Lecture Description
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Course Index
- Fundamental Theorems Connected with Zeros of Analytic Functions
- The Argument (Counting) Principle, Rouche's Theorem and The Fundamental Theorem
- Morera's Theorem and Normal Limits of Analytic Functions
- Hurwitz's Theorem and Normal Limits of Univalent Functions
- Local Constancy of Multiplicities of Assumed Values
- The Open Mapping Theorem
- Introduction to the Inverse Function Theorem
- Completion of the Proof of the Inverse Function Theorem: The Integral Inversion
- Univalent Analytic Functions have never-zero Derivatives and are Analytic Isomorphisms
- Introduction to the Implicit Function Theorem
- Proof of the Implicit Function Theorem: Topological Preliminaries
- Proof of the Implicit Function Theorem: The Integral Formula for & Analyticity
- Doing Complex Analysis on a Real Surface: The Idea of a Riemann Surface
- F(z,w)=0 is naturally a Riemann Surface
- Constructing the Riemann Surface for the Complex Logarithm
- Constructing the Riemann Surface for the m-th root function
- The Riemann Surface for the functional inverse of an analytic
- The Algebraic nature of the functional inverses of an analytic
- The Idea of a Direct Analytic Continuation or an Analytic Extension
- General or Indirect Analytic Continuation and the Lipschitz Nature of the Radius
- Analytic Continuation Along Paths via Power Series Part A
- Analytic Continuation Along Paths via Power Series Part B
- Continuity of Coefficients occurring in Families of Power Series defining Analytic
- Analytic Continuability along Paths: Dependence on the Initial Function
- Maximal Domains of Direct and Indirect Analytic Continuation: Second
- Deducing the Second (Simply Connected) Version of the Monodromy Theorem
- Existence and Uniqueness of Analytic Continuations on Nearby Paths
- Proof of the First (Homotopy) Version of the Monodromy Theorem
- Proof of the Algebraic Nature of Analytic Branches of the Functional Inverse
- The Mean-Value Property, Harmonic Functions and the Maximum Principle
- Proofs of Maximum Principles and Introduction to Schwarz's Lemma
- Proof of Schwarz's Lemma and Uniqueness of Riemann Mappings
- Reducing Existence of Riemann Mappings to Hyperbolic Geometry of Sub-domains
- Part A: Differential or Infinitesimal Schwarz's Lemma, Pick's Lemma, Hyperbolic
- Part B: Differential or Infinitesimal Schwarz's Lemma, Pick's Lemma, Hyperbolic
- Hyperbolic Geodesics for the Hyperbolic Metric on the Unit Disc
- Schwarz-Pick Lemma for the Hyperbolic Metric on the Unit Disc
- Arzela-Ascoli Theorem: Under Uniform Boundedness, Equicontinuity and Uniform
- Completion of the Proof of the Arzela-Ascoli Theorem and Introduction
- The Proof of Montel's Theorem
- The Candidate for a Riemann Mapping
- Completion of Proof of The Riemann Mapping Theorem - Part A
- Completion of Proof of The Riemann Mapping Theorem - Part B
Course Description
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.
Check the official website for a detailed course outline: http://nptel.ac.in/syllabus/111106084/
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