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Math 428: Topics in Complex Analysis

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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

Email: hassett@math.rice.edu

Webpage:
http://www.math.rice.edu/~hassett