Homology and Winding Numbers A proof based on existence of primatives for analytic functions on the disc. Many proofs don’t use topology. Conformal Maps and Riemann Surfaces. Covering Spaces, Uniformization and Big Picard. We are using Rudin’s proof here to avoid the use of winding numbers.

McMullen page 5 He also also outlines Goursat and gives the basic proof. We are using Rudin’s proof here to avoid the use of winding numbers. Application of Cauchy Integral Formula: Holomorphic Forms and Cauchy’s Theorem immediate consequences section 8: General Cauchy Integral Formula. See John Loftin’s Notes.

We want to get rid of these assumptions. One could also use Morera’s theoremwhich I mentioned in emails but didn’t prove in class. Complex Numbers and Cauchy’s Theorem. Applications of Cauchy’s Theorem. Order of an entire function Hadamard’s Theorm. General Cauchy Integral Formula.

We are using Rudin’s proof here to avoid the use of winding numbers. Friday, March 3rd solutions scratch sage notebook I used to check stuff with– it is messy. Homology and Winding Numbers A proof based on existence of primatives for analytic functions on the disc.

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We eventually want to understand Schlag Theorem 1. Elliptic Function Notes See also: Application of Cauchy Integral Formula: Stoll’s Notes chapter 8: Official Definition of Riemann Surfaces Video.

MATH 8701 – Complex Analysis – Fall 2013

Green and Krantz postpone analytic continuation to Chapter 10, which is something we do not want to do. Path Integrals section 3: This says that if integrals around closed loops are zero, then you are analytic it is a converse of Cauchy’s Theorem. McMullen page 5 He also also outlines Goursat and gives the basic proof. Holomorphic Homswork and Cauchy’s Theorem immediate consequences section 8: Interlude on Topology and Categories.

This both assumed Green’s theorem and the Jordan Curve Theorem. I typed up special notes for this section: Conformal Maps and Riemann Surfaces.

MATH – Complex Analysis – Fall

See John Loftin’s Notes. Infinite Products and Partial Fractions. 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. Schlag section 2GK 1. This develops a minimal amount of Wedhorn’s Notes section 2: The proof in GK and other places uses winding numbers.

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McMullen has a discussion on the Uniformization Theorem.

Complex (Spring )

Weierstrass M-test WW 3. Modular Curves Modular Forms Modular lambda function. Many proofs don’t use topology. My Notes partial Glickenstein’s Notes Munkres is also good. Fundamental theorem of algebra.

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Whittaker and Watson chapter 6.

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