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

- Institution: IIT Madras
- Department: Mathematics
- Teacher: Dr. T.E. Venkata Balaji
- Level: Undergraduate
- Language: English
- Subjects: Mathematics
- Source: [NPTEL](https://nptel.ac.in/courses/111106094)
- Licence: [CC BY-NC-SA](https://nptel.ac.in/faq) (Non-commercial use only.)
- Syllabus: [at the source](https://nptel.ac.in/courses/111106094)
- Who can watch: Tweens profiles and older (sources give no ratings; placed by Invisible Network from the source's own words)

## Lectures

1. [Properties of the Image of an Analytic Function: Introduction to the Picard Theorems](https://invisiblenetwork.ai/learn/nptel/111106094/391516a745b6) (Video)
2. [Recalling Singularities of Analytic Functions: Non-isolated and Isolated Removable, Pole and Essential Singularities](https://invisiblenetwork.ai/learn/nptel/111106094/de7d26900919) (Video)
3. [Recalling Riemann's Theorem on Removable Singularities](https://invisiblenetwork.ai/learn/nptel/111106094/e2175f3a1776) (Video)
4. [Casorati-Weierstrass Theorem; Dealing with the Point at Infinity -- Riemann Sphere and Riemann Stereographic Projection](https://invisiblenetwork.ai/learn/nptel/111106094/f8f5039cfcd4) (Video)
5. [Neighborhood of Infinity, Limit at Infinity and Infinity as an Isolated Singularity](https://invisiblenetwork.ai/learn/nptel/111106094/20b479e08e56) (Video)
6. [Studying Infinity: Formulating Epsilon-Delta Definitions for Infinite Limits and Limits at Infinity](https://invisiblenetwork.ai/learn/nptel/111106094/23c7ce2ffe6d) (Video)
7. [When is a function analytic at infinity ?](https://invisiblenetwork.ai/learn/nptel/111106094/15d58ec30461) (Video)
8. [Laurent Expansion at Infinity and Riemann\'s Removable Singularities Theorem for the Point at Infinity](https://invisiblenetwork.ai/learn/nptel/111106094/3eff4b135208) (Video)
9. [The Generalized Liouville Theorem: Little Brother of Little Picard and Analogue of Casorati-Weierstrass; Failure of Cauchy\'s Theorem at Infinity](https://invisiblenetwork.ai/learn/nptel/111106094/cdccf11fa73e) (Video)
10. [Morera\'s Theorem at Infinity, Infinity as a Pole and Behaviour at Infinity of Rational and Meromorphic Functions](https://invisiblenetwork.ai/learn/nptel/111106094/3626c1267266) (Video)
11. [Residue at Infinity and Introduction to the Residue Theorem for the Extended Complex Plane: Residue Theorem for the Point at Infinity](https://invisiblenetwork.ai/learn/nptel/111106094/b3f939c0c898) (Video)
12. [Proofs of Two Avatars of the Residue Theorem for the Extended Complex Plane and Applications of the Residue at Infinity](https://invisiblenetwork.ai/learn/nptel/111106094/63d9b654b19b) (Video)
13. [Infinity as an Essential Singularity and Transcendental Entire Functions](https://invisiblenetwork.ai/learn/nptel/111106094/c269e4c0952f) (Video)
14. [Meromorphic Functions on the Extended Complex Plane are Precisely Quotients of Polynomials](https://invisiblenetwork.ai/learn/nptel/111106094/7888ad482a26) (Video)
15. [The Ubiquity of Meromorphic Functions: The Nerves of the Geometric Network Bridging Algebra, Analysis and Topology](https://invisiblenetwork.ai/learn/nptel/111106094/f01db2842bcc) (Video)
16. [Continuity of Meromorphic Functions at Poles and Topologies of Spaces of Functions](https://invisiblenetwork.ai/learn/nptel/111106094/0215b4f75f4a) (Video)
17. [Why Normal Convergence, but Not Globally Uniform Convergence, is the Inevitable in Complex Analysis](https://invisiblenetwork.ai/learn/nptel/111106094/258c530497f6) (Video)
18. [Measuring Distances to Infinity, the Function Infinity and Normal Convergence of Holomorphic Functions in the Spherical Metric](https://invisiblenetwork.ai/learn/nptel/111106094/b48bc9a292d0) (Video)
19. [The Invariance Under Inversion of the Spherical Metric on the Extended Complex Plane](https://invisiblenetwork.ai/learn/nptel/111106094/5c90076c910a) (Video)
20. [Introduction to Hurwitz\'s Theorem for Normal Convergence of Holomorphic Functions in the Spherical Metric](https://invisiblenetwork.ai/learn/nptel/111106094/350e9e925a42) (Video)
21. [Completion of Proof of Hurwitz\'s Theorem for Normal Limits of Analytic Functions in the Spherical Metric](https://invisiblenetwork.ai/learn/nptel/111106094/076af1c57364) (Video)
22. [Hurwitz\'s Theorem for Normal Limits of Meromorphic Functions in the Spherical Metric](https://invisiblenetwork.ai/learn/nptel/111106094/cf7baa7d1384) (Video)
23. [What could the Derivative of a Meromorphic Function Relative to the Spherical Metric Possibly Be ?](https://invisiblenetwork.ai/learn/nptel/111106094/a849548e1331) (Video)
24. [Defining the Spherical Derivative of a Meromorphic Function](https://invisiblenetwork.ai/learn/nptel/111106094/1b9210188508) (Video)
25. [Well-definedness of the Spherical Derivative of a Meromorphic Function at a Pole and Inversion-invariance of the Spherical Derivative](https://invisiblenetwork.ai/learn/nptel/111106094/816b0658ca16) (Video)
26. [Topological Preliminaries: Translating Compactness into Boundedness](https://invisiblenetwork.ai/learn/nptel/111106094/37e152832882) (Video)
27. [Introduction to the Arzela-Ascoli Theorem: Passing from abstract Compactness to verifiable Equicontinuity](https://invisiblenetwork.ai/learn/nptel/111106094/7dceeab24abb) (Video)
28. [Proof of the Arzela-Ascoli Theorem for Functions: Abstract Compactness Implies Equicontinuity](https://invisiblenetwork.ai/learn/nptel/111106094/1de38d8496cd) (Video)
29. [Proof of the Arzela-Ascoli Theorem for Functions: Equicontinuity Implies Compactness](https://invisiblenetwork.ai/learn/nptel/111106094/abcfcfa75bc9) (Video)
30. [Introduction to the Montel Theorem - the Holomorphic Avatar of the Arzela-Ascoli Theorem & Why you get Equicontinuity for Free](https://invisiblenetwork.ai/learn/nptel/111106094/d387a24b3905) (Video)
31. [Completion of Proof of the Montel Theorem - the Holomorphic Avatar of the Arzela-Ascoli Theorem](https://invisiblenetwork.ai/learn/nptel/111106094/f86e938ee0e5) (Video)
32. [Introduction to Marty\'s Theorem - the Meromorphic Avatar of the Montel & Arzela-Ascoli Theorems](https://invisiblenetwork.ai/learn/nptel/111106094/f10bd6d0e912) (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](https://invisiblenetwork.ai/learn/nptel/111106094/f506bed545ce) (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](https://invisiblenetwork.ai/learn/nptel/111106094/98c616eec414) (Video)
35. [Normal Convergence at Infinity and Hurwitz\'s Theorems for Normal Limits of Analytic and Meromorphic Functions at Infinity](https://invisiblenetwork.ai/learn/nptel/111106094/0aea221bd350) (Video)
36. [Normal Sequential Compactness, Normal Uniform Boundedness and Montel\'s & Marty\'s Theorems at Infinity](https://invisiblenetwork.ai/learn/nptel/111106094/9b48cf3d4a66) (Video)
37. [Local Analysis of Normality and the Zooming Process - Motivation for Zalcman\'s Lemma](https://invisiblenetwork.ai/learn/nptel/111106094/d38c626bc232) (Video)
38. [Characterizing Normality at a Point by the Zooming Process and the Motivation for Zalcman\'s Lemma](https://invisiblenetwork.ai/learn/nptel/111106094/b5293247ad6f) (Video)
39. [Local Analysis of Normality and the Zooming Process - Motivation for Zalcman\'s Lemma](https://invisiblenetwork.ai/learn/nptel/111106094/95187a90e078) (Video)
40. [Montel\'s Deep Theorem: The Fundamental Criterion for Normality or Fundamental Normality Test based on Omission of Values](https://invisiblenetwork.ai/learn/nptel/111106094/a58fa60d68ec) (Video)
41. [Proofs of the Great and Little Picard Theorems](https://invisiblenetwork.ai/learn/nptel/111106094/04e89229d7e4) (Video)
42. [Royden\'s Theorem on Normality Based On Growth Of Derivatives](https://invisiblenetwork.ai/learn/nptel/111106094/dfa6651f42ec) (Video)
43. [Schottky\'s Theorem: Uniform Boundedness from a Point to a Neighbourhood & Problem Solving Session](https://invisiblenetwork.ai/learn/nptel/111106094/bf3944ca996d) (Video)