This course covers the main topics in classical Complex Analysis at the graduate level. It begins with reviewing elementary properties of holomorphic functions, Cauchy's integral formula, Taylor and Laurent series, and residue calculus. Then it continues to cover the following:
- Harmonic functions. Poisson's integral formula and Dirichlet's problem.
- Conformal mapping, Riemann mapping theorem.
- Analytic continuation, Monodromy Theorem, Riemann surfaces.
- Modular functions and the Picard Theorems.
- Other topics are possible, like product theorems, elliptic functions, and non-isolated removability theorems.
Textbooks:
- Stein and Shakarch: Complex Analysis
- L. Ahlfors: Complex Analysis, 3rd Edition, McGraw-Hill, New York, 1966.
- W. Rudin: Real and Complex Analysis, 2nd Edition, McGraw-Hill, New York, 1974.