Complex Analysis, Second Course

Aim

To extend and enhance the complex analysis taught in the basic complex analysis course. In the course, the student also practices his/her ability to read and write mathematical text and mathematical proofs. After completing the course, the student should

• be able to determine when a polynomial has all its zeros in the left half plane and outside the unit disk, and understand the theory behind the corresponding results,
• have a good understanding of conformal mappings, complex analysis on the Riemann sphere, analytic continuation and branches of analytic functions,
• be able to cite and explain some essential definitions and theorems about analytic continuation and Riemann surfaces,
• know the explicit Riemann surfaces for some simple multi-valued functions.

Content

More about the argument principle: the criteria of Routh-Hurwitz and Schur-Cohn. More about conformal mappings with applications. More about residue calculus. Complex analysis on the Riemann sphere. Analytic and meromorphic continuation. Explicit Riemann surfaces. Abstract Riemann surfaces.