**# 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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