Advanced Complex Analysis I

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/

Advanced Complex Analysis I
The Monodromy theorem. Analytic continuation along a curve of the natural logarithm (the imaginary part of the logarithm is shown only). Source: Wikipedia
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Video Lectures & Study Materials

Visit the official course website for more study materials: 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.

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

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