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Advanced Complex Analysis Part 2

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Description

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)

Course Curriculum

Unit 1: Theorems of Picard, Casorati-Weierstrass and Riemann on Removable Singularities
- Lecture 01 – Properties of the Image of an Analytic Function Unlimited
- Lecture 02 – Recalling Singularities of Analytic Functions Unlimited
- Lecture 03 – Recalling Riemann’s Theorem on Removable Singularities Unlimited
- Lecture 04 – Casorati-Weierstrass Theorem; Dealing with the Point at Infinity … Unlimited
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 Unlimited
- Lecture 06 – Studying Infinity: Formulating Epsilon-Delta Definitions for Infinite Limits … Unlimited
Unit 3: Infinity as a Point of Analyticity
- Lecture 07 – When is a Function Analytic at Infinity? Unlimited
- Lecture 08 – Laurent Expansion at Infinity and Riemann’s Removable Singularities Theorem … Unlimited
- Lecture 09 – The Generalized Liouville Theorem: Little Brother of Little Picard and … Unlimited
- Lecture 10 – Morera’s Theorem at Infinity, Infinity as a Pole and Behaviour at Infinity of … Unlimited
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 … Unlimited
- Lecture 12 – Proofs of Two Avatars of the Residue Theorem for the Extended Complex Plane … Unlimited
Unit 5: The Behavior of Transcendental and Meromorphic Functions at Infinity
- Lecture 13 – Infinity as an Essential Singularity and Transcendental Entire Functions Unlimited
- Lecture 14 – Meromorphic Functions on the Extended Complex Plane are Precisely … Unlimited
- Lecture 15 – The Ubiquity of Meromorphic Functions Unlimited
- Lecture 16 – Continuity of Meromorphic Functions at Poles and Topologies of Spaces of Functions Unlimited
Unit 6: Normal Convergence in the Inversion-Invariant Spherical Metric on the Extended Plane
- Lecture 17 – Why Normal Convergence, but Not Globally Uniform Convergence, … Unlimited
- Lecture 18 – Measuring Distances to Infinity, the Function Infinity and Normal Convergence … Unlimited
- Lecture 19 – The Invariance under Inversion of the Spherical Metric on … Unlimited
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 … Unlimited
- Lecture 21 – Completion of Proof of Hurwitz’s Theorem for Normal Limits of Analytic … Unlimited
- Lecture 22 – Hurwitz’s Theorem for Normal Limits of Meromorphic Functions in … Unlimited
Unit 8: The Inversion-Invariant Spherical Derivative for Meromorphic Functions
- Lecture 23 – What could the Derivative of a Meromorphic Function Relative to … Unlimited
- Lecture 24 – Defining the Spherical Derivative of a Meromorphic Function Unlimited
- Lecture 25 – Well-definedness of the Spherical Derivative of a Meromorphic Function at … Unlimited
Unit 9: From Compactness to Boundedness via Equicontinuity
- Lecture 26 – Topological Preliminaries: Translating Compactness into Boundedness Unlimited
- Lecture 27 – Introduction to the Arzela-Ascoli Theorem Unlimited
- Lecture 28 – Proof of the Arzela-Ascoli Theorem for Functions Unlimited
- Lecture 29 – Proof of the Arzela-Ascoli Theorem for Functions Unlimited
Unit 10: The Montel Theorem - The Holomorphic Avatar of the Arzela-Ascoli Theorem
- Lecture 30 – Introduction to the Montel Theorem Unlimited
- Lecture 31 – Completion of Proof of the Montel Theorem Unlimited
Unit 11: The Marty Theorem - The Meromorphic Avatar of the Montel and Arzela-Ascoli Theorems
- Lecture 32 – Introduction to Marty’s Theorem Unlimited
- Lecture 33 – Proof of One Direction of Marty’s Theorem Unlimited
- Lecture 34 – Proof of the Other Direction of Marty’s Theorem Unlimited
Unit 12: The Hurwitz, Montel and Marty Theorems at Infinity
- Lecture 35 – Normal Convergence at Infinity and Hurwitz’s Theorems for Normal Limits of … Unlimited
- Lecture 36 – Normal Sequential Compactness, Normal Uniform Boundedness and … Unlimited
Unit 13: Local Analysis of Normality by the Zooming Process and Zalcman's Lemma
- Lecture 37 – Local Analysis of Normality and the Zooming Process Unlimited
- Lecture 38 – Characterizing Normality at a Point by the Zooming Process and … Unlimited
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 Unlimited
- Lecture 40 – Montel’s Deep Theorem: The Fundamental Criterion for Normality or … Unlimited
- Lecture 41 – Proofs of the Great and Little Picard Theorems Unlimited
- Lecture 42 – Royden’s Theorem on Normality based on Growth of Derivatives Unlimited
- Lecture 43 – Schottky’s Theorem: Uniform Boundedness from a Point to a Neighbourhood … Unlimited

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