Advanced Complex Analysis II

Video Lectures

Displaying all 43 video lectures.
Lecture 1
Properties of the Image of an Analytic Function: Introduction to the Picard Theorems
Play Video		 Properties of the Image of an Analytic Function: Introduction to the Picard Theorems
Lecture 2
Recalling Singularities of Analytic Functions: Non-isolated and Isolated Removable
Play Video		 Recalling Singularities of Analytic Functions: Non-isolated and Isolated Removable
Lecture 3
Recalling Riemann's Theorem on Removable Singularities
Play Video		 Recalling Riemann's Theorem on Removable Singularities
Lecture 4
Casorati-Weierstrass Theorem; Dealing with the Point at Infinity
Play Video		 Casorati-Weierstrass Theorem; Dealing with the Point at Infinity
Lecture 5
Neighborhood of Infinity, Limit at Infinity and Infinity as an Isolated Singularity
Play Video		 Neighborhood of Infinity, Limit at Infinity and Infinity as an Isolated Singularity
Lecture 6
Studying Infinity: Formulating Epsilon-Delta Definitions for Infinite Limits
Play Video		 Studying Infinity: Formulating Epsilon-Delta Definitions for Infinite Limits
Lecture 7
When is a function analytic at infinity?
Play Video		 When is a function analytic at infinity?
Lecture 8
Laurent Expansion at Infinity and Riemann's Removable Singularities Theorem
Play Video		 Laurent Expansion at Infinity and Riemann's Removable Singularities Theorem
Lecture 9
The Generalized Liouville Theorem: Little Brother of Little Picard
Play Video		 The Generalized Liouville Theorem: Little Brother of Little Picard
Lecture 10
Morera's Theorem at Infinity, Infinity as a Pole and Behaviour at Infinity
Play Video		 Morera's Theorem at Infinity, Infinity as a Pole and Behaviour at Infinity
Lecture 11
Residue at Infinity and Introduction to the Residue Theorem for the Extended
Play Video		 Residue at Infinity and Introduction to the Residue Theorem for the Extended
Lecture 12
Proofs of Two Avatars of the Residue Theorem for the Extended Complex Plane
Play Video		 Proofs of Two Avatars of the Residue Theorem for the Extended Complex Plane
Lecture 13
Infinity as an Essential Singularity and Transcendental Entire Functions
Play Video		 Infinity as an Essential Singularity and Transcendental Entire Functions
Lecture 14
Meromorphic Functions on the Extended Complex Plane
Play Video		 Meromorphic Functions on the Extended Complex Plane
Lecture 15
The Ubiquity of Meromorphic Functions
Play Video		 The Ubiquity of Meromorphic Functions
Lecture 16
Continuity of Meromorphic Functions at Poles and Topologies
Play Video		 Continuity of Meromorphic Functions at Poles and Topologies
Lecture 17
Why Normal Convergence, but Not Globally Uniform Convergence,
Play Video		 Why Normal Convergence, but Not Globally Uniform Convergence,
Lecture 18
Measuring Distances to Infinity, the Function Infinity and Normal Convergence
Play Video		 Measuring Distances to Infinity, the Function Infinity and Normal Convergence
Lecture 19
The Invariance Under Inversion of the Spherical Metric on the Extended Complex Plane
Play Video		 The Invariance Under Inversion of the Spherical Metric on the Extended Complex Plane
Lecture 20
Introduction to Hurwitz's Theorem for Normal Convergence of Holomorphic Functions
Play Video		 Introduction to Hurwitz's Theorem for Normal Convergence of Holomorphic Functions
Lecture 21
Completion of Proof of Hurwitz's Theorem for Normal Limits of Analytic Functions
Play Video		 Completion of Proof of Hurwitz\'s Theorem for Normal Limits of Analytic Functions
Lecture 22
Hurwitz's Theorem for Normal Limits of Meromorphic Functions in the Spherical Metric
Play Video		 Hurwitz's Theorem for Normal Limits of Meromorphic Functions in the Spherical Metric
Lecture 23
What could the Derivative of a Meromorphic Function
Play Video		 What could the Derivative of a Meromorphic Function
Lecture 24
Defining the Spherical Derivative of a Meromorphic Function
Play Video		 Defining the Spherical Derivative of a Meromorphic Function
Lecture 25
Well-definedness of the Spherical Derivative of a Meromorphic Function
Play Video		 Well-definedness of the Spherical Derivative of a Meromorphic Function
Lecture 26
Topological Preliminaries: Translating Compactness into Boundedness
Play Video		 Topological Preliminaries: Translating Compactness into Boundedness
Lecture 27
Introduction to the Arzela-Ascoli Theorem
Play Video		 Introduction to the Arzela-Ascoli Theorem
Lecture 28
Proof of the Arzela-Ascoli Theorem for Functions
Play Video		 Proof of the Arzela-Ascoli Theorem for Functions
Lecture 29
Proof of the Arzela-Ascoli Theorem for Functions
Play Video		 Proof of the Arzela-Ascoli Theorem for Functions
Lecture 30
Introduction to the Montel Theorem
Play Video		 Introduction to the Montel Theorem
Lecture 31
Completion of Proof of the Montel Theorem
Play Video		 Completion of Proof of the Montel Theorem
Lecture 32
Introduction to Marty's Theorem
Play Video		 Introduction to Marty's Theorem
Lecture 33
Proof of one direction of Marty's Theorem
Play Video		 Proof of one direction of Marty's Theorem
Lecture 34
Proof of the other direction of Marty's Theorem
Play Video		 Proof of the other direction of Marty's Theorem
Lecture 35
Normal Convergence at Infinity and Hurwitz's Theorems
Play Video		 Normal Convergence at Infinity and Hurwitz's Theorems
Lecture 36
Normal Sequential Compactness, Normal Uniform Boundedness
Play Video		 Normal Sequential Compactness, Normal Uniform Boundedness
Lecture 37
Local Analysis of Normality and the Zooming Process - Motivation for Zalcman's Lemma
Play Video		 Local Analysis of Normality and the Zooming Process - Motivation for Zalcman's Lemma
Lecture 38
Characterizing Normality at a Point by the Zooming Process
Play Video		 Characterizing Normality at a Point by the Zooming Process
Lecture 39
Local Analysis of Normality and the Zooming Process - Motivation for Zalcman's Lemma
Play Video		 Local Analysis of Normality and the Zooming Process - Motivation for Zalcman\'s Lemma
Lecture 40
Montel's Deep Theorem: The Fundamental Criterion for Normality
Play Video		 Montel's Deep Theorem: The Fundamental Criterion for Normality
Lecture 41
Proofs of the Great and Little Picard Theorems
Play Video		 Proofs of the Great and Little Picard Theorems
Lecture 42
Royden's Theorem on Normality Based On Growth Of Derivatives
Play Video		 Royden's Theorem on Normality Based On Growth Of Derivatives
Lecture 43
Schottky's Theorem: Uniform Boundedness from a Point to a Neighbourhood
Play Video		 Schottky's Theorem: Uniform Boundedness from a Point to a Neighbourhood