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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.

Unit Number / Title

Lecture Number / Title

UNIT 1: Theorems of Picard, Casorati-Weierstrass and Riemann on Removable Singularities



Lecture 1:
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, Pole and Essential Singularities

Lecture 3:
Recalling Riemann's Theorem on Removable Singularities

Lecture 4:
Casorati-Weierstrass Theorem; Dealing with the Point at Infinity -- Riemann Sphere and Riemann Stereographic Projection

UNIT 2: Neighborhoods of Infinity, Limits at Infinity and Infinite Limits



Lecture 5:
Neighborhood of Infinity, Limit at Infinity and Infinity as an Isolated Singularity

Lecture 6:
Studying Infinity: Formulating Epsilon-Delta Definitions for Infinite Limits and Limits at Infinity

UNIT 3: Infinity as a Point of Analyticity



Lecture 7:
When is a function analytic at infinity ?

Lecture 8:
Laurent Expansion at Infinity and Riemann's Removable Singularities Theorem for the Point at Infinity

Lecture 9:
The Generalized Liouville Theorem: Little Brother of Little Picard and Analogue of Casorati-Weierstrass;
Failure of Cauchy's Theorem at Infinity

Lecture 10:
Morera's Theorem at Infinity, Infinity as a Pole and Behaviour at Infinity of
Rational and Meromorphic Functions

UNIT 4: Residue at Infinity and Residue Theorem for the Extended Complex Plane



Lecture 11:
Residue at Infinity and Introduction to the
Residue Theorem for the Extended Complex Plane:
Residue Theorem for the Point at Infinity

Lecture 12:
Proofs of Two Avatars of the Residue Theorem
for the Extended Complex Plane and
Applications of the Residue at Infinity

UNIT 5: The Behavior of Transcendental and Meromorphic Functions at Infinity



Lecture 13:
Infinity as an Essential Singularity and Transcendental Entire Functions

Lecture 14:
Morera's Theorem at Infinity, Infinity as a Pole and Behaviour at Infinity of
Rational and Meromorphic Functions

Lecture 15:
The Ubiquity of Meromorphic Functions: The Nerves of the Geometric
Network Bridging Algebra, Analysis and Topology

Lecture 16:
Continuity of Meromorphic Functions at Poles and Topologies of Spaces of Functions

UNIT 6: Normal Convergence In The Inversion-Invariant Spherical Metric on the Extended Plane



Lecture 17:
Why Normal Convergence, but Not Globally Uniform Convergence, is the Inevitable in Complex Analysis

Lecture 18:
Measuring Distances to Infinity, the Function Infinity and Normal Convergence of Holomorphic Functions
in the Spherical Metric

Lecture 19:
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 in the Spherical Metric

UNIT 7: Hurwitz Theorems on Normal Limits of Holomorphic and Meromorphic Functions under the Spherical Metric



Lecture 21:
Completion of Proof of Hurwitz's Theorem for Normal Limits of Analytic Functions in the Spherical Metric

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

UNIT 8: Hurwitz Theorems on Normal Limits of Holomorphic and Meromorphic Functions under the Spherical Metric



Lecture 23:
What could the Derivative of a Meromorphic Function Relative to the Spherical Metric Possibly Be ?

Lecture 24: Defining the Spherical Derivative of a Meromorphic Function

Lecture 25: Well-definedness of the Spherical Derivative of a Meromorphic Function at a Pole and
Inversion-invariance of the Spherical Derivative

UNIT 9: From Compactness to Boundedness via Equicontinuity - The Arzela Ascoli Uniform Boundedness Principle



Lecture 26:
Topological Preliminaries: Translating Compactness into Boundedness

Lecture 27: Introduction to the Arzela-Ascoli Theorem: Passing from abstract Compactness to verifiable Equicontinuity

Lecture 28: Proof of the Arzela-Ascoli Theorem for Functions: Abstract Compactness Implies Equicontinuity

Lecture 29: Proof of the Arzela-Ascoli Theorem for Functions: Equicontinuity Implies Compactness

UNIT 10: The Montel Theorem - The Holomorphic Avatar of the Arzela-Ascoli Theorem



Lecture 30:
Introduction to the Montel Theorem - the Holomorphic Avatar of the Arzela-Ascoli Theorem & Why you get Equicontinuity for Free

Lecture 31: Completion of Proof of the Montel Theorem -
the Holomorphic Avatar of the Arzela-Ascoli Theorem

UNIT 11: The Marty Theorem - The Meromorphic Avatar of the Montel and Arzela-Ascoli Theorems



Lecture 32:
Introduction to Marty's Theorem -
the Meromorphic Avatar of the Montel & Arzela-Ascoli Theorems

Lecture 33: Proof of one direction of Marty's Theorem -
the Meromorphic Avatar of the Montel & Arzela-Ascoli Theorems -
Normal Uniform Boundedness of Spherical Derivatives Implies Normal Sequential Compactness

Lecture 34: Proof of the other direction of Marty's Theorem - the Meromorphic Avatar of the Montel & Arzela-Ascoli Theorems - Normal Sequential Compactness Implies Normal Uniform Boundedness of Spherical Derivatives

UNIT 12: The Hurwitz, Montel and Marty Theorems at Infinity



Lecture 35:
Normal Convergence at Infinity and Hurwitz's Theorems for Normal Limits of Analytic and Meromorphic Functions at Infinity

Lecture 36: Normal Sequential Compactness, Normal Uniform Boundedness and Montel's & Marty's Theorems at Infinity

UNIT 13: Local Analysis of Normality by the Zooming Process and Zalcman's Lemma



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

Lecture 38: Characterizing Normality at a Point by the Zooming Process and the Motivation for Zalcman's Lemma

UNIT 14: Zalcman's Lemma, Montel's Normality Criterion and Theorems of Picard, Royden and Schottky



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

Lecture 40: Montel's Deep Theorem: The Fundamental Criterion for Normality or Fundamental Normality Test based on Omission of Values

Lecture 41: Proofs of the Great and Little Picard Theorems

Lecture 42: Characterizing Normality at a Point by the Zooming Process and the Motivation for Zalcman's Lemma

Lecture 43: Schottky's Theorem: Uniform Boundedness from a Point to a Neighbourhood
& Problem Solving Session

 

A first course in Topology covering the euclidean spaces (real line and real plane), and a first course in Complex Analysis covering Cauchy's Integration theory, Taylor series, Laurent series and the Residue theorem.


  1. Complex Variables with Applications, by Saminathan Ponnusamy & Herb Silverman, 2006, 524 pp, Birkhaeuser, Boston.
  2. Complex Analysis (UTM) by Theodore Gamelin, Springer, 2003.
  3. NPTEL Video Course on Advanced Complex Analysis - Part I athttp://nptel.ac.in/syllabus/111106084/
  4. NPTEL Video Course on Riemann Surfaces athttp://nptel.ac.in/syllabus/111106044/


http://webdoc.sub.gwdg.de/univerlag/2010/balaji.pdf



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