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Advanced Complex Analysis - Part 1:Zeros of Analytic Functions,Analytic continuation, Monodromy, Hyperbolic Geometry and the Reimann Mapping Theorem A8PAHKN6QQ

URIm2://res/A8PAHKN6QQ
Kindcourse
CreatorsT.E. Venkata Balaji (instructor)
Date2015-07-31
MSC30-XX
Scopein
Idscourse 03v0r5n49:111106084, course 03v0r5n49:111106094

Expressions

e1: edition

Languageen
Date2015-07-31
Contents
  • part 1: UNIT 1: Theorems of Rouche and Hurwitz
  • session 1: Fundamental Theorems Connected with Zeros of Analytic Functions
  • session 2: The Argument (Counting) Principle, Rouche's Theorem and The Fundamental Theorem of Algebra
  • session 3: Morera's Theorem and Normal Limits of Analytic Functions
  • session 4: Hurwitz's Theorem and Normal Limits of Univalent Functions
  • part 2: UNIT 2: Open Mapping Theorem
  • session 5: Local Constancy of Multiplicities of Assumed Values
  • session 6: The Open Mapping Theorem
  • part 3: UNIT 3: Inverse Function Theorem
  • session 7: Introduction to the Inverse Function Theorem
  • session 8: Completion of the Proof of the Inverse Function Theorem: The Integral Inversion Formula for the Inverse Function
  • session 9: Univalent Analytic Functions have never-zero Derivatives and are Analytic Isomorphisms
  • part 4: UNIT 4: Implicit Function Theorem
  • session 10: Introduction to the Implicit Function Theorem
  • session 11: Proof of the Implicit Function Theorem: Topological Preliminaries
  • session 12: Proof of the Implicit Function Theorem: The Integral Formula for & Analyticity of the Explicit Function
  • part 5: UNIT 5: Riemann Surfaces for Multi-Valued Functions
  • session 13: Doing Complex Analysis on a Real Surface: The Idea of a Riemann Surface
  • session 14: F(z,w)=0 is naturally a Riemann Surface
  • session 15: Constructing the Riemann Surface for the Complex Logarithm
  • session 16: Constructing the Riemann Surface for the m-th root function
  • session 17: The Riemann Surface for the functional inverse of an analytic mapping at a critical point
  • session 18: The Algebraic nature of the functional inverses of an analytic mapping at a critical point
  • part 6: UNIT 6: Analytic Continuation
  • session 19: The Idea of a Direct Analytic Continuation or an Analytic Extension
  • session 20: General or Indirect Analytic Continuation and the Lipschitz Nature of the Radius of Convergence
  • session 21: Analytic Continuation Along Paths via Power Series Part A
  • session 22: Analytic Continuation Along Paths via Power Series Part B
  • session 23: Continuity of Coefficients occurring in Families of Power Series defining Analytic Continuations along Paths
  • part 7: UNIT 7: Monodromy
  • session 24: Analytic Continuability along Paths: Dependence on the Initial Function and on the Path - First Version of the Monodromy Theorem
  • session 25: Maximal Domains of Direct and Indirect Analytic Continuation: SecondVersion of the Monodromy Theorem
  • session 26: Deducing the Second (Simply Connected) Version of the Monodromy Theorem from the First (Homotopy) Version
  • session 27: Existence and Uniqueness of Analytic Continuations on Nearby Paths
  • session 28: Proof of the First (Homotopy) Version of the Monodromy Theorem
  • session 29: Proof of the Algebraic Nature of Analytic Branches of the Functional Inverse of an Analytic Function at a Critical Point
  • part 8: Unit 8: Harmonic Functions, Maximum Principles, Schwarz Lemma and Uniqueness of Riemann Mappings
  • session 30: The Mean-Value Property, Harmonic Functions and the Maximum Principle
  • session 31: Proofs of Maximum Principles and Introduction to Schwarz Lemma
  • session 32: Proof of Schwarz Lemma and Uniqueness of Riemann Mappings
  • session 33: Reducing Existence of Riemann Mappings to Hyperbolic Geometry of Sub-domains of the Unit Disc
  • part 9: Unit 9: Pick Lemma and Hyperbolic Geometry on the Unit Disc
  • session 34: Differential or Infinitesimal Schwarzs Lemma, Picks Lemma, Hyperbolic Arclengths, Metric and Geodesics on the Unit Disc
  • session 35: Differential or Infinitesimal Schwarzs Lemma, Picks Lemma, Hyperbolic Arclengths, Metric and Geodesics on the Unit Disc.
  • session 36: Hyperbolic Geodesics for the Hyperbolic Metric on the Unit Disc
  • session 37: Schwarz-Pick Lemma for the Hyperbolic Metric on the Unit Disc
  • part 10: Unit 10: Theorems of Arzela-Ascoli and Montel
  • session 38: Arzela-Ascoli Theorem: Under Uniform Boundedness, Equicontinuity and Uniform Sequential Compactness are Equivalent
  • session 39: Completion of the Proof of the Arzela-Ascoli Theorem and Introduction to Montels Theorem
  • session 40: The Proof of Montels Theorem
  • part 11: Unit 11: Existence of a Riemann Mapping
  • session 41: The Candidate for a Riemann Mapping
  • session 42: Completion of Proof of The Riemann Mapping Theorem
  • session 43: Completion of Proof of The Riemann Mapping Theorem.

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