You will not share information about the quiz with classmates before they’ve taken it.
There will also be an opportunity for revisions on the first two quizzes (but not on the third), giving you the chance to earn back up to 10% of the maximum number of points on the quiz. After the quiz is graded, figure out what you did wrong on any of the problems you missed points on. Then stop by my office hours on Wednesday afternoon and talk to me about what you did wrong and how you’d correct it. I may ask you some follow-up questions during our conversation to make sure you understand your revision.
Assessment
Grades will be calculated as follows.
| Component | % |
|---|---|
| Daily Discord | 10% |
| Comprehension Checks | 15% |
| Reflection Assignments | 10% |
| Quizzes | 45% |
| Project | 10% |
| Participation | 10% |
Here are further details about some of these components:
Daily Discord. You get 1 point for each day that you post to Discord for the reading assignments, up to a maximum of \(n-2\) where \(n\) is the number of days a reading is assigned. In other words, you don’t need to post every day to get full credit, and there’s no extra credit for doing so.
Comprehension Checks. You get 1 point for each CC problem you attempt from the reading assignments, up to a maximum of \(n-10\), where \(n\) is the total number of comprehension check problems that are assigned. In other words, you don’t need to submit comprehension checks every day to get full credit, and there’s no extra credit for doing so.
Participation. You’ll get a full score for the participation component of your grade if you show me somehow that you’re putting in a good faith effort to engage with the class.
Accommodations
If you anticipate or experience any disability-related barriers to your learning in this course, please contact the office of Accessibility Resources right away and have them send me an official letter detailing what accommodations you require.
Covid-19
To protect the health of our community, masking is required in class.
The class is set up so that you’ll be able to submit everything you need to submit remotely, in case you find yourself need to isolate. You should also prepare for the possibility of your isolation by making sure you have contact information for a few students in class who can share their class notes with you. If you’d like to attend class virtually during your isolation, let me know an I’ll work with tech to set this up, but it’s possible that it might take a couple of days for this to happen so you should make sure you have a support network among your classmates.
If you find yourself faced with a different Covid-related situation that might affect your classwork, please let me know and I’ll do my best to accommodate your situation. Also know that a variety of offices on campus are available to help you through difficult situations. For instance, you might get in touch with the Counseling Center, or with Campus Safety. You can find more information and resources on the Coronavirus Updates & Resources page.
Honor Code
Please make sure that you are familiar with the Honor Code at CC. Violations of the Honor Code will have to be reported to the Honor Council, which is really no fun for anyone.