Advanced Complex Analysis II

Course Description

This is the second part of a series of lectures on advanced topics in Complex Analysis. By advanced, we mean topics that are not (or just barely) touched upon in a first course on Complex Analysis. The theme of the course is to study compactness and convergence in families of analytic (or holomorphic) functions and in families of meromorphic functions. The compactness we are interested herein is the so-called sequential compactness, and more specifically it is normal convergence -- namely convergence on compact subsets. The final objective is to prove the Great or Big Picard Theorem and deduce the Little or Small Picard Theorem. This necessitates studying the point at infinity both as a value or limit attained, and as a point in the domain of definition of the functions involved. This is done by thinking of the point at infinity as the north pole on the sphere, by appealing to the Riemann Stereographic Projection from the Riemann Sphere. Analytic properties are tied to the spherical metric on the Riemann Sphere. The notion of spherical derivative is introduced for meromorphic functions. Infinity is studied as a singular point. Laurent series at infinity, residue at infinity and a version of the Residue theorem for domains including the point at infinity are explained. In later lectures, Marty's theorem -- a version of the Montel theorem for meromorphic functions, Zalcman's Lemma -- a fundamental theorem on the local analysis of non-normality, Montel's theorem on normality, Royden's theorem and Schottky's theorem are proved. For more details on what is covered lecturewise, please look at the titles, goals and keywords which are given for each lecture.

Advanced Complex Analysis II
The Residue Theorem
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Video Lectures & Study Materials

Visit the official course website for more study materials:

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


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