Advanced Complex Analysis - Part 1:Zeros of Analytic Functions,Analytic continuation, Monodromy, Hyperbolic Geometry and the Reimann Mapping Theorem

Lectures

UNIT 1: Theorems of Rouche and Hurwitz

1. Fundamental Theorems Connected with Zeros of Analytic Functions Video
2. The Argument (Counting) Principle, Rouche's Theorem and The Fundamental Theorem of Algebra Video
3. Morera's Theorem and Normal Limits of Analytic Functions Video
4. Hurwitz's Theorem and Normal Limits of Univalent Functions Video

UNIT 2: Open Mapping Theorem

5. Local Constancy of Multiplicities of Assumed Values Video
6. The Open Mapping Theorem Video

UNIT 3: Inverse Function Theorem

7. Introduction to the Inverse Function Theorem Video
8. Completion of the Proof of the Inverse Function Theorem: The Integral Inversion Formula for the Inverse Function Video
9. Univalent Analytic Functions have never-zero Derivatives and are Analytic Isomorphisms Video

UNIT 4: Implicit Function Theorem

10. Introduction to the Implicit Function Theorem Video
11. Proof of the Implicit Function Theorem: Topological Preliminaries Video
12. Proof of the Implicit Function Theorem: The Integral Formula for & Analyticity of the Explicit Function Video

UNIT 5: Riemann Surfaces for Multi-Valued Functions

13. Doing Complex Analysis on a Real Surface: The Idea of a Riemann Surface Video
14. F(z,w)=0 is naturally a Riemann Surface Video
15. Constructing the Riemann Surface for the Complex Logarithm Video
16. Constructing the Riemann Surface for the m-th root function Video
17. The Riemann Surface for the functional inverse of an analytic mapping at a critical point Video
18. The Algebraic nature of the functional inverses of an analytic mapping at a critical point Video

UNIT 6: Analytic Continuation

19. The Idea of a Direct Analytic Continuation or an Analytic Extension Video
20. General or Indirect Analytic Continuation and the Lipschitz Nature of the Radius of Convergence Video
21. Analytic Continuation Along Paths via Power Series Part A Video
22. Analytic Continuation Along Paths via Power Series Part B Video
23. Continuity of Coefficients occurring in Families of Power Series defining Analytic Continuations along Paths Video

UNIT 7: Monodromy

24. Analytic Continuability along Paths: Dependence on the Initial Function and on the Path - First Version of the Monodromy Theorem Video
25. Maximal Domains of Direct and Indirect Analytic Continuation: SecondVersion of the Monodromy Theorem Video
26. Deducing the Second (Simply Connected) Version of the Monodromy Theorem from the First (Homotopy) Version Video
27. Existence and Uniqueness of Analytic Continuations on Nearby Paths Video
28. Proof of the First (Homotopy) Version of the Monodromy Theorem Video
29. Proof of the Algebraic Nature of Analytic Branches of the Functional Inverse of an Analytic Function at a Critical Point Video

Unit 8: Harmonic Functions, Maximum Principles, Schwarz Lemma and Uniqueness of Riemann Mappings

30. The Mean-Value Property, Harmonic Functions and the Maximum Principle Video
31. Proofs of Maximum Principles and Introduction to Schwarz Lemma Video
32. Proof of Schwarz Lemma and Uniqueness of Riemann Mappings Video
33. Reducing Existence of Riemann Mappings to Hyperbolic Geometry of Sub-domains of the Unit Disc Video

Unit 9: Pick Lemma and Hyperbolic Geometry on the Unit Disc

34. Differential or Infinitesimal Schwarzs Lemma, Picks Lemma, Hyperbolic Arclengths, Metric and Geodesics on the Unit Disc Video
35. Differential or Infinitesimal Schwarzs Lemma, Picks Lemma, Hyperbolic Arclengths, Metric and Geodesics on the Unit Disc. Video
36. Hyperbolic Geodesics for the Hyperbolic Metric on the Unit Disc Video
37. Schwarz-Pick Lemma for the Hyperbolic Metric on the Unit Disc Video

Unit 10: Theorems of Arzela-Ascoli and Montel

38. Arzela-Ascoli Theorem: Under Uniform Boundedness, Equicontinuity and Uniform Sequential Compactness are Equivalent Video
39. Completion of the Proof of the Arzela-Ascoli Theorem and Introduction to Montels Theorem Video
40. The Proof of Montels Theorem Video

Unit 11: Existence of a Riemann Mapping

41. The Candidate for a Riemann Mapping Video
42. Completion of Proof of The Riemann Mapping Theorem Video
43. Completion of Proof of The Riemann Mapping Theorem. Video