Complex Analysis (AKA Functions of a Complex Variable) is one of the most beautiful areas of mathematics and, fortunately, also extremely useful in applications. One of the most intriguing aspects of the subject is how different it is in flavour from Real Analysis, even though the fundamental definitions are formally the same. For example, a function of a complex variable is said to be differentiable at if the limit exists. So far, no different from the real case— but it turns out that if a function of a complex variable is differentiable in some open set in the complex plane, the derivative is itself differentiable. By induction, the original function must actually be infinitely differentiable! This is obviously in sharp contrast to the real case, where derivatives need not even be continuous, and where functions can be differentiable times but not times, for any .
The core of MATH 713 will consist of (most of) the first 7 chapters of the text. (I intend to cover Chapters I and II lightly, but we can go into them in more depth if you think it is necessary.) By the end of this semester, you should have a good understanding of the basics of complex analysis. The goal for the year is to cover essentially the whole text—and also an important classical theorem which Conway does not prove although he builds all the ingredients for the proof—by the end of MATH 714, but it is more important to understand the material well than to cover it quickly.
I will try to keep lecturing to a minimum, and devote class time to discussion and problem solving. As a result, you will be expected to read the textbook in a timely fashion, as necessary to participate in the class discussions. When I assign reading for a class, I will usually expect you to e-mail me, at least 24 hours prior to the class, with one or more questions you have about the reading. I will review those e-mails before the class and prepare a brief lecture or other activity for any common questions, as I feel necessary. I will also save those e-mails; they will be used at the end of the semester as evidence that you were completing the reading assignments, and will count towards the homework portion of your grade.
Almost all class information will be posted on the class website, https://sites.uwm.edu/kevinm/math-713-complex-analysis/. All class information (homework, class cancellations, etc.) will be posted on the website; some will be posted only there. If I find useful and relevant links during the semester, I will post them as well; if you find some yourself, please let me know. You are responsible for any information posted on the website, so please check it frequently.
Your grade for the course will be based on the following factors:
- Homework In addition to the textbook (and possibly other) readings, you will be assigned written homework regularly, some of which you will be expected to hand in. Of the homework you hand in, approximately half will be treated summatively; i.e. you will be given a grade which will contribute to your final grade for the course. (The other half will be treated formatively: you will be given feedback, but no score.) 25%.
- Class participation You will be expected to contribute to the class discussion, to the extent of leading discussions on topics from the text or homework problems. (For this reason alone, regular class attendance will be essential.) 25%.
- Exams There will be two exams, a midterm and the final exam. 25% each.
Average Time Investment
The amount of time that an average student should expect to spend on this class is as follows:
- Classroom time (face to face instruction): 45 hours
- Time taking exams (midterm, final exam): 15 hours
- Time completing reading and other homework assignments: 80 hours
- Time for preparation and study for exams: 10 hours
Total number of hours: 150.
Students with disabilities
If you feel you are a student with a disability, please feel free to contact me early in the semester for any help or accommodation you may need.
The Secretary of the University has a page dedicated to university policies for religious observances, grade appeal procedures, military service and other matters. You should also familiarize yourself with the information on the Dean of Students Office webpage concerning proper student conduct at the university, both academic misconduct and non-academic misconduct. You will be held responsible for the information and policies contained at these links.