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Math 132H: General Course Outline

Catalog 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

Complex Analysis by Stein and Shakarchi.

Schedule of Lectures

Lecture Section Topics

1-3

1.1-1.2

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)
-Basic properties, convergence, sets in the complex plane

4-6

1.3

Integration along curves

7-8

2.1-2.2

Goursat's theorem; Local existence of primitives and Cauchy's theorem in a disc

9-11

2.3-2.4

Evaluation of some integrals; Cauchy's integral formulas

12-14

3.1-3.2

Zeros and poles; The residue formula

15-16

Midterm/Continuation

17-18

3.3

Singularities and meromorphic functions

19-21

3.4-3.6

The argument principle and applications; Homotopies and simply connected domains; The complex algorithm

22-24

8.1-8.4

Conformal equivalence and examples (the disc and upper half-place, further examples, the Dirichlet problem in a strip); The Schwarz lemma and automorphisms of the disc and upper half-place (Automorphisms of the disc, automorphisms of the upper half-place); 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 Schwarz-Christoffel integral, boundary behavior, the mapping formula, return to elliptic integrals)

25-27

TBA

Catch-up, Review