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Course Co-ordinated by :
IIT Madras
Course Available from :
02-March-2016
NPTEL
Mathematics
Advanced Complex Analysis - Part 2: Compactness of Meromorphic Functions in the Spherical Metric, Spherical Derivative, Normality, Theorems of Marty-Zalcman-Montel-Picard-Royden-Schottky (Video)
Under Review
Properties of the Image of an Analytic Function: Introduction to the Picard Theorems
Modules / Lectures
Unit 1: Theorems of Picard, Casorati-Weierstrass and Riemann on Removable Singularities
Properties of the Image of an Analytic Function: Introduction to the Picard Theorems
Recalling Singularities of Analytic Functions: Non-isolated and Isolated Removable, Pole and Essential Singularities
Recalling Riemann's Theorem on Removable Singularities
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
Neighborhood of Infinity, Limit at Infinity and Infinity as an Isolated Singularity
Studying Infinity: Formulating Epsilon-Delta Definitions for Infinite Limits and Limits at Infinity
Unit 3: Infinity as a Point of Analyticity
When is a function analytic at infinity ?
Laurent Expansion at Infinity and Riemann\'s Removable Singularities Theorem for the Point at Infinity
The Generalized Liouville Theorem: Little Brother of Little Picard and Analogue of Casorati-Weierstrass; Failure of Cauchy\'s Theorem at Infinity
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
Residue at Infinity and Introduction to the Residue Theorem for the Extended Complex Plane: Residue Theorem for the Point at Infinity
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
Infinity as an Essential Singularity and Transcendental Entire Functions
Meromorphic Functions on the Extended Complex Plane are Precisely Quotients of Polynomials
The Ubiquity of Meromorphic Functions: The Nerves of the Geometric Network Bridging Algebra, Analysis and Topology
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
Why Normal Convergence, but Not Globally Uniform Convergence, is the Inevitable in Complex Analysis
Measuring Distances to Infinity, the Function Infinity and Normal Convergence of Holomorphic Functions in the Spherical Metric
The Invariance Under Inversion of the Spherical Metric on the Extended Complex Plane
Unit 7: Hurwitz Theorems on Normal Limits in the Spherical Metric
Introduction to Hurwitz\'s Theorem for Normal Convergence of Holomorphic Functions in the Spherical Metric
Completion of Proof of Hurwitz\'s Theorem for Normal Limits of Analytic Functions in the Spherical Metric
Hurwitz\'s Theorem for Normal Limits of Meromorphic Functions in the Spherical Metric
Unit 8: The Inversion-invariant Spherical Derivative for Meromorphic Functions
What could the Derivative of a Meromorphic Function Relative to the Spherical Metric Possibly Be ?
Defining the Spherical Derivative of a Meromorphic Function
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
Topological Preliminaries: Translating Compactness into Boundedness
Introduction to the Arzela-Ascoli Theorem: Passing from abstract Compactness to verifiable Equicontinuity
Proof of the Arzela-Ascoli Theorem for Functions: Abstract Compactness Implies Equicontinuity
Proof of the Arzela-Ascoli Theorem for Functions: Equicontinuity Implies Compactness
Unit 10: The Montel Theorem - The Holomorphic Avatar of the Arzela-Ascoli Theorem
Introduction to the Montel Theorem - the Holomorphic Avatar of the Arzela-Ascoli Theorem & Why you get Equicontinuity for Free
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
Introduction to Marty\'s Theorem - the Meromorphic Avatar of the Montel & Arzela-Ascoli Theorems
Proof of one direction of Marty\'s Theorem - the Meromorphic Avatar of the Montel & Arzela-Ascoli Theorems - Normal Uniform Boundedness of Spherical Derivatives Implies Normal Sequential Compactness
Proof of the other direction of Marty\'s Theorem - the Meromorphic Avatar of the Montel & Arzela-Ascoli Theorems - Normal Sequential Compactness Implies Normal Uniform Boundedness of Spherical Derivatives
Unit 12: The Hurwitz, Montel and Marty Theorems at Infinity
Normal Convergence at Infinity and Hurwitz\'s Theorems for Normal Limits of Analytic and Meromorphic Functions at Infinity
Normal Sequential Compactness, Normal Uniform Boundedness and Montel\'s & Marty\'s Theorems at Infinity
Unit 13: Local Analysis of Normality by the Zooming Process and Zalcman\'s Lemma
Local Analysis of Normality and the Zooming Process - Motivation for Zalcman\'s Lemma
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
Local Analysis of Normality and the Zooming Process - Motivation for Zalcman\'s Lemma
Montel\'s Deep Theorem: The Fundamental Criterion for Normality or Fundamental Normality Test based on Omission of Values
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 & Problem Solving Session
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