Lecture Description
- The CosmoLearning Team
Course Index
- Properties of the Image of an Analytic Function: Introduction to the Picard Theorems
- Recalling Singularities of Analytic Functions: Non-isolated and Isolated Removable
- Recalling Riemann's Theorem on Removable Singularities
- Casorati-Weierstrass Theorem; Dealing with the Point at Infinity
- Neighborhood of Infinity, Limit at Infinity and Infinity as an Isolated Singularity
- Studying Infinity: Formulating Epsilon-Delta Definitions for Infinite Limits
- When is a function analytic at infinity?
- Laurent Expansion at Infinity and Riemann's Removable Singularities Theorem
- The Generalized Liouville Theorem: Little Brother of Little Picard
- Morera's Theorem at Infinity, Infinity as a Pole and Behaviour at Infinity
- Residue at Infinity and Introduction to the Residue Theorem for the Extended
- Proofs of Two Avatars of the Residue Theorem for the Extended Complex Plane
- Infinity as an Essential Singularity and Transcendental Entire Functions
- Meromorphic Functions on the Extended Complex Plane
- The Ubiquity of Meromorphic Functions
- Continuity of Meromorphic Functions at Poles and Topologies
- Why Normal Convergence, but Not Globally Uniform Convergence,
- Measuring Distances to Infinity, the Function Infinity and Normal Convergence
- The Invariance Under Inversion of the Spherical Metric on the Extended Complex Plane
- Introduction to Hurwitz's Theorem for Normal Convergence of Holomorphic Functions
- Completion of Proof of Hurwitz\'s Theorem for Normal Limits of Analytic Functions
- Hurwitz's Theorem for Normal Limits of Meromorphic Functions in the Spherical Metric
- What could the Derivative of a Meromorphic Function
- Defining the Spherical Derivative of a Meromorphic Function
- Well-definedness of the Spherical Derivative of a Meromorphic Function
- Topological Preliminaries: Translating Compactness into Boundedness
- Introduction to the Arzela-Ascoli Theorem
- Proof of the Arzela-Ascoli Theorem for Functions
- Proof of the Arzela-Ascoli Theorem for Functions
- Introduction to the Montel Theorem
- Completion of Proof of the Montel Theorem
- Introduction to Marty's Theorem
- Proof of one direction of Marty's Theorem
- Proof of the other direction of Marty's Theorem
- Normal Convergence at Infinity and Hurwitz's Theorems
- Normal Sequential Compactness, Normal Uniform Boundedness
- Local Analysis of Normality and the Zooming Process - Motivation for Zalcman's Lemma
- Characterizing Normality at a Point by the Zooming Process
- Local Analysis of Normality and the Zooming Process - Motivation for Zalcman\'s Lemma
- Montel's Deep Theorem: The Fundamental Criterion for Normality
- Proofs of the Great and Little Picard Theorems
- Royden's Theorem on Normality Based On Growth Of Derivatives
- 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.