722 - Complex Analysis

Math 722 is the introductory graduate level course in complex analysis. As with all such courses, we will study:
·      Fractional linear transformations
·      Cauchy’s theorem and its applications
·      The calculus of residues
·      Series and product expansions
·      Entire functions
·      Conformal mapping and the Riemann mapping theorem
This will also be a course on the `theory of functions’. In particular, we will study the basic properties of
·      The Gamma function
·      The Zeta function
·      Elliptic functions
·      Theta functions
As time permits, we will consider applications to number theory and Riemann surfaces. The official text will be Complex Analysis, by Elias M. Stein and Rami Shakarchi, Princeton University Press, ISBN 0-691-11385-8. However, there are many other good references for this material, including:
·      Complex Analysis by Lars V. Ahlfors, McGraw-Hill, Inc.
·      The Theory of Functions by E.C. Titchmarsh, Oxford University Press.
·       Analytic Functions by S. Saks and A. Zygmund