The course is devoted to the theory of functions of one complex variable.

Prerequisites: Real Analysis of One Variable, including integration; Basic Algebra

Curriculum:

1. Complex-valued and holomorphic functions.
2. Cauchy theorem.
3. Integral Cauchy formula.
4. Taylor series and holomorphness test.
5. Laurent series and singular points.
6. Residues and the argument principle.
7. Topological properties of holomorphic functions.
8. Compact families of holomorphic functions.
9. Hurwitz theorem and one-sheeted functions.
10. Analytic continuation.
11. Riemann's theorem.
12. Riemann surfaces and Fuchsian groups.
13. Moduli spaces of complex tori.
14. Analytic functions and algebraic curves.

Textbooks

• S.Lang, Complex analysis, 2d ed., New York: Springer, 1985.
• J.Bak, D.J.Newman, Complex Analysis, Springer-Verlag, 1982.
• L. Ahlfors, Complex Analysis, 3rd ed, McGraw-Hill, 1979.