Constructing the Riemann Surface for the Complex Logarithm 
Constructing the Riemann Surface for the Complex Logarithm
by IIT Madras
Video Lecture 15 of 43
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Date Added: April 2, 2016

Lecture Description

This video lecture, part of the series Advanced Complex Analysis I by Prof. , does not currently have a detailed description and video lecture title. If you have watched this lecture and know what it is about, particularly what Mathematics topics are discussed, please help us by commenting on this video with your suggested description and title. Many thanks from,

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Course Index

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