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MATH 331 Theory of Func of Complex Var 15048 LEC A 4.00 23 8 15 08:30 09:20 M W F VOTEY 223 Dupuy T Open to Degree and CDE students;
11:40 12:55 T VOTEY 223 Prereqs enforced by the system: MATH
242
| Homework 1 (tex) |
Due: Friday, Jan 27th (solutions) |
| Homework 2 (tex) |
Due: Friday, Feb 10th (solutions) |
| Homework 3 (tex) |
Due: Friday Feb 24th (solutions) |
| Homework 4 (tex) |
Due: Friday, March 3rd (solutions)
(scratch sage notebook I used to check stuff with-- it is messy) |
| Homework 5 (tex) |
Due: Monday, March 13th (solutions) |
| Homework 6 (tex) |
Due: Monday, April 3rd (solutions) |
| Homework 7 (tex) |
Due: Friday, May 5th (solutions) |
| Topics |
References and
Specifics |
| Complex
Numbers and Cauchy's Theorem |
|
| Complex Numbers and Complex Functions |
Background on uniform
convergence, switching integrals and limits, switching derivatives
and
limits Remember, complex functions are just functions from \RR^2 \to
RR^2. See John
Loftin's Notes. Weierstrass M-test (WW 3.34, page 49) |
| Analytic Functions |
Hadamard's root test GK (Lemma 3.2.6) GK (3.2) |
| Holomorphic Functions |
Schlag (section 2), GK (1.3-1.4) Cauchy-Riemann equations |
| Complex Integration |
McMullen
(page 5) (He also also outlines Goursat and gives the basic proof) |
| Cauchy's Theorem (simple regions) |
Basic
Green's Theorem Proof We also talked about the Jordan Curve Theorem Simple path independence (for closed curves) Simple homotopy independence |
| **Cauchy-Goursat (which doesn't
assume continuity of the derivative) --- this was an exercise |
Proof
using triangles - from Ernie Croot's webpage Proof using squares - Alfors, or Green and Krantz (appendix B) |
| Applications
of Cauchy's Theorem |
|
| Integration/Cauchy Integral Formula +
Consequences: |
McMullen (section 1), G&K
(chapters 2 and 3) |
| Application of Cauchy Integral
Formula: Analytic = Holomorphic (This also gave the Cauchy Integral Formula for derivatives) |
To get analytic =>
holomorphic one can work directly with power series as in (say)
Whittaker and Watson (2.61 page 31), we also did this in homework
02. The hard direction is holomorphic => analytic, GK (3.3) |
| **Limits of Holomorphic Functions are
Holomorphic |
GK (3.5), this was also on HW01 Remark: One could also use Morera's theorem, which I mentioned in emails but didn't prove in class. This says that if integrals around closed loops are zero, then you are analytic (it is a converse of Cauchy's Theorem). |
| **Application of Cauchy 2: Taylor
Series Estimate |
GK (3.4) |
| Application of Taylor Series:
Liouville's Theorem |
GK Theorem 3.4.3 (page 86) |
| Application of Liouville: Fundamental
theorem of algebra |
GK, Theorem 3.4.5 (page 87) (There is another proof that uses winding numbers later on) |
| Interlude on Topology and Categories
|
Open
maps Compact Spaces Proper Maps Separated Spaces Eventually, this subject gets to |
| Existence of Logarithms | Schlag (Proposition 1.21) |
| Isolation of Zeros Local Structure of Analytic Mappings (Ramification points) Analytic Continuation |
Rudin - Real and Complex Analysis
(10.31) Schlag (page 16, Corollary 3.12) Remarks: *In the factorization we saw an example of a conformal map *Analytic continuation leads naturally to the idea of a Riemann Surface via the phenomena of Monodromy. Green and Krantz postpone analytic continuation to Chapter 10, which is something we do not want to do. *The Several variable version of the local structure theorem is the Weierstrass Preparation Theorem. *We also talked about Branch Points and Ramification (WARNING - terminology can vary) |
| Open Mapping Theorem | Rudin - Real and Complex Analysis
(10.31) Remark: We are using Rudin's proof here to avoid the use of winding numbers. The proof in GK and other places uses winding numbers. |
| Mean Value Theorem | Mathworld |
|
Maximum Modulus |
McMullen
(Corollary 1.10) (I had built towards this theorem for a long time---this was the original reason the open mapping theorem) |
| Residues
Integrals |
|
| Laurent Series Residues |
G&K (4.3 - Existence of Laurent
Expansions) Basic Complex Analysis - Marsden-Hoffman (chapter 4) Remark: we have been avoiding the later part of this chapter |
| Residue Theorem |
We stated this without proof. |
| Application of Residue Theorem to
integrals |
Whittaker and Watson (chapter 6.1, page 111 --- This is the treatment most modern books are based off of this chapter because of the use of winding numbers) |
| Infinite
Products and Partial Fractions |
|
| Partial Fraction Expansion = Mittag-Leffler Theorem | GK (8.3) Ahlfors (Ch 4, sec 2, subsection 1) |
| Elliptic Functions |
I typed up
special notes for this section: Elliptic Function Notes See also: GK (10.6) WW (sections 20.1 and 20.2) |
| Infinite Products = Weierstrass
Factorization Theorem |
WW 2.7 page 32 (has some weird
examples) GK (8.1-8.2) Ahlfors (Ch 4, sec 2, subsection 2) |
| General
Cauchy Integral Formula |
|
| Winding Numbers and Cauchy's Integral
Formula Theorem |
Winding
Numbers Video General Cauchy Theorem Video Arguement Principal Video Rouche's Theorem was done when Christelle visited. **************** Remarks: **************** When we did our proof so simple regions we assumed Green's theorem for simple regions. This both assumed Green's theorem and the Jordan Curve Theorem. We want to get rid of these assumptions. Terry Tao's Notes (Notes 3, section 1) - Gives a polygonal path approach to the general Cauchy's Theorem. Stoll's Notes (chapter 8): This develops a minimal amount of Wedhorn's Notes section 2: Path Integrals section 3: Homotopy section 4A,4B: Holomorphic Forms and Cauchy's Theorem (immediate consequences) section 8: Homology and Winding Numbers A proof based on existence of primatives for analytic functions on the disc. We eventually want to understand Schlag Theorem 1.16, page 14, this proof is the most enlightening as it uses general machinery. Many proofs don't use topology. The downside of these are that they are only good for one theorem, which the general machine of algebraic topology is good for many many many things. |
| Conformal
Maps and Riemann Surfaces |
|
| PP^1 = Riemann Sphere |
PP^1
video Holomorphicity at Infinity video Functions to PP^1 video Endomorphisms of PP^1 video Bonus: Official Definition of Riemann Surfaces Video |
| Mobius Transformations (Automorphisms of PP^1) |
GK (6.3), Also see Homework 05 |
| Schwarz Lemma |
GK (5.5) |
| Entire
Functions of Finite Order And Hadamard's Theorem |
|
| Order
of an entire function Hadamard's Theorm |
Course Notes
on Hadamard's Theorem See also GK (9.3) |
| Riemann
Mapping Theorem |
|
| Montel's Theorem (w/out proof) |
|
| Proof of Riemann Mapping Theorem (via
"square root trick") |
GK (6.7) |
| Covering
Spaces, Uniformization and Big Picard |
|
| Fundamental Groups and Covering
Spaces |
My Notes (partial) Glickenstein's Notes Munkres is also good. McMullen has a discussion on the Uniformization Theorem. |
| Modular Forms |
Modular
Curves Modular Forms Modular lambda function |
| Picard Theorems |
GK (10.5), Proof
Wiki, McMullen page 89 (which may be hard to read) |