Advanced Complex Analysis - Part 2: Compactness of Meromorphic Functions in the Spherical Metric, Spherical Derivative, Normality, Theorems of Marty-Zalcman-Montel-Picard-Royden-Schottky

Lectures

Unit 1: Theorems of Picard, Casorati-Weierstrass and Riemann on Removable Singularities

1. Properties of the Image of an Analytic Function: Introduction to the Picard Theorems Video
2. Recalling Singularities of Analytic Functions: Non-isolated and Isolated Removable, Pole and Essential Singularities Video
3. Recalling Riemann's Theorem on Removable Singularities Video
4. Casorati-Weierstrass Theorem; Dealing with the Point at Infinity -- Riemann Sphere and Riemann Stereographic Projection Video

Unit 2: Neighborhoods of Infinity, Limits at Infinity and Infinite Limits

5. Neighborhood of Infinity, Limit at Infinity and Infinity as an Isolated Singularity Video
6. Studying Infinity: Formulating Epsilon-Delta Definitions for Infinite Limits and Limits at Infinity Video

Unit 3: Infinity as a Point of Analyticity

7. When is a function analytic at infinity ? Video
8. Laurent Expansion at Infinity and Riemann\'s Removable Singularities Theorem for the Point at Infinity Video
9. The Generalized Liouville Theorem: Little Brother of Little Picard and Analogue of Casorati-Weierstrass; Failure of Cauchy\'s Theorem at Infinity Video
10. Morera\'s Theorem at Infinity, Infinity as a Pole and Behaviour at Infinity of Rational and Meromorphic Functions Video

Unit 4: Residue at Infinity and Residue Theorem for the Extended Complex Plane

11. Residue at Infinity and Introduction to the Residue Theorem for the Extended Complex Plane: Residue Theorem for the Point at Infinity Video
12. Proofs of Two Avatars of the Residue Theorem for the Extended Complex Plane and Applications of the Residue at Infinity Video

Unit 5: The Behavior of Transcendental and Meromorphic Functions at Infinity

13. Infinity as an Essential Singularity and Transcendental Entire Functions Video
14. Meromorphic Functions on the Extended Complex Plane are Precisely Quotients of Polynomials Video
15. The Ubiquity of Meromorphic Functions: The Nerves of the Geometric Network Bridging Algebra, Analysis and Topology Video
16. Continuity of Meromorphic Functions at Poles and Topologies of Spaces of Functions Video

Unit 6: Normal Convergence In The Inversion-Invariant Spherical Metric on the Extended Plane

17. Why Normal Convergence, but Not Globally Uniform Convergence, is the Inevitable in Complex Analysis Video
18. Measuring Distances to Infinity, the Function Infinity and Normal Convergence of Holomorphic Functions in the Spherical Metric Video
19. The Invariance Under Inversion of the Spherical Metric on the Extended Complex Plane Video

Unit 7: Hurwitz Theorems on Normal Limits in the Spherical Metric

20. Introduction to Hurwitz\'s Theorem for Normal Convergence of Holomorphic Functions in the Spherical Metric Video
21. Completion of Proof of Hurwitz\'s Theorem for Normal Limits of Analytic Functions in the Spherical Metric Video
22. Hurwitz\'s Theorem for Normal Limits of Meromorphic Functions in the Spherical Metric Video

Unit 8: The Inversion-invariant Spherical Derivative for Meromorphic Functions

23. What could the Derivative of a Meromorphic Function Relative to the Spherical Metric Possibly Be ? Video
24. Defining the Spherical Derivative of a Meromorphic Function Video
25. Well-definedness of the Spherical Derivative of a Meromorphic Function at a Pole and Inversion-invariance of the Spherical Derivative Video

Unit 9: From Compactness to Boundedness via Equicontinuity

26. Topological Preliminaries: Translating Compactness into Boundedness Video
27. Introduction to the Arzela-Ascoli Theorem: Passing from abstract Compactness to verifiable Equicontinuity Video
28. Proof of the Arzela-Ascoli Theorem for Functions: Abstract Compactness Implies Equicontinuity Video
29. Proof of the Arzela-Ascoli Theorem for Functions: Equicontinuity Implies Compactness Video

Unit 10: The Montel Theorem - The Holomorphic Avatar of the Arzela-Ascoli Theorem

30. Introduction to the Montel Theorem - the Holomorphic Avatar of the Arzela-Ascoli Theorem & Why you get Equicontinuity for Free Video
31. Completion of Proof of the Montel Theorem - the Holomorphic Avatar of the Arzela-Ascoli Theorem Video

Unit 11: The Marty Theorem - The Meromorphic Avatar of the Montel and Arzela-Ascoli Theorems

32. Introduction to Marty\'s Theorem - the Meromorphic Avatar of the Montel & Arzela-Ascoli Theorems Video
33. Proof of one direction of Marty\'s Theorem - the Meromorphic Avatar of the Montel & Arzela-Ascoli Theorems - Normal Uniform Boundedness of Spherical Derivatives Implies Normal Sequential Compactness Video
34. Proof of the other direction of Marty\'s Theorem - the Meromorphic Avatar of the Montel & Arzela-Ascoli Theorems - Normal Sequential Compactness Implies Normal Uniform Boundedness of Spherical Derivatives Video

Unit 12: The Hurwitz, Montel and Marty Theorems at Infinity

35. Normal Convergence at Infinity and Hurwitz\'s Theorems for Normal Limits of Analytic and Meromorphic Functions at Infinity Video
36. Normal Sequential Compactness, Normal Uniform Boundedness and Montel\'s & Marty\'s Theorems at Infinity Video

Unit 13: Local Analysis of Normality by the Zooming Process and Zalcman\'s Lemma

37. Local Analysis of Normality and the Zooming Process - Motivation for Zalcman\'s Lemma Video
38. Characterizing Normality at a Point by the Zooming Process and the Motivation for Zalcman\'s Lemma Video

Unit 14: Zalcman\'s Lemma, Montel\'s Normality Criterion and Theorems of Picard, Royden and Schottky

39. Local Analysis of Normality and the Zooming Process - Motivation for Zalcman\'s Lemma Video
40. Montel\'s Deep Theorem: The Fundamental Criterion for Normality or Fundamental Normality Test based on Omission of Values Video
41. Proofs of the Great and Little Picard Theorems Video
42. Royden\'s Theorem on Normality Based On Growth Of Derivatives Video
43. Schottky\'s Theorem: Uniform Boundedness from a Point to a Neighbourhood & Problem Solving Session Video