Math 132H: General Course Outline
Course Description
Lecture, three hours; discussion, one hour. Requisites: courses 32B, 33B, and 131A with grades of B or better. This course is specifically designed for students who have strong commitment to pursue graduate studies in mathematics. Introduction to complex analysis with more emphasis on proofs. Honors course parallel to course 132. P/NP or letter grading.
Textbook(s)
Complex Analysis by Stein and Shakarchi.
Schedule of Lectures
Lecture  Section  Topics 

13 
1.11.2 
Complex numbers and the complex plane (Basic properties, convergence, sets in the complex plane); Functionas on the complex plane (continuous functions, holomorphic functions, power series) 
46 
1.3 
Integration along curves 
78 
2.12.2 
Goursat's theorem; Local existence of primitives and Cauchy's theorem in a disc 
911 
2.32.4 
Evaluation of some integrals; Cauchy's integral formulas 
1214 
3.13.2 
Zeros and poles; The residue formula 
1516 
Midterm/Continuation 

1718 
3.3 
Singularities and meromorphic functions 
1921 
3.43.6 
The argument principle and applications; Homotopies and simply connected domains; The complex algorithm 
2224 
8.18.4 
Conformal equivalence and examples (the disc and upper halfplace, further examples, the Dirichlet problem in a strip); The Schwarz lemma and automorphisms of the disc and upper halfplace (Automorphisms of the disc, automorphisms of the upper halfplace); The Riemann mapping thoerem (Necessary conditions and statement of theorem, Montel's theorem, proof of Riemann mapping theorem; Conformal mappings onto polygons (Some examples, the SchwarzChristoffel integral, boundary behavior, the mapping formula, return to elliptic integrals) 
2527 
TBA 
Catchup, Review 