Complex Analysis I and II

Complex Analysis I, II – 60370, 60380

1. Winding number, integral along curves. Various definitions of a holomorphic function. Connection with harmonic functions. Cauchy Integral Theorems and Cauchy Integral Formula for closed curves in a domain, and for the boundary of a domain, Poisson Formula. The integral of a holomorphic function and its dependence on the path of integration. Open Mapping Theorem, Inverse Function Theorem, maximum and minimum principle, Liouville’s Theorem. Uniform convergence of holomorphic functions. Normal families of holomorphic functions. Montel and Vitali Theorems, Picard’s Theorem. Power series, Laurent series. Residues and classification of isolated singularities, meromorphic functions. Divisor of a meromorphic function. Residue Theorem, argument principle, Rouché’s Theorem, computation of integrals. Riemann Mapping Theorem, argument principle. Möbius maps. Schwartz Lemma. Theorems of Mittag-Leffler and Weierstrass. Gamma Function, Riemann Zeta Function, Weierstrass Ã-function.
2. Definition of complex manifolds and examples. Riemann surfaces. The concepts of divisors, line bundles, differential forms and Chern forms. The Riemann-Roch Theorem. The Dirichlet problem for harmonic functions. The concept of genus of a Riemann surface.

 

References

• Ahlfors, Complex Analysis.
• Burchel, An Introduction to Classical Complex Analysis I. Conway, Functions of One Complex Variable.
• Forster, Lectures on Riemann Surfaces.
• Gunning, Lectures on Riemann Surfaces.
• Knopp, Theory of Functions I, II, and Problem Books.

Complex Analysis I covers approximately Chapters 1-6 of Ahlfors. Complex Analysis II covers the remaining material.