| Lecture |
Date |
Topics |
| 1 |
1/13/97 |
algebra and analytic geometry of complex numbers, the Riemann sphere
|
| 2 |
1/15/97 |
power series and their convergence
|
| 3 |
1/17/97 |
differentiability and analyticity, differentiability of power
series
|
| 4 |
1/20/97 |
elementary functions in the complex domain, mapping properties
|
| 5 |
1/22/97 |
conformality
|
| 6 |
1/24/97 |
Cauchy-Riemann equations, partial z and partial z bar, harmonic
functions, existence of harmonic conjugates
|
| 7 |
1/27/97 |
Mobius transformations
|
| 8 |
1/29/97 |
Path integrals, Cauchy's formula and theorem for a disc, analytic
implies power series at each point,
|
| 9 |
1/31/97 |
Cauchy's estimate, Maximum Modulus Principle, Liouville's Theorem,
Fundamental Theorem of Algebra, isolation and finite multiplicity
of roots
|
| 10 |
2/3/97 |
homotopy of paths and Cauchy's theorem, motivation for winding numbers
|
| 11 |
2/5/97 |
winding numbers and Cauchy's integral formula, zero counting, Open
Mapping Theorem
|
| 12 |
2/7/97 |
Morera's Theorem, theorem on existence of primitives, Goursat's Theorem,
isolated singularities
|
| 13 |
2/10/97 |
poles, essential singularities, Casorati-Weierstrass Theorem, Laurent
expansion
|
| 14 |
2/12/97 |
The Residue Theorem, residue integrals
|
| 15 |
2/14/97 |
Argument Principle, Rouche's Theorem, Weierstrass Theorem
|
| 16 |
2/17/97 |
Hurwitz's Theorem and Montel's Theorem
|
| 17 |
2/19/97 |
Schwarz's Lemma and applications
|
| 18 |
2/21/97 |
The Riemann Mapping Theorem
|
| 19 |
2/24/97 |
Complements on conformal mapping, mean value property and
maximum priniciple harmonic functions on a disk
|
| 20 |
2/26/97 |
digression: statement of the little and great Picard
theorems and integrability of functions with isolated singularities,
Poisson kernel, Dirichlet problem on a disk
|
| 21 |
2/28/97 |
Harnack's inequality and principle, subharmonic and superharmonic
functions
|
| 22 |
3/3/97 |
Perron's approach to the solution of the Dirichlet problem
|
| 23 |
3/5/97 |
Schwarz reflection principle, some loose ends, midsemester evaluation
|
| 24 |
3/7/97 |
Midterm Exam
|
