# playlist of the 43 videos (click the up-left corner of the video)
source: nptelhrd 2015年7月31日
Mathematics - Advanced Complex Analysis - Part 1 by Dr. T.E. Venkata Balaji, Department of Mathematics, IIT Madras. For more details on NPTEL visit http://nptel.ac.in
Lec-01 Fundamental Theorems Connected with Zeros of Analytic Functions 58:10
Lec-02 The Argument (Counting) Principle, Rouche's Theorem and The Fundamental Theorem 52:29
Lec-03 Morera's Theorem and Normal Limits of Analytic Functions 52:09
Lec-04 Hurwitz's Theorem and Normal Limits of Univalent Functions 56:54
Lec-05 Local Constancy of Multiplicities of Assumed Values 52:00
Lec-06 The Open Mapping Theorem 1:08:37
Lec-07 Introduction to the Inverse Function Theorem 45:21
Lec-08 Completion of the Proof of the Inverse Function Theorem: The Integral Inversion 44:37
Lec-09 Univalent Analytic Functions have never-zero Derivatives and are Analytic Isomorphisms 47:53
Lec-10 Introduction to the Implicit Function Theorem 40:31
Lec-11 Proof of the Implicit Function Theorem: Topological Preliminaries 47:52
Lec-12 Proof of the Implicit Function Theorem: The Integral Formula for & Analyticity 53:50
Lec-13 Doing Complex Analysis on a Real Surface: The Idea of a Riemann Surface 55:45
Lec-14 F(z,w)=0 is naturally a Riemann Surface 47:11
Lec-15 Constructing the Riemann Surface for the Complex Logarithm 1:03:23
Lec-16 Constructing the Riemann Surface for the m-th root function 1:00:27
Lec-17 The Riemann Surface for the functional inverse of an analytic 48:06
Lec-18 The Algebraic nature of the functional inverses of an analytic 48:25
Lec-19 The Idea of a Direct Analytic Continuation or an Analytic Extension 51:31
Lec-20 General or Indirect Analytic Continuation and the Lipschitz Nature of the Radius 36:57
Lec-21A Analytic Continuation Along Paths via Power Series Part A 30:33
Lec-21B Analytic Continuation Along Paths via Power Series Part B 35:38
Lec-22 Continuity of Coefficients occurring in Families of Power Series defining Analytic 52:15
Lec-23 Analytic Continuability along Paths: Dependence on the Initial Function 44:40
Lec-24 Maximal Domains of Direct and Indirect Analytic Continuation: Second 42:26
Lec-25 Deducing the Second (Simply Connected) Version of the Monodromy Theorem 49:40
Lec-27 Existence and Uniqueness of Analytic Continuations on Nearby Paths 47:50
Lec-28 Proof of the First (Homotopy) Version of the Monodromy Theorem 48:06
Lec-30 Proof of the Algebraic Nature of Analytic Branches of the Functional Inverse 45:10
Lec-31 The Mean-Value Property, Harmonic Functions and the Maximum Principle 1:02:23
Lec-32 Proofs of Maximum Principles and Introduction to Schwarz's Lemma 48:49
Lec-33 Proof of Schwarz's Lemma and Uniqueness of Riemann Mappings 46:50
Lec-34 Reducing Existence of Riemann Mappings to Hyperbolic Geometry of Sub-domains 48:56
Lec 35 Part A Differential or Infinitesimal Schwarz's Lemma, Pick's Lemma, Hyperbolic 54:57
Lec-35 Part B Differential or Infinitesimal Schwarz's Lemma, Pick's Lemma, Hyperbolic 53:16
Lec-36 Hyperbolic Geodesics for the Hyperbolic Metric on the Unit Disc 48:58
Lec-37 Schwarz-Pick Lemma for the Hyperbolic Metric on the Unit Disc 53:27
Lec-38 Arzela-Ascoli Theorem: Under Uniform Boundedness, Equicontinuity and Uniform 46:10
Lec-39 Completion of the Proof of the Arzela-Ascoli Theorem and Introduction 37:41
Lec-40 The Proof of Montel's Theorem 50:08
Lec-41 The Candidate for a Riemann Mapping 48:13
Lec-42A Completion of Proof of The Riemann Mapping Theorem 38:17
Lec-42B Completion of Proof of The Riemann Mapping Theorem 42:17
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