Xianghong Gong (Spring 2019)
This is a graduate course of complex analysis in one complex variable. The following topics will be covered: Elementary properties of analytic functions and harmonic functions. Cauchy’s theorem, power series and Laurent series expansions of analytic functions. Residues. Rauché’s theorem, Hurwitz’s theorem, and open mapping theorem. Schwarz lemma and the automorphsim group of the unit disc. Poisson integral formula and Schwarz reflection principle. Normal families, Riemann mapping theorem, and Picard's theorems. Runge's approximation theorem, Mittag-Leffler theorem and Weierstrass product theorem. Special functions and the prime number theorem. If time permits, select topics on Dirichlet problem via the Perron method, and Green’s functions and introduction to Riemann surfaces and the uniformization theorem.
Text: Theodore W. Gamelin: Complex Analysis, Springer-Verlag, New-York, 2001. ISBN 0-387-95093-1.