Proof of one direction of Marty's Theorem 
Proof of one direction of Marty's Theorem
by IIT Madras
Video Lecture 33 of 43
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Date Added: August 11, 2016

Lecture Description

This video lecture, part of the series Advanced Complex Analysis II by Prof. , does not currently have a detailed description and video lecture title. If you have watched this lecture and know what it is about, particularly what Mathematics topics are discussed, please help us by commenting on this video with your suggested description and title. Many thanks from,

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Course Index

  1. Properties of the Image of an Analytic Function: Introduction to the Picard Theorems
  2. Recalling Singularities of Analytic Functions: Non-isolated and Isolated Removable
  3. Recalling Riemann's Theorem on Removable Singularities
  4. Casorati-Weierstrass Theorem; Dealing with the Point at Infinity
  5. Neighborhood of Infinity, Limit at Infinity and Infinity as an Isolated Singularity
  6. Studying Infinity: Formulating Epsilon-Delta Definitions for Infinite Limits
  7. When is a function analytic at infinity?
  8. Laurent Expansion at Infinity and Riemann's Removable Singularities Theorem
  9. The Generalized Liouville Theorem: Little Brother of Little Picard
  10. Morera's Theorem at Infinity, Infinity as a Pole and Behaviour at Infinity
  11. Residue at Infinity and Introduction to the Residue Theorem for the Extended
  12. Proofs of Two Avatars of the Residue Theorem for the Extended Complex Plane
  13. Infinity as an Essential Singularity and Transcendental Entire Functions
  14. Meromorphic Functions on the Extended Complex Plane
  15. The Ubiquity of Meromorphic Functions
  16. Continuity of Meromorphic Functions at Poles and Topologies
  17. Why Normal Convergence, but Not Globally Uniform Convergence,
  18. Measuring Distances to Infinity, the Function Infinity and Normal Convergence
  19. The Invariance Under Inversion of the Spherical Metric on the Extended Complex Plane
  20. Introduction to Hurwitz's Theorem for Normal Convergence of Holomorphic Functions
  21. Completion of Proof of Hurwitz\'s Theorem for Normal Limits of Analytic Functions
  22. Hurwitz's Theorem for Normal Limits of Meromorphic Functions in the Spherical Metric
  23. What could the Derivative of a Meromorphic Function
  24. Defining the Spherical Derivative of a Meromorphic Function
  25. Well-definedness of the Spherical Derivative of a Meromorphic Function
  26. Topological Preliminaries: Translating Compactness into Boundedness
  27. Introduction to the Arzela-Ascoli Theorem
  28. Proof of the Arzela-Ascoli Theorem for Functions
  29. Proof of the Arzela-Ascoli Theorem for Functions
  30. Introduction to the Montel Theorem
  31. Completion of Proof of the Montel Theorem
  32. Introduction to Marty's Theorem
  33. Proof of one direction of Marty's Theorem
  34. Proof of the other direction of Marty's Theorem
  35. Normal Convergence at Infinity and Hurwitz's Theorems
  36. Normal Sequential Compactness, Normal Uniform Boundedness
  37. Local Analysis of Normality and the Zooming Process - Motivation for Zalcman's Lemma
  38. Characterizing Normality at a Point by the Zooming Process
  39. Local Analysis of Normality and the Zooming Process - Motivation for Zalcman\'s Lemma
  40. Montel's Deep Theorem: The Fundamental Criterion for Normality
  41. Proofs of the Great and Little Picard Theorems
  42. Royden's Theorem on Normality Based On Growth Of Derivatives
  43. Schottky's Theorem: Uniform Boundedness from a Point to a Neighbourhood

Course Description

This is the second part of a series of lectures on advanced topics in Complex Analysis. By advanced, we mean topics that are not (or just barely) touched upon in a first course on Complex Analysis. The theme of the course is to study compactness and convergence in families of analytic (or holomorphic) functions and in families of meromorphic functions. The compactness we are interested herein is the so-called sequential compactness, and more specifically it is normal convergence -- namely convergence on compact subsets. The final objective is to prove the Great or Big Picard Theorem and deduce the Little or Small Picard Theorem. This necessitates studying the point at infinity both as a value or limit attained, and as a point in the domain of definition of the functions involved. This is done by thinking of the point at infinity as the north pole on the sphere, by appealing to the Riemann Stereographic Projection from the Riemann Sphere. Analytic properties are tied to the spherical metric on the Riemann Sphere. The notion of spherical derivative is introduced for meromorphic functions. Infinity is studied as a singular point. Laurent series at infinity, residue at infinity and a version of the Residue theorem for domains including the point at infinity are explained. In later lectures, Marty's theorem -- a version of the Montel theorem for meromorphic functions, Zalcman's Lemma -- a fundamental theorem on the local analysis of non-normality, Montel's theorem on normality, Royden's theorem and Schottky's theorem are proved. For more details on what is covered lecturewise, please look at the titles, goals and keywords which are given for each lecture.

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