| Date | Topic | Homework | Comments |
| 1/10 1/12 1/14 |
Intro, Complex numbers as a field Geometric interpretations Continuity, differentiability |
Read chapter 1 reread chapter 1 read ahead... |
. Homework 1 . |
| 1/17 1/19 1/21 |
Martin Luther King Jr Day Riemann sphere, rational maps degree of a rational map |
No classes zeros, poles Mobius transformations |
No classes Homework 1 is due . |
| 1/24 1/26 1/28 |
Decomposition of rat. fun., power series Diff of power series, Abel's thm exp, sin, cos, sinh, cosh, tan |
Singular parts of poles, radius of convergence Term by term differentiation . |
. . Homework 2 is due |
| 1/31 2/2 2/4 |
Line integrals Line integrals (cont.), Index |
Open sets, paths, arcs, line integrals Snow day! Properties of line integral, Winding number |
. Problem session 2-3pm |
| 2/7 2/9 2/11 |
Index/winding number (cont.) Simply connected sets Anti-derivative and integration |
Ahlfors 2.1, p.114 Ahlfors 4.2, p.138 Ahlfors 2.2 |
Homework 3 is due (updated) Problem session 2-3pm . |
| 2/14 2/16 2/18 |
Cauchy theorem Cauchy integral formulae Zeros of holomorphic functions |
. analyticity using geometric series expansion Uniqueness of holomorphic extensions |
. Problem session 2-3pm . |
| 2/21 2/22 2/24 |
Poles and essential singularities Local behavior using local argument Open mapping theorem |
Isolated singularities Local argument principle, Ahlfors 4.3.3 Local inverse at non critical points |
Homework 4 is due Problem session 2-3pm Homework 5 is due |
| 2/28 3/2 3/4 |
Midterm Maximum modulus principle Weierstrass theorem |
. . around "uniformly on compacta" |
. Problem session 2-3pm . |
| 3/7 3/9 3/11 |
Schwarz lemma, General Cauchy formula General Cauchy formula (cont.) . |
Aut(D), chains, cycles homology of cycles Log and related functions |
Homework 6 is due . . |
| 3/14 3/16 3/18 |
Calculus of residues Argument principle, Rouche's theorem Some definite integrals |
. . . |
Homework 7 is due . . |
| 3/21-25 | Spring Break | Spring Break | Spring Break |
| 3/28 3/30 4/1 |
More definite integrals Arzela-Ascoli Arzela-Ascoli |
. completeness, totally bounded sets, compactness the diagonal argument |
. Homework 8 is due . |
| 4/4 4/6 4/8 |
Normal families Normal families, Riemann mapping Riemann mapping theorem |
metric defining uniform convergence on compacta conformal equivalence Hurwitz theorem, ... |
|
| 4/11 4/13 4/15 |
Harmonic functions Maximum Principle Poisson kernel |
. . Homework 9 is due |
|
| 4/18 4/20 4/22 |
Poisson integral Schwarz reflection principle Harnack inequality |
Dirichlet problem in the disc + criterion for harmonicity . |
. Problem session 2-3pm . |
| 4/25 4/27 4/29 |
Harnack's theorem Analytic continuation No class |
. . . |
. Homework 10 is due No class |
| 5/2 | Final 1-3 pm |
Homeworks: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.