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Advanced Complex Analysis Part 2

Advanced Complex Analysis Part 2. Instructor: Dr. T.E. Venkata Balaji, Department of Mathematics, IIT Madras. 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. (from nptel.ac.in)

Properties of the Image of an Analytic Function


Unit 1: Theorems of Picard, Casorati-Weierstrass and Riemann on Removable Singularities
Lecture 01 - Properties of the Image of an Analytic Function: Introduction to the Picard Theorems
Lecture 02 - Recalling Singularities of Analytic Functions: Non-isolated and Isolated Removable, Pole and Essential Singularities
Lecture 03 - Recalling Riemann's Theorem on Removable Singularities
Lecture 04 - Casorati-Weierstrass Theorem; Dealing with the Point at Infinity - Riemann Sphere and Riemann Stereographic Projection
Unit 2: Neighborhoods of Infinity, Limits at Infinity and Infinite Limits
Lecture 05 - Neighborhood of Infinity, Limit at Infinity and Infinity as an Isolated Singularity
Lecture 06 - Studying Infinity: Formulating Epsilon-Delta Definitions for Infinite Limits and Limits at Infinity
Unit 3: Infinity as a Point of Analyticity
Lecture 07 - When is a Function Analytic at Infinity?
Lecture 08 - Laurent Expansion at Infinity and Riemann's Removable Singularities Theorem for the Point at Infinity
Lecture 09 - The Generalized Liouville Theorem: Little Brother of Little Picard and Analogue of Casorati-Weierstrass; Failure of Cauchy's Theorem at Infinity
Lecture 10 - Morera's Theorem at Infinity, Infinity as a Pole and Behaviour at Infinity of Rational and Meromorphic Functions
Unit 4: Residue at Infinity and Residue Theorem for the Extended Complex Plane
Lecture 11 - Residue at Infinity and Introduction to the Residue Theorem for the Extended Complex Plane: Residue Theorem for the Point at Infinity
Lecture 12 - Proofs of Two Avatars of the Residue Theorem for the Extended Complex Plane and Applications of the Residue at Infinity
Unit 5: The Behavior of Transcendental and Meromorphic Functions at Infinity
Lecture 13 - Infinity as an Essential Singularity and Transcendental Entire Functions
Lecture 14 - Meromorphic Functions on the Extended Complex Plane are Precisely Quotients of Polynomials
Lecture 15 - The Ubiquity of Meromorphic Functions: The Nerves of the Geometric Network Bridging Algebra, Analysis and Topology
Lecture 16 - Continuity of Meromorphic Functions at Poles and Topologies of Spaces of Functions
Unit 6: Normal Convergence in the Inversion-Invariant Spherical Metric on the Extended Plane
Lecture 17 - Why Normal Convergence, but Not Globally Uniform Convergence, is the Inevitable in Complex Analysis
Lecture 18 - Measuring Distances to Infinity, the Function Infinity and Normal Convergence of Holomorphic Functions in the Spherical Metric
Lecture 19 - The Invariance under Inversion of the Spherical Metric on the Extended Complex Plane
Unit 7: Hurwitz Theorems on Normal Limits of Holomorphic and Meromorphic Functions under the Spherical Metric
Lecture 20 - Introduction to Hurwitz's Theorem for Normal Convergence of Holomorphic Functions in the Spherical Metric
Lecture 21 - Completion of Proof of Hurwitz's Theorem for Normal Limits of Analytic Functions in the Spherical Metric
Lecture 22 - Hurwitz's Theorem for Normal Limits of Meromorphic Functions in the Spherical Metric
Unit 8: The Inversion-Invariant Spherical Derivative for Meromorphic Functions
Lecture 23 - What could the Derivative of a Meromorphic Function Relative to the Spherical Metric Possibly Be?
Lecture 24 - Defining the Spherical Derivative of a Meromorphic Function
Lecture 25 - Well-definedness of the Spherical Derivative of a Meromorphic Function at a Pole and Inversion-invariance of the Spherical Derivative
Unit 9: From Compactness to Boundedness via Equicontinuity
Lecture 26 - Topological Preliminaries: Translating Compactness into Boundedness
Lecture 27 - Introduction to the Arzela-Ascoli Theorem: Passing from Abstract Compactness to Verifiable Equicontinuity
Lecture 28 - Proof of the Arzela-Ascoli Theorem for Functions: Abstract Compactness Implies Equicontinuity
Lecture 29 - Proof of the Arzela-Ascoli Theorem for Functions: Equicontinuity Implies Compactness
Unit 10: The Montel Theorem - The Holomorphic Avatar of the Arzela-Ascoli Theorem
Lecture 30 - Introduction to the Montel Theorem - the Holomorphic Avatar of the Arzela-Ascoli Theorem and Why you get Equicontinuity for Free
Lecture 31 - Completion of Proof of the Montel Theorem - the Holomorphic Avatar of the Arzela-Ascoli Theorem
Unit 11: The Marty Theorem - The Meromorphic Avatar of the Montel and Arzela-Ascoli Theorems
Lecture 32 - Introduction to Marty's Theorem - the Meromorphic Avatar of the Montel and Arzela-Ascoli Theorems
Lecture 33 - Proof of One Direction of Marty's Theorem - the Meromorphic Avatar of the Montel and Arzela-Ascoli Theorems - Normal Uniform Boundedness of Spherical Derivatives Implies Normal Sequential Compactness
Lecture 34 - Proof of the Other Direction of Marty's Theorem - the Meromorphic Avatar of the Montel and Arzela-Ascoli Theorems - Normal Sequential Compactness Implies Normal Uniform Boundedness of Spherical Derivatives
Unit 12: The Hurwitz, Montel and Marty Theorems at Infinity
Lecture 35 - Normal Convergence at Infinity and Hurwitz's Theorems for Normal Limits of Analytic and Meromorphic Functions at Infinity
Lecture 36 - Normal Sequential Compactness, Normal Uniform Boundedness and Montel's and Marty's Theorems at Infinity
Unit 13: Local Analysis of Normality by the Zooming Process and Zalcman's Lemma
Lecture 37 - Local Analysis of Normality and the Zooming Process - Motivation for Zalcman's Lemma
Lecture 38 - Characterizing Normality at a Point by the Zooming Process and the Motivation for Zalcman's Lemma
Unit 14: Zalcman's Lemma, Montel's Normality Criterion and Theorems of Picard, Royden and Schottky
Lecture 39 - Local Analysis of Normality and the Zooming Process - Motivation for Zalcman's Lemma
Lecture 40 - Montel's Deep Theorem: The Fundamental Criterion for Normality or Fundamental Normality Test based on Omission of Values
Lecture 41 - Proofs of the Great and Little Picard Theorems
Lecture 42 - Royden's Theorem on Normality based on Growth of Derivatives
Lecture 43 - Schottky's Theorem: Uniform Boundedness from a Point to a Neighbourhood and Problem Solving Session

References
Advanced Complex Analysis Part 2
Instructor: Dr. T.E. Venkata Balaji, Department of Mathematics, IIT Madras. This is the second part of a series of lectures on advanced topics in Complex Analysis.