T.E. Venkata Balaji: Mathematics - Advanced Complex Analysis - Part 2 (IT Madras)

# Click the up-left corner for the playlist of the 43 videos 

source: nptelhrd    2016年3月2日
Advanced Complex Analysis - Part 2 by Dr. T.E. Venkata Balaji, Department of Mathematics, IIT Madras. For more details on NPTEL visit http://nptel.ac.in

Lec-01 Properties of the Image of an Analytic Function: Introduction to the Picard Theorems 43:05
Lec-02 Recalling Singularities of Analytic Functions: Non-isolated and Isolated Removable 44:16
Lec-03 Recalling Riemann\'s Theorem on Removable Singularities 52:33
Lec-04 Casorati-Weierstrass Theorem; Dealing with the Point at Infinity 47:40
Lec-05 Neighborhood of Infinity, Limit at Infinity and Infinity as an Isolated Singularity 31:17
Lec-06 Studying Infinity: Formulating Epsilon-Delta Definitions for Infinite Limits 48:00
Lec-07 When is a function analytic at infinity ? 42:31
Lec-08 Laurent Expansion at Infinity and Riemann's Removable Singularities Theorem 40:33
Lec-09 The Generalized Liouville Theorem: Little Brother of Little Picard 47:28
Lec-10 Morera's Theorem at Infinity, Infinity as a Pole and Behaviour at Infinity 47:12
Lec-11 Residue at Infinity and Introduction to the Residue Theorem for the Extended 41:01
Lec-12 Proofs of Two Avatars of the Residue Theorem for the Extended Complex Plane 51:28
Lec-13 Infinity as an Essential Singularity and Transcendental Entire Functions 40:59
Lec-14 Meromorphic Functions on the Extended Complex Plane 47:24
Lec-15 The Ubiquity of Meromorphic Functions 58:07
Lec-16 Continuity of Meromorphic Functions at Poles and Topologies 41:49
Lec-17 Why Normal Convergence, but Not Globally Uniform Convergence, 45:23
Lec-18 Measuring Distances to Infinity, the Function Infinity and Normal Convergence 46:30
Lec-19 The Invariance Under Inversion of the Spherical Metric on the Extended Complex Plane 51:30
Lec-20 Introduction to Hurwitz\'s Theorem for Normal Convergence of Holomorphic Functions 43:55
Lec-21 Completion of Proof of Hurwitz\'s Theorem for Normal Limits of Analytic Functions 37:54
Lec-22 Hurwitz's Theorem for Normal Limits of Meromorphic Functions in the Spherical Metric 44:10
Lec-23 What could the Derivative of a Meromorphic Function 44:40
Lec-24 Defining the Spherical Derivative of a Meromorphic Function 36:15
Lec-25 Well-definedness of the Spherical Derivative of a Meromorphic Function 43:42
Lec-26 Topological Preliminaries: Translating Compactness into Boundedness 33:16
Lec-27 Introduction to the Arzela-Ascoli Theorem 41:04
Lec-28 Proof of the Arzela-Ascoli Theorem for Functions 43:20
Lec-29 Proof of the Arzela-Ascoli Theorem for Functions 44:00
Lec-30 Introduction to the Montel Theorem 39:46
Lec-31 Completion of Proof of the Montel Theorem 43:54
Lec-32 Introduction to Marty's Theorem 44:08
Lec-33 Proof of one direction of Marty's Theorem 44:48
Lec-34 Proof of the other direction of Marty's Theorem 41:07
Lec-35 Normal Convergence at Infinity and Hurwitz's Theorems 44:07
Lec-36 Normal Sequential Compactness, Normal Uniform Boundedness 42:12
Lec-37 Local Analysis of Normality and the Zooming Process - Motivation for Zalcman's Lemma 42:42
Lec-38 Characterizing Normality at a Point by the Zooming Process 38:25
Lec-39 Local Analysis of Normality and the Zooming Process - Motivation for Zalcman\'s Lemma 43:26
Lec-40 Montel\'s Deep Theorem: The Fundamental Criterion for Normality 48:16
Lec-41 Proofs of the Great and Little Picard Theorems 38:23
Lec-42 Royden\'s Theorem on Normality Based On Growth Of Derivatives 36:04
Lec-43 Schottky\'s Theorem: Uniform Boundedness from a Point to a Neighbourhood 41:44

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