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Course Co-ordinated by :
IIT Madras
Course Available from :
03-August-2015
NPTEL
Mathematics
Advanced Complex Analysis - Part 1:Zeros of Analytic Functions,Analytic continuation, Monodromy, Hyperbolic Geometry and the Reimann Mapping Theorem (Video)
Under Review
Fundamental Theorems Connected with Zeros of Analytic Functions
Modules / Lectures
UNIT 1: Theorems of Rouche and Hurwitz
Fundamental Theorems Connected with Zeros of Analytic Functions
The Argument (Counting) Principle, Rouche's Theorem and The Fundamental Theorem of Algebra
Morera's Theorem and Normal Limits of Analytic Functions
Hurwitz's Theorem and Normal Limits of Univalent Functions
UNIT 2: Open Mapping Theorem
Local Constancy of Multiplicities of Assumed Values
The Open Mapping Theorem
UNIT 3: Inverse Function Theorem
Introduction to the Inverse Function Theorem
Completion of the Proof of the Inverse Function Theorem: The Integral Inversion Formula for the Inverse Function
Univalent Analytic Functions have never-zero Derivatives and are Analytic Isomorphisms
UNIT 4: Implicit Function Theorem
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 of the Explicit Function
UNIT 5: Riemann Surfaces for Multi-Valued Functions
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 mapping at a critical point
The Algebraic nature of the functional inverses of an analytic mapping at a critical point
UNIT 6: Analytic Continuation
The Idea of a Direct Analytic Continuation or an Analytic Extension
General or Indirect Analytic Continuation and the Lipschitz Nature of the Radius of Convergence
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 Continuations along Paths
UNIT 7: Monodromy
Analytic Continuability along Paths: Dependence on the Initial Function and on the Path - First Version of the Monodromy Theorem
Maximal Domains of Direct and Indirect Analytic Continuation: SecondVersion of the Monodromy Theorem
Deducing the Second (Simply Connected) Version of the Monodromy Theorem from the First (Homotopy) Version
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 of an Analytic Function at a Critical Point
Unit 8: Harmonic Functions, Maximum Principles, Schwarz Lemma and Uniqueness of Riemann Mappings
The Mean-Value Property, Harmonic Functions and the Maximum Principle
Proofs of Maximum Principles and Introduction to Schwarz Lemma
Proof of Schwarz Lemma and Uniqueness of Riemann Mappings
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
Differential or Infinitesimal Schwarzs Lemma, Picks Lemma, Hyperbolic Arclengths, Metric and Geodesics on the Unit Disc
Differential or Infinitesimal Schwarzs Lemma, Picks Lemma, Hyperbolic Arclengths, Metric and Geodesics on the Unit Disc.
Hyperbolic Geodesics for the Hyperbolic Metric on the Unit Disc
Schwarz-Pick Lemma for the Hyperbolic Metric on the Unit Disc
Unit 10: Theorems of Arzela-Ascoli and Montel
Arzela-Ascoli Theorem: Under Uniform Boundedness, Equicontinuity and Uniform Sequential Compactness are Equivalent
Completion of the Proof of the Arzela-Ascoli Theorem and Introduction to Montels Theorem
The Proof of Montels Theorem
Unit 11: Existence of a Riemann Mapping
The Candidate for a Riemann Mapping
Completion of Proof of The Riemann Mapping Theorem
Completion of Proof of The Riemann Mapping Theorem.
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