Math 428: Topics in Complex Analysis

MWF 2:00-2:50, Herman Brown 453

This is a second course in complex analysis. I have two overlapping objectives: To expose you to advanced techniques of complex function theory and to core examples of complex geometry. The course will divide naturally into two parts. The first will be devoted to brief surveys of a range of complex-analytic topics:

Harmonic functions
Review of infinite products with applications
Analytic continuation
Rational approximation
Special functions: Gamma, Beta, and Zeta functions
Prime number theorem
The main source will be Function Theory of One Complex Variable, by Robert E. Greene and Steven G. Krantz, published by the American Mathematical Society. This should be available in the bookstore.

The second half will be a more in-depth discussion of elliptic functions:

Doubly-periodic functions
Weierstrass P-functions
Modular group and modular functions
Theta functions
Applications to number theory, e.g., representing integers as sums of squares
The main reference here will be K. Chandrasekharan, Elliptic functions, published by Springer Verlag. I have not ordered this book through the bookstore because of its extremely high cost. However, there are multiple copies on reserve in Fondren.

Prerequisites: A semester course in complex analysis. (We will define this more precisely in the first week of class.)

Assessment: Weekly problem sets (40%): Due each Wednesday starting September 3. They should be turned in during class. Homework assignments are not pledged. You are strongly encouraged to work together, though each student should write up her/his own submission.

Midterm exam (20%): A closed-book 80-minute take-home test, to be taken sometime between Friday, October 3 and Monday, October 6.

Final exam (40%): A closed-book three-hour take-home exam.

ADA statement: If you have a documented disability that will impact your work in this class, please contact me to discuss necessary accomodations. Additionally, you will need to register with the Disability Support Services Office in the Ley Student Center.

Contact information:
Brendan Hassett
Office: Herman Brown 402
Phone: (713) 348-5261