Course No.：S070100XJ009
Course Category：Basic Course of Subject
Period/Credits：40/2
Prerequisites：basic theory of one complex variable, basic topology.
Aims & Requirements：
Complex analysis I is a basic course of graduate students of all majors in mathematics, it also an optional course for physical students. The course is aimed to lay part of the foundation for doing research work in related areas. After learning this course, students are hoped to be able to understand basic ideas and concepts of modern complex analysis and know well the history of developments of this subject.
Primary Coverage：
Chapter 1. Schwarz’s lemma and its applications
The Schwarz derivatives, Poincare metric, Schwarz-Pick’s lemma, the ultra-hyperbolic metric, Poincare metrics on planar domains, Picard’s theorem, Bloch’s theorem.
Chapter 2. Normal family and Riemann’s mapping theorem
Normal families, Riemann’s mapping theorem, prime ends and limits.
Chapter 3. Univalent functions
Classical distortion theorems, sequences of univalent functions, Grunsky inequality.
Chapter 4. Conformal mappings on multi-connected domains
Domains with parallel line segments deleted, circular domains.
Chapter 5. Harmonic functions
Poisson’s formula, the maximum principle, sequences of harmonic functions and Harnack’s inequality, subharmonic functions, Dirichlet’s problem and Green’s functions, harmonic measures.
Chapter 6. Extremal length
Extremal lengths, modular of annuli and quadrilaterals, extremal annuli, elliptic and modular functions, extremal distances and harmonic functions, reduced extremal distance.
Chapter 7. Capacity
The transfinite diameter, potentials, capacity and the transfinite diameter.
References:
1. Lars V. Ahlfors, Conformal Invariants–Topics in Geometric Function Theory, McGraw - Hill Book Company, 1973.
Author：Gui-zhen Cui（Academy of Mathematics and Systems Science）
Date：June, 2009
