Course
Outline - Mat 70300 - Fall 2012
Course meetings, Tues
-Thurs 4:15 PM - 5:45 PM pm Room 6496
Texts: Functions
of One Complex Variable I, John B. Conway, Springer, Complex Analysis by L. Ahlfors, McGraw Hill.
Other classical texts you may want to consult are: Theory of Functions by Caratheodory, Chelsea, The five small volumes on Theory of
Functions by K. Knopp, dover
reprint. Some of the material can also be found in Hyperbolic Geometry from a
local viewpoint, L. Keen, N. Lakic, Cambridge_ Other
texts
Instructor:
Prof. Keen. Office Room 4208. Phone 212
817 8531 or email lkeen@gc.cuny.edu Office hours: By
appointment
Outline
I plan to give a
standard beginning complex analysis course. I will assign exercises from Conway’s
book. I suggest you read the
material as presented in Ahlfors’ book as well. I will assume the material in the
first chapter on the algebraic properties of complex numbers and their
geometric representation.
Contents of the course will include:
Complex numbers, Topology of
the complex plane, power series and analytic functions.
Properties of analytic functions: Conformal mapping. Linear
fractional transformations. The exponential and
logarithmic functions.
Complex Integration: Cauchy’s theorem, Cauchy’s integral formulas, Liouville theorem, principle of analytic continuation, Morera’s theorem,
Singularites:
Poles, zeros, residues, the argument principle, Rouche’s
theorem
Maximum modulus
theorem, Schwarz’s Lemma
Riemann Mapping
Theorem, Schwarz reflection principle, Harmonic functions, Poisson
kernel,
Entire functions, Mittag-Leffler theorem, Jensen’s theorem
Homework assignments will appear on this page
approximately every week. Students are strongly advised to work on all the
homework problems to make sure they are keeping pace with the class.
The final grade will be based only on the homework grades.
Class and Homework Assignments
(Exercises from
Conway)
|
|
||
|
Date |
Class
Topic |
Reading
and Assignment |
|
Aug 28 |
Introduction
- review of complex numbers |
Chapter
1 p.4/3, p.6/7, p.10/4 due 9/6 |
|
Aug 30 |
Metric Spaces and Topology |
Chapter 2 p.13/2,5,10 20/1,2,6,7 due 9/11 |
|
Sept 4 |
Metric spaces continued |
Chapter 2 p.24/2,3,5 due 9/13 |
|
Sept 6 |
Metric spaces continued |
Chapter 2 p.28/3,4,5,7,9 p.29/1 due
9/20 |
|
Sept
11 |
Power
Series, |
Chapter
3 p.33/1,3,6 p.43/2,3,11
due 9/27 |
|
Sept
13 |
Analytic
functions |
|
|
Sept
18 |
No
Class |
|
|
Sept
20 |
C-R
equations |
|
|
Sept
25 |
No
Class |
|
|
Sept
27 |
Mobius transformations |
Chapter 3 p. 54/1,3,4,6,9,15 due Oct. 2 |
| Oct 2 | Complex Integration | Chapter 3 p. 55-57/13,14,15,18 due Oct 9 |
| Oct 4 | Complex Integration Con'td | Chapter 4 p. 67/9,12 13,19 due Oct 11 |
| Oct 9 | Cauchy's Theorem | Exercise sheet 1 due Oct 16 |
| Oct 11 | Cauchy's Theorem and Integral Formula | Oct 16 | Applications of Cauchy Integral Formula | Chapter 4 p. 83/1,2,3,4 due Oct 23 | Oct 18 | Isolated singularities | Exercise sheet 2 due Oct 25 | Oct 23 | Local Properties |
| Oct 25 | Cauchy theorem in general | Exercise sheet 3 due Nov 6 |
| Oct 30, Nov 1 | Storm closed | |
| Nov 6 | Argument principle, Roche's theorem | |
| Nov 8 | Residues | Conway p. 121/ 1 a,d 2/b,d,h,6 due Nov 15 | <\tr>
| Nov 13 | Laurent Series | |
| Nov 15 | Mittag-Leffler and Weierstrass Preparation theorem | Conway p. 173 Ex. 4,5,6,7 due Nov 29 |
| Nov 20 | Weierstrass Preparation theorem | |
| Nov 27 | Gamma Function | Conway p. 185 1,5,8 due Dec 6 |
| Nov 29 | Stirling's Formula, | |
| Dec 4 | Harmonic functions | |
| Dec 6 | Poisson - Jensen formulas | Conway p. 255/8,9, p. 262/1,2 Due Dec 13 |
| Dec 11 | Symmetry and subharmonic functions | |
| Dec 13 | Dirichlet's Problem | |