Last semester, in MATH 713, we covered the essential core of the theory of analytic functions of a complex variable: Cauchy’s Theorem, the Cauchy Integral Formula, the power series expansion of analytic functions, and the uniqueness of analytic continuation. The last of these is the theorem that if two functions are analytic in a region , and are equal on some subset of which has a limit point in , then they are equal everywhere in . A corollary of this result is that if is analytic in , , and is a path from to some point (which may or may not lie in ), then there is at most one analytic continuation of along . Note however, that this does not imply that the value of is uniquely determined by the given data, because analytic continuations along different paths from to may lead to different values . This observation will lead us to the concepts of a complete analytic function, and a Riemann surface, and a large portion of MATH 714 will be devoted to the study of Riemann surfaces.
We will start the semester, however, with a brief review of the core material from MATH 713, and applications to some important functions, particularly the -function and the Riemann -function. (These applications are in Chapter VII of the textbook.) Since simple-connectedness will be an important ingredient in our discussion of Riemann surfaces, we also finally need to learn the homotopy version of Cauchy’s Theorem from Section IV.6.
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-714-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.