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Advanced Complex Analysis Part 1

Advanced Complex Analysis Part 1. Instructor: Dr. T.E. Venkata Balaji, Department of Mathematics, IIT Madras. This is the first 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 zeros of analytic (or holomorphic) functions and related theorems. These include the theorems of Hurwitz and Rouche, the open mapping theorem, the inverse and implicit function theorems, applications of those theorems, behaviour at a critical point, analytic branches, constructing Riemann surfaces for functional inverses, analytic continuation and monodromy, hyperbolic geometry and the Riemann mapping theorem. (from nptel.ac.in)

Fundamental Theorems Connected with Zeroes of Analytic Functions


Unit 1: Theorems of Rouche and Hurwitz
Lecture 01 - Fundamental Theorems Connected with Zeroes of Analytic Functions
Lecture 02 - The Argument (Counting) Principle, Rouche's Theorem and the Fundamental Theorem of Algebra
Lecture 03 - Morera's Theorem and Normal Limits of Analytic Functions
Lecture 04 - Hurwitz's Theorem and Normal Limits of Univalent Functions
Unit 2: Open Mapping Theorem
Lecture 05 - Local Constancy of Multiplicities of Assumed Values
Lecture 06 - The Opening Mapping Theorem
Unit 3: Inverse Function Theorem
Lecture 07 - Introduction to the Inverse Function Theorem
Lecture 08 - Completion of the Proof of the Inverse Function Theorem: The Integral Inversion Formula for the Inverse Function
Lecture 09 - Univalent Analytic Functions have Never-Zero Derivatives and are Analytic Isomorphisms
Unit 4: Implicit Function Theorem
Lecture 10 - Introduction to the Implicit Function Theorem
Lecture 11 - Proof of the Implicit Function Theorem: Topological Preliminaries
Lecture 12 - Proof of the Implicit Function Theorem: The Integral Formula for Analyticity of the Explicit Function
Unit 5: Riemann Surfaces for Multi-valued Functions
Lecture 13 - Doing Complex Analysis on a Real Surface: The Idea of a Riemann Surface
Lecture 14 - F(z,w)=0 is Naturally a Riemann Surface
Lecture 15 - Constructing the Riemann Surface for the Complex Logarithm
Lecture 16 - Constructing the Riemann Surface for the m-th Root Function
Lecture 17 - The Riemann Surface for the Functional Inverse of an Analytic Mapping at a Critical Point
Lecture 18 - The Algebraic Nature of the Functional Inverse of an Analytic Mapping at a Critical Point
Unit 6: Analytic Continuation
Lecture 19 - The Idea of a Direct Analytic Continuation or an Analytic Extension
Lecture 20 - General or Indirect Analytic Continuation and the Lipschitz Nature of the Radius of Convergence
Lecture 21a - Analytic Continuation along Paths via Power Series Part A
Lecture 21b - Analytic Continuation along Paths via Power Series Part B
Lecture 22 - Continuity of Coefficients Occurring in Families of Power Series defining Analytic Continuations along Paths
Unit 7: Monodromy
Lecture 23 - Analytic Continuability along Paths: Dependence on the Initial Function and on the Path - First Version of the Monodromy Theorem
Lecture 24 - Maximal Domains of Direct and Indirect Analytic Continuation - Second Version of the Monodromy Theorem
Lecture 25 - Deducing the Second Version of the Monodromy Theorem from the First (Homotopy) Version
Lecture 27 - Existence and Uniqueness of Analytic Continuations on Nearby Paths
Lecture 28 - Proof of the First (Homotopy) Version of the Monodromy Theorem
Lecture 30 - Proof of the Algebraic Nature of Analytic Branches of the Functional Inverse of an Analytic Function at a Critical Point
Unit 8: Harmonic Functions, Maximum Principles, Schwarz Lemma and Uniqueness of Riemann Mappings
Lecture 31 - The Mean Value Property, Harmonic Functions and the Maximum Principle
Lecture 32 - Proofs of Maximum Principles and Introduction to Schwarz Lemma
Lecture 33 - Proof of Schwarz Lemma and Uniqueness of Riemann Mappings
Lecture 34 - Reducing Existence of Riemann Mappings to Hyperbolic Geometry of Sub-domains of the Unit Disc
Unit 9: Pick Lemma and Hyperbolic Geometry on the Unit Disc
Lecture 35a - Differential and Infinitesimal Schwarz's Lemma, Pick's Lemma, Hyperbolic Arc Lengths, Metric and Geodesics on the Unit Disc
Lecture 35b - Differential and Infinitesimal Schwarz's Lemma, Pick's Lemma, Hyperbolic Arc Lengths, Metric and Geodesics on the Unit Disc (cont.)
Lecture 36 - Hyperbolic Geodesics for the Hyperbolic Metric on the Unit Disc
Lecture 37 - Schwarz-Pick Lemma for the Hyperbolic Metric on the Unit Disc
Unit 10: Theorems of Arzela-Ascoli and Montel
Lecture 38 - Arzela-Ascoli Theorem: Under Uniform Boundedness, Equicontinuity and Uniform Sequential Compactness are Equivalent
Lecture 39 - Completion of the Proof of the Arzela-Ascoli Theorem and Introduction to Montel's Theorem
Lecture 40 - The Proof of Montel's Theorem
Unit 11: Existence of a Riemann Mapping
Lecture 41 - The Candidate for a Riemann Mapping
Lecture 42a - Completion of Proof of the Riemann Mapping Theorem
Lecture 42b - Completion of Proof of the Riemann Mapping Theorem (cont.)

References
Advanced Complex Analysis Part 1
Instructor: Dr. T.E. Venkata Balaji, Department of Mathematics, IIT Madras. The theme of the course is to study zeros of analytic (or holomorphic) functions and related theorems.