Mathematics - Advanced Complex Analysis - Part 2 (English)

   Friday, May 16, 2025   

Starting a new Lecture Notes Series on Mathematics - Advanced Complex Analysis - Part 2

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Lecture 1: Mod-01 Lec-01 Properties of the Image of an Analytic Function: Introduction to the Picard Theorems

Lecture 2: Mod-01 Lec-02 Recalling Singularities of Analytic Functions: Non-isolated and Isolated Removable

Lecture 3: Mod-01 Lec-03 Recalling Riemann\'s Theorem on Removable Singularities

Lecture 4: Mod-01 Lec-04 Casorati-Weierstrass Theorem; Dealing with the Point at Infinity

Lecture 5: Mod-02 Lec-05 Neighborhood of Infinity, Limit at Infinity and Infinity as an Isolated Singularity

Lecture 6: Mod-02 Lec-06 Studying Infinity: Formulating Epsilon-Delta Definitions for Infinite Limits

Lecture 7: Mod-03 Lec-07 When is a function analytic at infinity ?

Lecture 8: Mod-03 Lec-08 Laurent Expansion at Infinity and Riemann's Removable Singularities Theorem

Lecture 9: Mod-03 Lec-09 The Generalized Liouville Theorem: Little Brother of Little Picard

Lecture 10: Mod-03 Lec-10 Morera's Theorem at Infinity, Infinity as a Pole and Behaviour at Infinity

Lecture 11: Mod-04 Lec-11 Residue at Infinity and Introduction to the Residue Theorem for the Extended

Lecture 12: Mod-04 Lec-12 Proofs of Two Avatars of the Residue Theorem for the Extended Complex Plane

Lecture 13: Mod-05 Lec-13 Infinity as an Essential Singularity and Transcendental Entire Functions

Lecture 14: Mod-05 Lec-14 Meromorphic Functions on the Extended Complex Plane

Lecture 15: Mod-05 Lec-15 The Ubiquity of Meromorphic Functions

Lecture 16: Mod-05 Lec-16 Continuity of Meromorphic Functions at Poles and Topologies

Lecture 17: Mod-06 Lec-17 Why Normal Convergence, but Not Globally Uniform Convergence,

Lecture 18: Mod-06 Lec-18 Measuring Distances to Infinity, the Function Infinity and Normal Convergence

Lecture 19: Mod-06 Lec-19 The Invariance Under Inversion of the Spherical Metric on the Extended Complex Plane

Lecture 20: Mod-07 Lec-20 Introduction to Hurwitz\'s Theorem for Normal Convergence of Holomorphic Functions

Lecture 21: Mod-07 Lec-21 Completion of Proof of Hurwitz\'s Theorem for Normal Limits of Analytic Functions

Lecture 22: Mod-07 Lec-22 Hurwitz's Theorem for Normal Limits of Meromorphic Functions in the Spherical Metric

Lecture 23: Mod-08 Lec-23 What could the Derivative of a Meromorphic Function

Lecture 24: Mod-08 Lec-24 Defining the Spherical Derivative of a Meromorphic Function

Lecture 25: Mod-08 Lec-25 Well-definedness of the Spherical Derivative of a Meromorphic Function

Lecture 26: Mod-09 Lec-26 Topological Preliminaries: Translating Compactness into Boundedness

Lecture 27: Mod-09 Lec-27 Introduction to the Arzela-Ascoli Theorem

Lecture 28: Mod-09 Lec-28 Proof of the Arzela-Ascoli Theorem for Functions

Lecture 29: Mod-09 Lec-29 Proof of the Arzela-Ascoli Theorem for Functions

Lecture 30: Mod-10 Lec-30 Introduction to the Montel Theorem

Lecture 31: Mod-10 Lec-31 Completion of Proof of the Montel Theorem

Lecture 32: Mod-11 Lec-32 Introduction to Marty's Theorem

Lecture 33: Mod-11 Lec-33 Proof of one direction of Marty's Theorem

Lecture 34: Mod-11 Lec-34 Proof of the other direction of Marty's Theorem

Lecture 35: Mod-12 Lec-35 Normal Convergence at Infinity and Hurwitz's Theorems

Lecture 36: Mod-12 Lec-36 Normal Sequential Compactness, Normal Uniform Boundedness

Lecture 37: Mod-13 Lec-37 Local Analysis of Normality and the Zooming Process - Motivation for Zalcman's Lemma

Lecture 38: Mod-13 Lec-38 Characterizing Normality at a Point by the Zooming Process

Lecture 39: Mod-14 Lec-39 Local Analysis of Normality and the Zooming Process - Motivation for Zalcman\'s Lemma

Lecture 40: Mod-14 Lec-40 Montel\'s Deep Theorem: The Fundamental Criterion for Normality

Lecture 41: Mod-14 Lec-41 Proofs of the Great and Little Picard Theorems

Lecture 42: Mod-14 Lec-42 Royden\'s Theorem on Normality Based On Growth Of Derivatives

Lecture 43: Mod-14 Lec-43 Schottky\'s Theorem: Uniform Boundedness from a Point to a Neighbourhood