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Introduction to complex variables
(Math 3450, Spring 2025)

Logistics

Class meets MWF, 1:25–2:15pm in ECCR 108.

You can contact me by e-mail: jonathan.wise@colorado.edu. You can also send an anonymous message.

Office hours

My office is Room 204 in the Math Department. My office hours sometimes change, so I maintain a calendar showing the times I will be available. You can also make an appointment or drop in without an appointment.

Topics

The following are the main topics of the course, and an approximate schedule in which we will cover them :

  1. Week 1 : Arithmetic of the complex numbers, exponential function (ch. 1)
  2. Weeks 2-3 : Power series, radius of convergence, exponential and trigonometric functions, multiply valued functions, logarithm (ch. 2)
  3. Weeks 4-6 : Complex differentiation, Cauchy Riemann equations (ch. 4-5)
  4. Week 7 : Winding number, fundamental theorem of algebra, maximum principle (ch. 7)
  5. Weeks 8-10 : Complex integration, Cauchy's theorem, Cauchy's integral formula (ch. 8-9)
  6. Weeks 11-12 : Möbius transformations, Riemann sphere (ch. 3)
  7. Weeks 13-15 : TBD

Cauchy's integral formula is the highlight of this course, and is the most important piece to understand. All applications of complex analysis use it in an essential way.

We will probably fall behind the above schedule, but if we stick to it, there will be time at the end for advanced topics (possibilities include the Riemann zeta function and Riemann surfaces).

Goals

I like to imagine a hierarchy of understanding of a mathematical concept, such as a definition or theorem. These correspond relatively well to grades:

  1. Know the statements.
  2. Recognize situations in which the definition or theorem can be applied.
  3. Explain the definition or theorem and apply it to solve problems.
  4. Adapt the reasoning to novel situations and apply the definition or theorem in more sophisticated settings.

When I grade your work, I will compare the understanding you demonstrate to this list and assign a grade accordingly. This is not a proofs-based course, so it is possible to achieve an A without writing a rigorous proof. However, we will still achieve the same depth as a proofs-based class would, and there will be opportunities to write proofs for those who desire them.

Textbook

The textbooks for this course are :

Visual Complex Analysis by Tristan Needham

Complex analysis by John M. Howie

There are many other textbooks that cover introductory complex analysis, but Needham's has a unique and beautiful visual approach. We will therefore rely primarily on Needham's book. However, the presentation is a little unusual, and the exercises are on the more difficult side. We may therefore need to make reference to Howie's book from time to time. Please note : Howie's book is available for free through the university's subscription to SpringerLink.

Homework and participation

There will be an assignment in between every two classes. You will be expected to spend at least an hour on this assignment, but not necessarily to complete it. When you submit your work, you will also give yourself a grade indicating how much time you spent on the assignment :

  1. About one hour (or more).
  2. About 45 minutes.
  3. About 30 minutes.
  4. About 15 minutes.

Grading

Your grade will be based on the following components :

The midterm exams will be multimodal : there will be an in-class component and an oral component. The final will only have a written component. The exams will be assessed according to the rubric in the Course Goals. All graded assignments will be posted and submitted through Canvas.

Grading is done on a GPA scale :

Because the GPA scale is very forgiving of missed assignments, there is no special policy to accommodate missed homework or in-class work. The policy for missed exams and significant amounts of missed homework will depend on circumstances.

When submitting work that you prepare at home (this includes homework and oral exams) you must include a bibliography citing all outside resources consulted. You do not have to include a bibliography if you only used the textbooks and lectures. You may wish to model your bibliographies on this example.

Submit your solutions on Canvas, in PDF format. The best way to accomplish this is to write up your work in Latex and compile it to PDF (for example using Overleaf). You can also write your solutions by hand and scan them. If you prefer to do things this way, be sure to write neatly and scan in an efficient and readable way (use a tool like CamScanner).

Etiquette and responsibilities

Engagement

Learning math requires active engagement. No matter how clearly or beautifully presented a lecture might be, it can never substitute for wrestling with mathematical ideas on your own. My role in this course will be to provide a stucture for your exploration of complex analysis, and to suggest ways you might organize the ideas in your minds. That exploring and organizing will be your responsibility, and a lot of it will have to occur outside of class. The more of it you do, and the more energy and enthusiasm you bring to it, the deeper your understanding will be later on.

If you are doing this right, you will have questions. Some will be deep and some will be trivial, but it is impossible to engage actively with math without asking them. You don't have to ask all of your questions in class — although I do want to hear some of them — but you can use this as a guide to make sure you are doing math in a way that deepens your understanding.

After each class, I will suggest some work to solidify the ideas from that class and the prepare for the next one. I expect you to spend one hour in between classes working on this material. You do not have to complete all of the exercises or tasks suggested! After you've done an hour of work, turn in what you have done; you will get full credit for the assignment.

Academic honesty

If you make use of any resources other than the lectures and official textbooks of the course — including texts, lectures, other people, websites, videos, generative AIs, or any other sources — you should cite those works in a bibliography. Your sources should be listed in a format that allows a reader to look up the reference : include page numbers, theorem statements, URLs, timestamps, etc., as necessary. Failure to do so may have a significant effect on your grade.

Required syllabus statements

The university has several required syllabus statements. These are fully supported in this class. In particular, missed assignments due to illness or religious observance will be dealt with by the revision policy discussed above. If anything comes up that you would like to discuss, or would like me to be aware of, please contact me, either by e-mail or anonymously.