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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 socalled 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

 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 – CasoratiWeierstrass Theorem; Dealing with the Point at Infinity … Unlimited

 Lecture 05 – Neighborhood of Infinity, Limit at Infinity and Infinity as an Isolated Singularity Unlimited
 Lecture 06 – Studying Infinity: Formulating EpsilonDelta Definitions for Infinite Limits … Unlimited

 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

 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

 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

 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

 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

 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 – Welldefinedness of the Spherical Derivative of a Meromorphic Function at … Unlimited

 Lecture 26 – Topological Preliminaries: Translating Compactness into Boundedness Unlimited
 Lecture 27 – Introduction to the ArzelaAscoli Theorem Unlimited
 Lecture 28 – Proof of the ArzelaAscoli Theorem for Functions Unlimited
 Lecture 29 – Proof of the ArzelaAscoli Theorem for Functions Unlimited

 Lecture 30 – Introduction to the Montel Theorem Unlimited
 Lecture 31 – Completion of Proof of the Montel Theorem Unlimited

 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

 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

 Lecture 37 – Local Analysis of Normality and the Zooming Process Unlimited
 Lecture 38 – Characterizing Normality at a Point by the Zooming Process and … Unlimited

 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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