The course is devoted to the theory of functions of one complex variable.
Prerequisites: Real Analysis of One Variable, including integration; Basic Algebra
Curriculum:
- Complex-valued and holomorphic functions.
- Cauchy theorem.
- Integral Cauchy formula.
- Taylor series and holomorphness test.
- Laurent series and singular points.
- Residues and the argument principle.
- Topological properties of holomorphic functions.
- Compact families of holomorphic functions.
- Hurwitz theorem and one-sheeted functions.
- Analytic continuation.
- Riemann’s theorem.
- Riemann surfaces and Fuchsian groups.
- Moduli spaces of complex tori.
- 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.